Research Article Mathematical Modelling of the Spread and...
13
Research Article Mathematical Modelling of the Spread and Control of Onchocerciasis in Tropical Countries: Case Study in Nigeria Ikechukwu Chiwueze Oguoma and Thomas Mbah Acho Department of Mathematics and Applied Mathematics, Faculty of Natural and Agricultural Sciences, University of the Free State, Bloemfontein 9300, South Africa Correspondence should be addressed to Ikechukwu Chiwueze Oguoma; [email protected] Received 2 May 2014; Accepted 2 June 2014; Published 21 October 2014 Academic Editor: Abdon Atangana Copyright © 2014 I. C. Oguoma and T. M. Acho. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Onchocerciasis, also known as river blindness and Robles disease, is a parasitic disease caused by Onchocerca volvulus, a nematode (roundworm), and it is endemic in tropical countries like Nigeria. e objective of this paper is to discuss the mathematical formulation underpinning the spread and control of this disease on one hand. On the other hand, we make use of some new analytical methods to derive the solution of the resulting set of equations. e numerical results are presented to test the efficiency and the accuracy of both methods. e techniques used for solving these problems are friendly, very easy, and less time consuming. e numerical solutions in both cases display the biological behaviour of the real life situation. 1. Introduction Onchocerciasis is a country-wide public health problem in Nigeria. e present estimates had suggested that 7–10 million Nigerians are infected with Onchocerca volvulus, and approximately 40 million are at risk of the disease [1]. More than 120,000 cases of Onchocerciasis related blindness have been reported [2–4], and thousands of people suffer from disabling complications of this disease [5]. Onchocerciasis, which is the world’s second-leading infectious cause of blind- ness, is a parasitic disease caused by the filarial nematode, Onchocerca volvulus. It is termed as “river blindness” because it is spread by the bite of the blackfly vector that breads near oxygen-high fast-flowing streams and rivers people rely on for washing, drinking, and farming which results in depopulation of the fertile river valleys. e lifespan of blackflies is short, lasting only 2-3 weeks. e disease affects rural communities in Nigeria and is the major cause of blindness and skin disease in endemic areas with serious socioeconomic effects. It has been identified by the World Health Organization as one of the neglected tropical diseases (NTD) [6, 7], “Neglected” because they are not mentioned in the millennium Development Goals. is implied that they are not usually included among the important development discussions, and as such they do not receive adequate atten- tion or funding. Figure 1 explains the life cycle of the parasite Onchocerca volvulus. Onchocerca volvulus is transmitted to humans through the bite of a blackfly which introduces immature larva forms of the parasite (infective larvae) into their human host. e larvae migrate to the subcutaneous tissue where they undergo further development and form skin nodules as they mature into adult worms (macrofilaria). e adult worms mate and produce millions of microscopic larvae (microfilaria). is continues for years of the adult worm’s life. Although the infection with larvae begins immediately, the disease may not become apparent in an individual for months to years. In most individuals, it develops slowly in the skin, while some patients may show symptoms initially with eye problems, ocular involvement, and general debilitation. e blackflies absorb the microfilariae as they feed during the day, which further undergo developmental stages within the blackflies into infective larvae, which are then transmitted to the next human victim. A study was carried out in Imo River Basin, Imo State, Southeastern Nigeria, to assess the prevalence and intensity of Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2014, Article ID 631658, 12 pages http://dx.doi.org/10.1155/2014/631658
Transcript of Research Article Mathematical Modelling of the Spread and...
Research ArticleMathematical Modelling of the Spread and Control ofOnchocerciasis in Tropical Countries Case Study in Nigeria
Ikechukwu Chiwueze Oguoma and Thomas Mbah Acho
Department of Mathematics and Applied Mathematics Faculty of Natural and Agricultural Sciences University of the Free StateBloemfontein 9300 South Africa
Correspondence should be addressed to Ikechukwu Chiwueze Oguoma ikeoguomayahoocom
Received 2 May 2014 Accepted 2 June 2014 Published 21 October 2014
Academic Editor Abdon Atangana
Copyright copy 2014 I C Oguoma and T M AchoThis is an open access article distributed under theCreativeCommonsAttributionLicense which permits unrestricted use distribution and reproduction in anymedium provided the originalwork is properly cited
Onchocerciasis also known as river blindness and Robles disease is a parasitic disease caused byOnchocerca volvulus a nematode(roundworm) and it is endemic in tropical countries like Nigeria The objective of this paper is to discuss the mathematicalformulation underpinning the spread and control of this disease on one hand On the other hand we make use of some newanalytical methods to derive the solution of the resulting set of equations The numerical results are presented to test the efficiencyand the accuracy of both methodsThe techniques used for solving these problems are friendly very easy and less time consumingThe numerical solutions in both cases display the biological behaviour of the real life situation
1 Introduction
Onchocerciasis is a country-wide public health problemin Nigeria The present estimates had suggested that 7ndash10million Nigerians are infected with Onchocerca volvulus andapproximately 40 million are at risk of the disease [1] Morethan 120000 cases of Onchocerciasis related blindness havebeen reported [2ndash4] and thousands of people suffer fromdisabling complications of this disease [5] Onchocerciasiswhich is the worldrsquos second-leading infectious cause of blind-ness is a parasitic disease caused by the filarial nematodeOnchocerca volvulus It is termed as ldquoriver blindnessrdquo becauseit is spread by the bite of the blackfly vector that breadsnear oxygen-high fast-flowing streams and rivers peoplerely on for washing drinking and farming which resultsin depopulation of the fertile river valleys The lifespan ofblackflies is short lasting only 2-3 weeks The disease affectsrural communities in Nigeria and is the major cause ofblindness and skin disease in endemic areas with serioussocioeconomic effects It has been identified by the WorldHealth Organization as one of the neglected tropical diseases(NTD) [6 7] ldquoNeglectedrdquo because they are not mentioned inthe millennium Development Goals This implied that theyare not usually included among the important development
discussions and as such they do not receive adequate atten-tion or funding
Figure 1 explains the life cycle of the parasite Onchocercavolvulus
Onchocerca volvulus is transmitted to humans throughthe bite of a blackfly which introduces immature larva formsof the parasite (infective larvae) into their human host
The larvae migrate to the subcutaneous tissue wherethey undergo further development and form skin nodulesas they mature into adult worms (macrofilaria) The adultworms mate and produce millions of microscopic larvae(microfilaria)This continues for years of the adultwormrsquos lifeAlthough the infection with larvae begins immediately thediseasemay not become apparent in an individual formonthsto years In most individuals it develops slowly in the skinwhile some patients may show symptoms initially with eyeproblems ocular involvement and general debilitation
The blackflies absorb themicrofilariae as they feed duringthe day which further undergo developmental stages withinthe blackflies into infective larvae which are then transmittedto the next human victim
A study was carried out in Imo River Basin Imo StateSoutheasternNigeria to assess the prevalence and intensity of
Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 631658 12 pageshttpdxdoiorg1011552014631658
2 Abstract and Applied Analysis
Blackfly stages Blackfly (genus Simulium)takes a blood meal
(L3 larvae enter bite wound)Migrate to head
and blackflyrsquos proboscis
L3 larvae
L1 larvae Blackfly takesa blood meal
(ingests microfilariae)
Microfilariae penetrateblackflyrsquos midgutand migrate to
thoracic muscles
Human stages
Subcutaneous tissues
Adults in subcutaneous nodule
Adults produce unsheathedmicrofilariae that typically arefound in skin and lymphaticsof connective tissues but also
occasionally in peripheralblood urine and sputum
12
3
45
6
7
8
9
di Infective stage
Diagnostic stage
i
d
Figure 1 Life cycle of Onchocerca volvulus [8]
microfilaria of Onchocerca volvulus in the area [9]The surveycoverage was about 918 of the study population and it wasfound by EC Uttah found that thirty-seven percent of thoseexamined was positive for Onchocerca volvulusmicrofilariae(392 ofmales and 349 of females)Themicrofilariae (mf)prevalence increased steadily with increasing age to reach704 in the oldest age group
Going towards the northern part of Nigeria the endemic-ity of human onchocerciasis was assessed by Anosike andOnwuliri [10] in eight rural communities at risk in NingiLocal Government Area Bauchi State Nigeria between July1990 and March 1991 Of the 1536 subjects skin-snipped 334(217) were positive for Onchocerca volvulusmicrofilariae
Wogu and Okaka investigated the overall prevalence ofonchocerciasis in Okpuje Owan West LGA of Edo StateNigeriaThe figure recorded in their study of 475 comparesfavourably with the 486 prevalence reported by Edungbolaand Asaolu [11] in Kwara State Nigeria but higher than the269 recorded by Nwaorgu et al [12] in an onchocerciasismesoendemic area of Enugu State Nigeria However Akinboand Okaka [13] reported a high overall prevalence (83) ofthe disease in Ovia North East LGA Edo State Nigeria[14] These reports suggest that we need a well-thoughtmathematical formulation of a simple and appropriate modelfor the spread and control of this disease
2 Model Formulation
In this section we define the parameters used in the modelbelow as follows 120573
1is the contact rate between 119867
119878and 119867
119864
1205732is the contact rate between 119867
119878and 119867
119868 1205733is the contact
rate between 119867119864and 119867
119868 1205734is the contact rate between 119867
119868
120588120577
1205731HS
1205732HS
1205733HE
1205734HI1205735VS
120575120582
120574VI
HS HE
VI
HI
VS
Figure 2 Model transfer diagram for onchocerciasis
and 119881119878 1205735is the contact rate between 119881
119878and 119881
119868 120574 is the
transmission rate from 119881119868to119867119878 120588 and 120575 are the proportions
of IvermectinMectizan treatment that cured the patient 120577is the rate of movement of humans from the infective 119867
119868
compartment into the human susceptible 119867119878compartment
120582 is the rate ofmovement of human from the human infective119867119868compartment into the latent (incubation) period 119867
119864
compartment
3 Model Diagram
Figure 2 shows the model diagram which consists of fivesets of ordinary differential equations highlighting the rate ofchange with respect to time of human susceptible compart-ment 119867
119878 human latent (incubation) period compartment
119867119864 human infective compartment 119867
119868 vector susceptible
compartment 119881119878 and the vector infective compartment 119881
119868
Abstract and Applied Analysis 3
4 Mathematical Formulation of the Model
From the model transfer diagram (Figure 2) we have thefollowing set of equations
119889119867119878
119889119905= 120588120577 + 120574119881
119868minus (1205731+ 1205732)119867119878 (1)
119889119867119864
119889119905= 120575120582 + 120573
1119867119878minus 1205733119867119864 (2)
119889119867119868
119889119905= 1205732119867119878+ 1205733119867119864minus 120588120577 minus 120575120582 minus 120573
4119867119868 (3)
119889119881119878
119889119905= 1205734119867119868minus 1205735119881119878 (4)
119889119881119868
119889119905= 1205735119881119878minus 120574119881119868 (5)
To find the steady state we assume that the compartments119881119878
119881119868119867119864119867119878 and119867
119868do not depend on time Therefore
119889119867119878
119889119905=119889119867119864
119889119905=119889119867119868
119889119905=119889119881119878
119889119905=119889119881119868
119889119905= 0 (6)
We now solve the differential equations 119889119867119878119889119905 = 119889119867
119864119889119905 =
119889119867119868119889119905 = 119889119881
119878119889119905 = 119889119881
119868119889119905 = 0 to find the value for119867
119878119867119864
119867119868 119881119878 and 119881
119868
From (4)1205734119867119868minus 1205735119881119878= 0
119867119868=1205735
1205734
119881119878
119867119868= 119860119881119878
(7)
From (5)1205735119881119878minus 120574119881119868= 0
119881119868=1205735119881119878
120574= 119861119881119878
(8)
From (1)120588120577 + 120574119881
119868minus (1205731+ 1205732)119867119878= 0
(1205731+ 1205732)119867119878= 120588120577 + 120574119881
119868
(9)
But
119881119868=1205735119881119878
120574
(1205731+ 1205732)119867119878= 120588120577 +
1205741205735119881119878
120574
119867119878=120588120577 + 120573
5119881119878
1205731+ 1205732
= 119862119881119878
(10)
From (2)120575120582 + 120573
1119867119878minus 1205733119867119864= 0
1205733119867119864= 120575120582 + 120573
1119867119878
but 119867119878=120588120577 + 120573
5119881119878
1205731+ 1205732
1205733119867119864= 120575120582 + 120573
1(120588120577 + 120573
5119881119878
1205731+ 1205732
)
119867119864=120575120582 (1205731+ 1205732) + 1205731120588120577 + 120573
11205735119881119878
1205733
= 119863119881119878
(11)
Lastly from (3)
1205732119867119878+ 1205733119867119864minus 120588120577 minus 120575120582 minus 120573
4119867119868= 0
1205732119867119878+ 1205733119867119864minus 120588120577 minus 120575120582 minus 120573
4119867119868= 0
1205732119862119881119878+ 1205733119863119881119878minus 120588120577 minus 120575120582 minus 120573
4119860119881119878= 0
(12)
We have that
119881119878=
120588120577 + 120575120582
1205732119862 + 120573
3119863 minus 120573
4119860 (13)
provided that 1205732119862 + 120573
3119863 gt 120573
4119860
Therefore the disease free equilibrium (DFE) is given as
1198640= (
120588120577
1205731+ 1205732
0 0 0 0) (14)
and the endemic equilibrium is given as
119867119878=120588120577 + 120573
5
1205731+ 1205732
((120588120577 + 120575120582)
times (1205732(120588120577 + 120573
5
1205731+ 1205732
) + 1205733
times (120575120582 (1205731+ 1205732) + 1205731120588120577 + 120573
11205735
1205733
)
minus1205734(1205735
1205734
))
minus1
)
= ((120588120577 + 1205735) (120588120577 + 120575120582))
times (1205732(120588120577 + 120573
5) + (120573
1+ 1205732)
times [120575120582 (1205731+ 1205732) + 1205731120588120577 + 120573
11205735]
minus (1205731+ 1205732) 1205735)minus1
(15)
119867119864=120575120582 (1205731+ 1205732) + 1205731120588120577 + 120573
11205735
1205733
times ((120588120577 + 120575120582) (1205732(120588120577 + 120573
5
1205731+ 1205732
) + 120575120582 (1205731+ 1205732)
+ 1205731120588120577 + 120573
11205735minus 1205735)minus1
)
4 Abstract and Applied Analysis
= ([120575120582 (1205731+ 1205732) + 1205731120588120577 + 120573
11205735] (120588120577 + 120575120582))
times ((12057321205733) (120588120577 + 120573
5)
1205731+ 1205732
+ 120575120582 (1205731+ 1205732) 1205733
+ 12057311205733120588120577 + 120573
112057331205735minus 12057331205735)
minus1
(16)
119867119868=1205735
1205734
((120588120577 + 120575120582)
times (1205732(120588120577 + 120573
5
1205731+ 1205732
)
+ 1205733(120575120582 (1205731+ 1205732) + 1205731120588120577 + 120573
11205735
1205733
)
minus1205734(1205735
1205734
))
minus1
)
= (1205735(120588120577 + 120575120582)) (120573
21205734(120588120577 + 120573
5
1205731+ 1205732
) + 1205734120575120582 (1205731+ 1205732)
+ 12057311205734120588120577 + 120573
112057341205735minus 12057341205735)minus1
(17)
119881119878= ((120588120577 + 120575120582)
times (1205732(120588120577 + 120573
5
1205731+ 1205732
)
+ 1205733(120575120582 (1205731+ 1205732) + 1205731120588120577 + 120573
11205735
1205733
)
minus1205734(1205735
1205734
))
minus1
)
(18)
119881119868=1205735
120574((120588120577 + 120575120582)
times (1205732(120588120577 + 120573
5
1205731+ 1205732
)
+ 1205733(120575120582 (1205731+ 1205732) + 1205731120588120577 + 120573
11205735
1205733
)
minus1205734(1205735
1205734
))
minus1
)
= (1205735(120588120577 + 120575120582))
times (1205741205732(120588120577 + 120573
5
1205731+ 1205732
) + 120574120575120582 (1205731+ 1205732)
+ 1205741205731120588120577 + 120574120573
11205735minus 1205741205735)minus1
(19)
5 Stability Analysis
To assess the stability of the (DFE) we solve the model char-acteristics equation using the Jacobean matrix formulatedfrom (1) (2) (3) (4) and (5) above as follows
119869 = (
minus(1205731+ 1205732) 0 0 0 120574
1205731
minus1205733
0 0 01205732
1205733minus1205734
0 00 0 120573
4minus12057350
0 0 0 1205735minus120574
) (20)
Applying the eigenvalue
120582 det (119869 minus 120582119868) = 0 (21)
we have
det[[[[[[
[
minus (1205731+ 1205732) minus 120582 0 0 0 120574
1205731
minus1205733minus 120582 0 0 0
1205732
1205733
minus1205734minus 120582 0 0
0 0 1205734
minus1205735minus 120582 0
0 0 0 1205735
minus120574 minus 120582
]]]]]]
]
= 0
det[[[[[[
[
minus (1205731+ 1205732+ 120582) 0 0 0 120574
1205731
minus (1205733+ 120582) 0 0 0
1205732
1205733
minus (1205734+ 120582) 0 0
0 0 1205734
minus (1205735+ 120582) 0
0 0 0 1205735
minus (120574 + 120582)
]]]]]]
]
= 0
minus (1205731+ 1205732+ 120582)
[[[[
[
minus (1205733+ 120582) 0 0 0
1205733
minus (1205734+ 120582) 0 0
0 1205734
minus (1205735+ 120582) 0
0 0 1205735
minus (120574 + 120582)
]]]]
]
+ 120574[[[[
[
1205731minus (1205733+ 120582) 0 0
1205732
1205733
minus (1205734+ 120582) 0
0 0 1205734
minus (1205735+ 120582)
0 0 0 1205735
]]]]
]
= 0
Abstract and Applied Analysis 5
minus (1205731+ 1205732+ 120582)
[[
[
minus (1205733+ 120582)
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
minus (1205734+ 120582) 0 0
1205734
minus (1205735+ 120582) 0
0 1205735
minus (120574 + 120582)
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
]]
]
+ 120574[[
[
1205731
[[
[
1205733minus (1205734+ 120582) 0
0 1205734
minus (1205735+ 120582)
0 0 1205735
]]
]
+ (1205733+ 120582)
[[
[
1205732minus (1205734+ 120582) 0
0 1205734
minus (1205735+ 120582)
0 0 1205735
]]
]
]]
]
= 0
minus (1205731+ 1205732+ 120582) [minus (120573
3+ 120582) [minus (120573
4+ 120582)
100381610038161003816100381610038161003816100381610038161003816
minus (1205735+ 120582) 0
1205735
minus (120574 + 120582)
100381610038161003816100381610038161003816100381610038161003816]] + 120574120573
1120573312057341205735+ 120574 (120573
3+ 120582) 120573
212057341205735= 0
minus [(1205731+ 1205732+ 120582) (120573
3+ 120582) (120573
4+ 120582) (120573
5+ 120582) (120574 + 120582)] + 120574120573
1120573312057341205735+ 1205741205732120573312057341205735+ 120574120582120573
212057341205735= 0
(22)
Hence we obtain from minus(1205731+ 1205732+ 120582) = 0
1205821= minus (120573
1+ 1205732) (23)
From minus(1205733+ 120582) = 0
1205822= minus1205733 (24)
From minus(1205734+ 120582) = 0
1205823= minus1205734 (25)
From (1205735+ 120582) = 0
1205824= minus1205735 (26)
From (120574 + 120582) = 0
1205825= minus120574 (27)
Given that the results of 1205821 1205822 1205823 1205824 and 120582
5are all negative
and considering the principles of linearized stability whichsays that the systems are asymptotically stable if and only ifall the eigenvalues of the system have negative real parts wesay that the disease-free equilibrium (DFE) is asymptoticallystable
6 Application of the HomotopyDecomposition Method (HDM)
In this section we apply the HDM [15ndash17] to solve thesemathematical sets of equations representing the onchocerci-asis disease model as follows
119867119878(119905) minus 119867
119878(0) = int
119905
0
(120588120577 + 120574119881119868(120591) minus (120573
1+ 1205732)119867119878(120591)) 119889120591
119867119864(119905) minus 119867
119864(0) = int
119905
0
(120575120582 + 1205731119867119878(120591) minus 120573
3119867119864(120591)) 119889120591
119867119868(119905) minus 119867
119868(0)
= int119905
0
(1205732119867119878(120591) + 120573
3119867119864(120591) minus 120588120577 minus 120575120582 minus 120573
4119867119868(120591)) 119889120591
119881119878(119905) minus 119881
119878(0) = int
119905
0
(1205734119867119868(120591) minus 120573
5119881119878(120591)) 119889120591
119881119868(119905) minus 119881
119868(0) = int
119905
0
(1205735119881119878(120591) minus 120574119881
119868(120591)) 119889120591
(28)
We then assume that the solutions of the above integralequations can be put in the following form for 119875 isin (0 1)
119867119878(119905) =
infin
sum119899=0
119875119899119867119878119899(119905) 119867
119864(119905) =
infin
sum119899=0
119875119899119867119864119899(119905)
119867119868(119905) =
infin
sum119899=0
119875119899119867119868119899(119905) 119881
119878(119905) =
infin
sum119899=0
119875119899119881119878119899(119905)
119881119868(119905) =
infin
sum119899=0
119875119899119881119868119899(119905)
(29)
Replacing the above expressions in the integral equations weobtain the following expressions
infin
sum119899=0
119875119899119867119878119899(119905) minus 119867
119878(0)
= 119875int119905
0
[(120588120577 + 120574infin
sum119899=0
119875119899119881119868119899(120591)
6 Abstract and Applied Analysis
minus (1205731+ 1205732)infin
sum119899=0
119875119899119867119878119899(120591)) 119889120591]
infin
sum119899=0
119875119899119867119864119899(119905) minus 119867
119864(0)
= 119875int119905
0
[(120575120582 + 1205731
infin
sum119899=0
119875119899119867119878119899(120591)
minus 1205733
infin
sum119899=0
119875119899119867119864119899(120591)) 119889120591]
infin
sum119899=0
119875119899119867119868119899(119905) minus 119867
119868(0)
= 119875int119905
0
[(1205732
infin
sum119899=0
119875119899119867119878119899(120591) + 120573
3
infin
sum119899=0
119875119899119867119864119899(120591)
minus 120588120577 minus 120575120582 minus 1205734
infin
sum119899=0
119875119899119867119868119899(120591)) 119889120591]
infin
sum119899=0
119875119899119881119878119899(119905) minus 119881
119878(0)
= 119875int119905
0
[(1205734
infin
sum119899=0
119875119899119867119868119899(120591) minus 120573
5
infin
sum119899=0
119875119899119881119878119899(120591)) 119889120591]
infin
sum119899=0
119875119899119881119868119899(119905) minus 119881
119868(0)
= 119875int119905
0
[(1205735
infin
sum119899=0
119875119899119881119878119899(120591) minus 120574
infin
sum119899=0
119875119899119881119868119899(120591)) 119889120591]
(30)
We compared the terms of the same power of 119875 and obtainedthe following
1198750 1198671198780(119905) = 119867
119878(0)
1198750 1198671198640(119905) = 119867
119864(0)
1198750 1198671198680(119905) = 119867
119868(0)
1198750 1198811198780(119905) = 119881
119878(0)
1198750 1198811198680(119905) = 119881
119868(0)
1198751 1198671198781(119905) = int
119905
0
(120588120577 + 1205741198811198680(120591) minus (120573
1+ 1205732)1198671198780(120591)) 119889120591
1198751 1198671198641(119905) = int
119905
0
(120575120582 + 12057311198671198780(120591) minus 120573
31198671198640(120591)) 119889120591
1198751 1198671198681(119905)
= int119905
0
(12057321198671198780(120591)+120573
31198671198640(120591)minus120588120577minus 120575120582 minus120573
41198671198680(120591)) 119889120591
1198751 1198811198781(119905) = int
119905
0
(12057341198671198680(120591) minus 120573
51198811198780(120591)) 119889120591
1198751 1198811198681(119905) = int
119905
0
(12057351198811198780(120591) minus 120574119881
1198680(120591)) 119889120591
(31)
Continuing for 119899 ge 2 we have
119867119878119899(119905) = int
119905
0
(120574119881119868(119899minus1)
(120591) minus (1205731+ 1205732)119867119878(119899minus1)
(120591)) 119889120591
119867119864119899(119905) = int
119905
0
(1205731119867119878(119899minus1)
(120591) minus 1205733119867119864(119899minus1)
(120591)) 119889120591
119867119868119899(119905) = int
119905
0
(1205732119867119878(119899minus1)
(120591) + 1205733119867119864(119899minus1)
(120591)
minus 1205734119867119868(119899minus1)
(120591)) 119889120591
119881119878119899(119905) = int
119905
0
(1205734119867119868(119899minus1)
(120591) minus 1205735119881119878(119899minus1)
(120591)) 119889120591
119881119868119899(119905) = int
119905
0
(1205735119881119878(119899minus1)
(120591) minus 120574119881119868(119899minus1)
(120591)) 119889120591
(32)
We now integrate the above equations to obtain the following
1198671198680(119905) = 0 119867
1198640(119905) = 0 119867
1198780 [119905] = 119873 1198811198780 [119905] = 119872
1198811198680 [119905] = 0 119867
1198681(119905) = 119905 (minus120575120582 minus 120577120588 + 119873120573
2)
1198671198781 [119905] = 119905 (120577120588 minus 119873 (1205731 + 1205732))
1198671198641(119905) = 119905 (120575120582 + 119873120573
1)
1198811198681 [119905] = 1198721199051205735 119881
1198781 [119905] = minus1198721199051205735
1198671198682(119905) =
1
21199052120577120588120573
2minus1
2119873119905212057311205732minus1
2119873119905212057322+1
21199052120575120582120573
3
+1
2119873119905212057311205733+1
21199052120575120582120573
4+1
21199052120577120588120573
4minus1
2119873119905212057321205734
1198671198642(119905) =
1
21199052120577120588120573
1minus1
2119873119905212057321minus1
2119873119905212057311205732
minus1
21199052120575120582120573
3minus1
2119873119905212057311205733
1198671198782 [119905] = minus
1
21199052120577120588120573
1+1
2119873119905212057321minus1
21199052120577120588120573
2
+ 119873119905212057311205732+1
2119873119905212057322+1
21198721199052120574120573
5
1198811198782 [119905] = minus
1
21199052120575120582120573
4minus1
21199052120577120588120573
4
+1
2119873119905212057321205734+1
211987211990521205732
5
1198811198682 [119905] = minus
1
21198721199052120574120573
5minus1
211987211990521205732
5
1198671198683(119905) = minus
1
61199053120577120588120573
11205732+1
61198731199053120573211205732minus1
611990531205771205881205732
2
+1
31198731199053120573112057322+1
6119873119905312057332+1
61199053120577120588120573
11205733
minus1
61198731199053120573211205733minus1
61198731199053120573112057321205733
minus1
611990531205751205821205732
3
minus1
61198731199053120573112057323minus1
61199053120577120588120573
21205734
+1
61198731199053120573112057321205734+1
61198731199053120573221205734
minus1
61199053120575120582120573
31205734minus1
61198731199053120573112057331205734
Abstract and Applied Analysis 7
minus1
611990531205751205821205732
4minus1
611990531205771205881205732
4+1
61198731199053120573212057324
+1
61198721199053120574120573
21205735
1198671198643(119905) = minus
1
611990531205771205881205732
1+1
6119873119905312057331minus1
61199053120577120588120573
11205732
+1
31198731199053120573211205732+1
61198731199053120573112057322
minus1
61199053120577120588120573
11205733+1
61198731199053120573211205733
+1
61198731199053120573112057321205733+1
611990531205751205821205732
3
+1
61198731199053120573112057323+1
61198721199053120574120573
11205735
1198671198783 [119905] =
1
611990531205771205881205732
1minus1
6119873119905312057331+1
31199053120577120588120573
11205732
minus1
21198731199053120573211205732+1
611990531205771205881205732
2minus1
21198731199053120573112057322
minus1
6119873119905312057332minus1
611987211990531205742120573
5
minus1
61198721199053120574120573
11205735
minus1
61198721199053120574120573
21205735minus1
611987211990531205741205732
5
1198811198783 [119905] =
1
61199053120577120588120573
21205734minus1
61198731199053120573112057321205734minus1
61198731199053120573221205734
+1
61199053120575120582120573
31205734+1
61198731199053120573112057331205734+1
611990531205751205821205732
4
+1
611990531205771205881205732
4minus1
61198731199053120573212057324+1
61199053120575120582120573
41205735
+1
61199053120577120588120573
41205735minus1
61198731199053120573212057341205735minus1
611987211990531205733
5
1198811198683 [119905] =
1
611987211990531205742120573
5minus1
61199053120575120582120573
41205735minus1
61199053120577120588120573
41205735
+1
61198731199053120573212057341205735+1
611987211990531205741205732
5+1
611987211990531205733
5
1198671198784 [119905] = minus
1
2411990541205771205881205733
1+1
24119873119905412057341minus1
811990541205771205881205732
11205732
+1
61198731199054120573311205732minus1
81199054120577120588120573
112057322+1
411987311990541205732112057322
minus1
2411990541205771205881205733
2+1
61198731199054120573112057332+1
24119873119905412057342
+1
2411987211990541205743120573
5+1
2411987211990541205742120573
11205735
+1
2411987211990541205741205732
11205735+1
2411987211990541205742120573
21205735
+1
121198721199054120574120573
112057321205735+1
2411987211990541205741205732
21205735
minus1
241199054120574120575120582120573
41205735minus1
241199054120574120577120588120573
41205735
+1
241198731199054120574120573
212057341205735+1
24119872119905412057421205732
5
+1
241198721199054120574120573
112057325+1
241198721199054120574120573
212057325+1
2411987211990541205741205733
5
1198671198644(119905) =
1
2411990541205771205881205733
1minus1
24119873119905412057341+1
1211990541205771205881205732
11205732
minus1
81198731199054120573311205732+1
241199054120577120588120573
112057322minus1
811987311990541205732112057322
minus1
241198731199054120573112057332+1
2411990541205771205881205732
11205733minus1
241198731199054120573311205733
+1
241199054120577120588120573
112057321205733minus1
1211987311990541205732112057321205733minus1
2411987311990541205731120573221205733
+1
241199054120577120588120573
112057323minus1
2411987311990541205732112057323minus1
2411987311990541205731120573212057323
minus1
2411990541205751205821205733
3minus1
241198731199054120573112057333minus1
2411987211990541205742120573
11205735
minus1
2411987211990541205741205732
11205735minus1
241198721199054120574120573
112057321205735
minus1
241198721199054120574120573
112057331205735minus1
241198721199054120574120573
112057325
1198671198684(119905) =
1
2411990541205771205881205732
11205732minus1
241198731199054120573311205732+1
121199054120577120588120573
112057322
minus1
811987311990541205732112057322+1
2411990541205771205881205733
2minus1
81198731199054120573112057332
minus1
24119873119905412057342minus1
2411990541205771205881205732
11205733+1
241198731199054120573311205733
minus1
241199054120577120588120573
112057321205733+1
1211987311990541205732112057321205733
+1
2411987311990541205731120573221205733
minus1
241199054120577120588120573
112057323+1
2411987311990541205732112057323+1
2411987311990541205731120573212057323
+1
2411990541205751205821205733
3+1
241198731199054120573112057333+1
241199054120577120588120573
112057321205734
minus1
2411987311990541205732112057321205734+1
2411990541205771205881205732
21205734minus1
1211987311990541205731120573221205734
minus1
241198731199054120573321205734minus1
241199054120577120588120573
112057331205734
+1
2411987311990541205732112057331205734+1
2411987311990541205731120573212057331205734
+1
2411990541205751205821205732
31205734+1
2411987311990541205731120573231205734
+1
241199054120577120588120573
212057324minus1
2411987311990541205731120573212057324minus1
2411987311990541205732212057324
+1
241199054120575120582120573
312057324+1
2411987311990541205731120573312057324+1
2411990541205751205821205733
4
+1
2411990541205771205881205733
4minus1
241198731199054120573212057334minus1
2411987211990541205742120573
21205735
minus1
241198721199054120574120573
112057321205735minus1
2411987211990541205741205732
21205735
+1
241198721199054120574120573
112057331205735
minus1
241198721199054120574120573
212057341205735minus1
241198721199054120574120573
212057325
1198811198784 [119905] = minus
1
241199054120577120588120573
112057321205734+1
2411987311990541205732112057321205734
minus1
2411990541205771205881205732
21205734+1
1211987311990541205731120573221205734
8 Abstract and Applied Analysis
+1
241198731199054120573321205734+1
241199054120577120588120573
112057331205734
minus1
2411987311990541205732112057331205734minus1
2411987311990541205731120573212057331205734
minus1
2411990541205751205821205732
31205734
minus1
2411987311990541205731120573231205734minus1
241199054120577120588120573
212057324
+1
2411987311990541205731120573212057324+1
2411987311990541205732212057324
minus1
241199054120575120582120573
312057324minus1
2411987311990541205731120573312057324
minus1
2411990541205751205821205733
4minus1
2411990541205771205881205733
4
+1
241198731199054120573212057334
+1
241198721199054120574120573
212057341205735minus1
241199054120577120588120573
212057341205735
+1
2411987311990541205731120573212057341205735+1
2411987311990541205732212057341205735
minus1
241199054120575120582120573
312057341205735minus1
2411987311990541205731120573312057341205735
minus1
2411990541205751205821205732
41205735minus1
2411990541205771205881205732
41205735+1
2411987311990541205732120573241205735
minus1
241199054120575120582120573
412057325minus1
241199054120577120588120573
412057325
+1
2411987311990541205732120573412057325+1
2411987211990541205734
5
1198811198684 [119905] = minus
1
2411987211990541205743120573
5+1
241199054120574120575120582120573
41205735+1
241199054120574120577120588120573
41205735
minus1
241198731199054120574120573
212057341205735+1
241199054120577120588120573
212057341205735
minus1
2411987311990541205731120573212057341205735minus1
2411987311990541205732212057341205735
+1
241199054120575120582120573
312057341205735+1
2411987311990541205731120573312057341205735
+1
2411990541205751205821205732
41205735+1
2411990541205771205881205732
41205735
minus1
2411987311990541205732120573241205735minus1
24119872119905412057421205732
5
+1
241199054120575120582120573
412057325+1
241199054120577120588120573
412057325
minus1
2411987311990541205732120573412057325minus1
2411987211990541205741205733
5minus1
2411987211990541205734
5
1198671198785 [119905] =
1
12011990551205771205881205734
1minus
1
120119873119905512057351+1
3011990551205771205881205733
11205732
minus1
241198731199055120573411205732+1
2011990551205771205881205732
112057322minus1
1211987311990551205733112057322
+1
301199055120577120588120573
112057332minus1
1211987311990551205732112057332+
1
12011990551205771205881205734
2
minus1
241198731199055120573112057342minus
1
120119873119905512057352minus
1
12011987211990551205744120573
5
minus1
12011987211990551205743120573
11205735minus
1
120119872119905512057421205732
11205735
minus1
12011987211990551205741205733
11205735minus
1
12011987211990551205743120573
21205735
minus1
6011987211990551205742120573
112057321205735minus1
4011987211990551205741205732
112057321205735
minus1
120119872119905512057421205732
21205735minus1
401198721199055120574120573
1120573221205735
minus1
12011987211990551205741205733
21205735+
1
12011990551205742120575120582120573
41205735
+1
12011990551205742120577120588120573
41205735+
1
1201199055120574120575120582120573
112057341205735
+1
1201199055120574120577120588120573
112057341205735minus
1
12011987311990551205742120573
212057341205735
+1
1201199055120574120575120582120573
212057341205735+1
601199055120574120577120588120573
212057341205735
minus1
601198731199055120574120573
1120573212057341205735minus1
6011987311990551205741205732
212057341205735
+1
1201199055120574120575120582120573
312057341205735+
1
1201198731199055120574120573
1120573312057341205735
+1
12011990551205741205751205821205732
41205735+
1
12011990551205741205771205881205732
41205735
minus1
1201198731199055120574120573
2120573241205735minus
1
120119872119905512057431205732
5
minus1
12011987211990551205742120573
112057325minus
1
12011987211990551205741205732
112057325
minus1
12011987211990551205742120573
212057325minus1
601198721199055120574120573
1120573212057325
minus1
12011987211990551205741205732
212057325+
1
1201199055120574120575120582120573
412057325
+1
1201199055120574120577120588120573
412057325minus
1
1201198731199055120574120573
2120573412057325
minus1
120119872119905512057421205733
5minus
1
1201198721199055120574120573
112057335
minus1
1201198721199055120574120573
212057335minus
1
12011987211990551205741205734
5
1198671198645 [119905] = minus
1
12011990551205771205881205734
1+
1
120119873119905512057351
minus1
4011990551205771205881205733
11205732+1
301198731199055120573411205732
minus1
4011990551205771205881205732
112057322+1
2011987311990551205733112057322
minus1
1201199055120577120588120573
112057332+1
3011987311990551205732112057332
+1
1201198731199055120573112057342
minus1
12011990551205771205881205733
11205733+
1
1201198731199055120573411205733
minus1
6011990551205771205881205732
112057321205733+1
4011987311990551205733112057321205733
minus1
1201199055120577120588120573
1120573221205733+1
40119873119905512057321120573221205733
+1
12011987311990551205731120573321205733minus
1
12011990551205771205881205732
112057323
+1
12011987311990551205733112057323minus
1
1201199055120577120588120573
1120573212057323
Abstract and Applied Analysis 9
+1
60119873119905512057321120573212057323+
1
120119873119905512057311205732212057323
minus1
1201199055120577120588120573
112057333+
1
12011987311990551205732112057333
+1
12011987311990551205731120573212057333+
1
12011990551205751205821205734
3
+1
1201198731199055120573112057343+
1
12011987211990551205743120573
11205735
+1
120119872119905512057421205732
11205735+
1
12011987211990551205741205733
11205735
+1
12011987211990551205742120573
112057321205735+1
6011987211990551205741205732
112057321205735
+1
1201198721199055120574120573
1120573221205735+
1
12011987211990551205742120573
112057331205735
+1
12011987211990551205741205732
112057331205735+
1
1201198721199055120574120573
1120573212057331205735
+1
1201198721199055120574120573
1120573231205735minus
1
1201199055120574120575120582120573
112057341205735
minus1
1201199055120574120577120588120573
112057341205735+
1
1201198731199055120574120573
1120573212057341205735
+1
12011987211990551205742120573
112057325+
1
12011987211990551205741205732
112057325
+1
1201198721199055120574120573
1120573212057325+
1
1201198721199055120574120573
1120573312057325
+1
1201198721199055120574120573
112057335
1198671198685 [119905]= minus
1
12011990551205771205881205733
11205732+
1
1201198731199055120573411205732minus1
4011990551205771205881205732
112057322
+1
3011987311990551205733112057322minus1
401199055120577120588120573
112057332+1
2011987311990551205732112057332
minus1
12011990551205771205881205734
2+1
301198731199055120573112057342+
1
120119873119905512057352
+1
12011990551205771205881205733
11205733minus
1
1201198731199055120573411205733
+1
6011990551205771205881205732
112057321205733minus1
4011987311990551205733112057321205733
+1
1201199055120577120588120573
1120573221205733minus1
40119873119905512057321120573221205733
minus1
12011987311990551205731120573321205733+
1
12011990551205771205881205732
112057323
minus1
12011987311990551205733112057323+
1
1201199055120577120588120573
1120573212057323
minus1
60119873119905512057321120573212057323minus
1
120119873119905512057311205732212057323
+1
1201199055120577120588120573
112057333minus
1
12011987311990551205732112057333
minus1
12011987311990551205731120573212057333minus
1
12011990551205751205821205734
3
minus1
1201198731199055120573112057343minus
1
12011990551205771205881205732
112057321205734
+1
12011987311990551205733112057321205734minus1
601199055120577120588120573
1120573221205734
+1
40119873119905512057321120573221205734minus
1
12011990551205771205881205733
21205734
+1
4011987311990551205731120573321205734+
1
1201198731199055120573421205734
+1
12011990551205771205881205732
112057331205734minus
1
12011987311990551205733112057331205734
+1
1201199055120577120588120573
1120573212057331205734minus1
60119873119905512057321120573212057331205734
minus1
120119873119905512057311205732212057331205734+
1
1201199055120577120588120573
1120573231205734
minus1
120119873119905512057321120573231205734minus
1
120119873119905512057311205732120573231205734
minus1
12011990551205751205821205733
31205734minus
1
12011987311990551205731120573331205734
minus1
1201199055120577120588120573
1120573212057324+
1
120119873119905512057321120573212057324
minus1
12011990551205771205881205732
212057324+1
60119873119905512057311205732212057324
+1
12011987311990551205733212057324+
1
1201199055120577120588120573
1120573312057324
minus1
120119873119905512057321120573312057324minus
1
120119873119905512057311205732120573312057324
minus1
12011990551205751205821205732
312057324minus
1
120119873119905512057311205732312057324
minus1
1201199055120577120588120573
212057334+
1
12011987311990551205731120573212057334
+1
12011987311990551205732212057334minus
1
1201199055120575120582120573
312057334
minus1
12011987311990551205731120573312057334minus
1
12011990551205751205821205734
4
minus1
12011990551205771205881205734
4+
1
1201198731199055120573212057344
+1
12011987211990551205743120573
21205735+
1
12011987211990551205742120573
112057321205735
+1
12011987211990551205741205732
112057321205735+
1
120119872119905512057421205732
21205735
+1
601198721199055120574120573
1120573221205735+
1
12011987211990551205741205733
21205735
minus1
12011987211990551205742120573
112057331205735minus
1
12011987211990551205741205732
112057331205735
minus1
1201198721199055120574120573
1120573212057331205735minus
1
1201198721199055120574120573
1120573231205735
+1
12011987211990551205742120573
212057341205735minus
1
1201199055120574120575120582120573
212057341205735
minus1
1201199055120574120577120588120573
212057341205735+
1
1201198721199055120574120573
1120573212057341205735
+1
12011987211990551205741205732
212057341205735+
1
12011987311990551205741205732
212057341205735
minus1
1201198721199055120574120573
1120573312057341205735+
1
1201198721199055120574120573
2120573241205735
+1
12011987211990551205742120573
212057325+
1
1201198721199055120574120573
1120573212057325
+1
12011987211990551205741205732
212057325minus
1
1201198721199055120574120573
1120573312057325
+1
1201198721199055120574120573
2120573412057325+
1
1201198721199055120574120573
212057335
10 Abstract and Applied Analysis
1198811198785 [119905] =
1
12011990551205771205881205732
112057321205734minus
1
12011987311990551205733112057321205734
+1
601199055120577120588120573
1120573221205734minus1
40119873119905512057321120573221205734
+1
12011990551205771205881205733
21205734minus1
4011987311990551205731120573321205734
minus1
1201198731199055120573421205734minus
1
12011990551205771205881205732
112057331205734
+1
12011987311990551205733112057331205734minus
1
1201199055120577120588120573
1120573212057331205734
+1
60119873119905512057321120573212057331205734+
1
120119873119905512057311205732212057331205734
minus1
1201199055120577120588120573
1120573231205734+
1
120119873119905512057321120573231205734
+1
120119873119905512057311205732120573231205734+
1
12011990551205751205821205733
31205734
+1
12011987311990551205731120573331205734+
1
1201199055120577120588120573
1120573212057324
minus1
120119873119905512057321120573212057324+
1
12011990551205771205881205732
212057324
minus1
60119873119905512057311205732212057324minus
1
12011987311990551205733212057324
minus1
1201199055120577120588120573
1120573312057324+
1
120119873119905512057321120573312057324
+1
120119873119905512057311205732120573312057324+
1
12011990551205751205821205732
312057324
+1
120119873119905512057311205732312057324+
1
1201199055120577120588120573
212057334
minus1
12011987311990551205731120573212057334minus
1
12011987311990551205732212057334
+1
1201199055120575120582120573
312057334+
1
12011987311990551205731120573312057334
+1
12011990551205751205821205734
4+
1
12011990551205771205881205734
4minus
1
1201198731199055120573212057344
minus1
12011987211990551205742120573
212057341205735minus
1
1201198721199055120574120573
1120573212057341205735
+1
1201199055120577120588120573
1120573212057341205735minus
1
120119873119905512057321120573212057341205735
minus1
12011987211990551205741205732
212057341205735+
1
12011990551205771205881205732
212057341205735
minus1
60119873119905512057311205732212057341205735minus
1
12011987311990551205733212057341205735
+1
1201198721199055120574120573
1120573312057341205735minus
1
1201199055120577120588120573
1120573312057341205735
+1
120119873119905512057321120573312057341205735+
1
120119873119905512057311205732120573312057341205735
+1
12011990551205751205821205732
312057341205735+
1
120119873119905512057311205732312057341205735
minus1
1201198721199055120574120573
2120573241205735+
1
1201199055120577120588120573
2120573241205735
minus1
120119873119905512057311205732120573241205735minus
1
120119873119905512057322120573241205735
+1
1201199055120575120582120573
3120573241205735+
1
120119873119905512057311205733120573241205735
+1
12011990551205751205821205733
41205735+
1
12011990551205771205881205733
41205735
minus1
12011987311990551205732120573341205735minus1
601198721199055120574120573
2120573412057325
+1
1201199055120577120588120573
2120573412057325minus
1
120119873119905512057311205732120573412057325
minus1
120119873119905512057322120573412057325+
1
1201199055120575120582120573
3120573412057325
+1
120119873119905512057311205733120573412057325+
1
12011990551205751205821205732
412057325
+1
12011990551205771205881205732
412057325minus
1
120119873119905512057321205732412057325
+1
1201199055120575120582120573
412057335+
1
1201199055120577120588120573
412057335
minus1
12011987311990551205732120573412057335minus
1
12011987211990551205735
5
1198811198685 [119905] =
1
12011987211990551205744120573
5minus
1
12011990551205742120575120582120573
41205735
minus1
12011990551205742120577120588120573
41205735+
1
12011987311990551205742120573
212057341205735
minus1
1201199055120574120577120588120573
212057341205735+
1
1201198731199055120574120573
1120573212057341205735
minus1
1201199055120577120588120573
1120573212057341205735+
1
120119873119905512057321120573212057341205735
+1
12011987311990551205741205732
212057341205735minus
1
12011990551205771205881205732
212057341205735
+1
60119873119905512057311205732212057341205735+
1
12011987311990551205733212057341205735
minus1
1201199055120574120575120582120573
312057341205735minus
1
1201198731199055120574120573
1120573312057341205735
+1
1201199055120577120588120573
1120573312057341205735minus
1
120119873119905512057321120573312057341205735
minus1
120119873119905512057311205732120573312057341205735minus
1
12011990551205751205821205732
312057341205735
minus1
120119873119905512057311205732312057341205735minus
1
12011990551205741205751205821205732
41205735
minus1
12011990551205741205771205881205732
41205735+
1
1201198731199055120574120573
2120573241205735
minus1
1201199055120577120588120573
2120573241205735+
1
120119873119905512057311205732120573241205735
+1
120119873119905512057322120573241205735minus
1
1201199055120575120582120573
3120573241205735
minus1
120119873119905512057311205733120573241205735minus
1
12011990551205751205821205733
41205735
minus1
12011990551205771205881205733
41205735+
1
12011987311990551205732120573341205735
+1
120119872119905512057431205732
5minus
1
1201199055120574120575120582120573
412057325
minus1
1201199055120574120577120588120573
412057325+
1
1201198721199055120574120573
2120573412057325
+1
1201198731199055120574120573
2120573412057325minus
1
1201199055120577120588120573
2120573412057325
+1
120119873119905512057311205732120573412057325+
1
120119873119905512057322120573412057325
Abstract and Applied Analysis 11
minus1
1201199055120575120582120573
3120573412057325minus
1
120119873119905512057311205733120573412057325
minus1
12011990551205751205821205732
412057325minus
1
12011990551205771205881205732
412057325
+1
120119873119905512057321205732412057325+
1
120119872119905512057421205733
5
minus1
1201199055120575120582120573
412057335minus
1
1201199055120577120588120573
412057335
+1
12011987311990551205732120573412057335+
1
12011987211990551205741205734
5
+1
12011987211990551205735
5
(33)
In general by the repeated application of the above proce-dure we have that
119867119878119899=119905119899
1198991198861119899 119867
119864119899=119905119899
1198991198862119899 119867
119868119899=119905119899
1198991198863119899
119881119878119899=119905119899
1198991198864119899 119881
119868119899=119905119899
1198991198865119899
(34)
where 1198861119899 1198862119899 1198863119899 1198864119899 1198865119899
depend on fixed set of empiricalparameters Hence the approximate solution of the modelabove is given as
119867119878(119905) =
infin
sum119899=0
119905119899
1198991198861119899
119867119864(119905) =
infin
sum119899=0
119905119899
1198991198862119899
119867119868(119905) =
infin
sum119899=0
119905119899
1198991198863119899
119881119878(119905) =
infin
sum119899=0
119905119899
1198991198864119899
119881119868(119905) =
infin
sum119899=0
119905119899
1198991198865119899
(35)
We therefore use the following theoretical parameters 1205731=
05 1205732= 05 120573
3= 05 120573
4= 05 120573
5= 05 120574 = 1 120577 = 1
119873 = 10 119872 = 10 120588 = 1 120575 = 1 and 120582 = 1 to examinethe behavior of the five different compartmental numericalsolutions in the system Figure 3 shows the behavior of thenumerical solution
7 Conclusion
Onchocerciasis model was examined and the mathematicalformulation underpinning the spread and control of thisdisease has been proposed on one hand On the otherhand we made use of some new analytical methods calledthe homotopy decomposition method (HDM) to derive thesolution of the resulting set of (1) to (5)The numerical resultswere presented to test the efficiency and the accuracy of themethod used The mathematical analysis of the resulting setof (1) to (5) above revealed that the stability analysis of the
05 10 15 20Time (per year)
2
4
6
8
10
Appr
oxim
ate s
olut
ions VS
HS
VI
HI
HE
Figure 3 Numerical solution of the model
five roots of the characteristics equations has all negativereal part In particular given that the stability analysis showsthat the DFE is asymptotically stable it would be possiblethat the parasite Onchocerca volvulus be eradicated in a finitetime On the other hand the numerical solution confirmedthat as the number of humans and vectors in the susceptiblecompartments reduces the number of infective humansand vectors in the infectious compartment increases Thenumerical solutions in both cases displayed the biologicalbehaviour of the real world situation Therefore in orderto build a healthy and a sustainable population treatmentwith IvermectinMectizan drugs and a more hygienic envi-ronment that does not allow the breeding and survival ofblackflies could lead to a total eradication of the disease
Conflict of Interests
The authors declare that they have no conflict of interestsregarding the publication of this paper
References
[1] J H F Remme The Global Burden of Onchocerciasis in 1990World Health Organization Geneva Switzerland 2004 httpwwwwhointhealthinfoglobal burden diseaseOnchocercia-sis201990pdf
[2] WorldHealthOrganization ldquoPrevention of blindness and visualimpairment priority eye diseasesrdquo 2014 httpwwwwhointblindnesscausespriorityenindex3html
[3] World Health Organization Certification of Elimination ofHuman Onchocerciasis Criteria and Procedures World HealthOrganizationGeneva Switzerland 2001 httpwhqlibdocwhointhq2001WHO CDS CPE CEE 200118bpdf
[4] K A Lindblade B Arana G Zea-Flores et al ldquoEliminationof Onchocerca volvulus transmission in the Santa Rosa focusof Guatemalardquo The American Journal of Tropical Medicine andHygiene vol 77 pp 334ndash341 2007
[5] J C Anosike and C O E Onwuliri ldquoStudies on filariasis inBauchi State Nigeria 1 Endemicity of human onchocerciasisin Ningi Local Government Areardquo Annals of Tropical Medicineand Parasitology vol 89 no 1 pp 31ndash38 1995
[6] WorldHealthOrganizationWeekly Epidemiological Records 71WHO Geneva Switzerland 1993
12 Abstract and Applied Analysis
[7] WorldHealthOrganization httpwwwwhointneglected dis-easesen
[8] Sightsaver httpwwwsightsaversorgabout uspublicationsonchocerciasis
[9] E C Uttah ldquoOnchocerciasis in the Upper Ima River BasinrdquoIranian Journal of Parasitology vol 5 no 2 pp 33ndash41 2010
[10] J C Anosike and C O E Onwuliri ldquoStudies on filariasis inBauchi State Nigeriamdash1 endemicity of human onchocerciasisin Ningi Local Government Areardquo Annals of Tropical Medicineand Parasitology vol 89 no 1 pp 31ndash38 1995
[11] L D Edungbola and S O Asaolu ldquoParasitologic survey ofonchocerciasis (river blindness) in Babana district Kwara StateNigeriardquo American Journal of Tropical Medicine and Hygienevol 33 no 6 pp 1147ndash1154 1984
[12] O C Nwaorgu A Ohaegbula I E Onweluzo et al ldquoResultsof a large scale onchocercosis survey in Enugu State NigeriardquoJournal of Helminthology vol 68 no 2 pp 155ndash159 1994
[13] F O Akinbo and C E Okaka ldquoPrevalence and socio-economiceffects of onchocerciasis in Ovia North East Local GovernmentArea of Edo State Nigeriardquo Rivista Di Parasitologie vol 22 no3 pp 215ndash221 2005
[14] M D Wogu and C E Okaka ldquoPrevalence and socio-economic effects of onchocerciasis in Okpuje Owan WestLocal Government Area Edo State Nigeriardquo InternationalJournal of Biomedical and Health Sciences vol 4 no 3 2008httpwwwasopahorg
[15] S C O Noutchie ldquoHomotopy Decomposition analysis of atuberculosis modelrdquo Life Science Journal vol 10 no 3 pp 459ndash465 2013
[16] A Atangana and A Secer ldquoThe time-fractional coupled-Korteweg-de-Vries equationsrdquo Abstract and Applied Analysisvol 2013 Article ID 947986 8 pages 2013
[17] A Atangana and N Bildik ldquoApproximate solution of tubercu-losis disease population dynamics modelrdquo Abstract and AppliedAnalysis vol 2013 Article ID 759801 8 pages 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Abstract and Applied Analysis
Blackfly stages Blackfly (genus Simulium)takes a blood meal
(L3 larvae enter bite wound)Migrate to head
and blackflyrsquos proboscis
L3 larvae
L1 larvae Blackfly takesa blood meal
(ingests microfilariae)
Microfilariae penetrateblackflyrsquos midgutand migrate to
thoracic muscles
Human stages
Subcutaneous tissues
Adults in subcutaneous nodule
Adults produce unsheathedmicrofilariae that typically arefound in skin and lymphaticsof connective tissues but also
occasionally in peripheralblood urine and sputum
12
3
45
6
7
8
9
di Infective stage
Diagnostic stage
i
d
Figure 1 Life cycle of Onchocerca volvulus [8]
microfilaria of Onchocerca volvulus in the area [9]The surveycoverage was about 918 of the study population and it wasfound by EC Uttah found that thirty-seven percent of thoseexamined was positive for Onchocerca volvulusmicrofilariae(392 ofmales and 349 of females)Themicrofilariae (mf)prevalence increased steadily with increasing age to reach704 in the oldest age group
Going towards the northern part of Nigeria the endemic-ity of human onchocerciasis was assessed by Anosike andOnwuliri [10] in eight rural communities at risk in NingiLocal Government Area Bauchi State Nigeria between July1990 and March 1991 Of the 1536 subjects skin-snipped 334(217) were positive for Onchocerca volvulusmicrofilariae
Wogu and Okaka investigated the overall prevalence ofonchocerciasis in Okpuje Owan West LGA of Edo StateNigeriaThe figure recorded in their study of 475 comparesfavourably with the 486 prevalence reported by Edungbolaand Asaolu [11] in Kwara State Nigeria but higher than the269 recorded by Nwaorgu et al [12] in an onchocerciasismesoendemic area of Enugu State Nigeria However Akinboand Okaka [13] reported a high overall prevalence (83) ofthe disease in Ovia North East LGA Edo State Nigeria[14] These reports suggest that we need a well-thoughtmathematical formulation of a simple and appropriate modelfor the spread and control of this disease
2 Model Formulation
In this section we define the parameters used in the modelbelow as follows 120573
1is the contact rate between 119867
119878and 119867
119864
1205732is the contact rate between 119867
119878and 119867
119868 1205733is the contact
rate between 119867119864and 119867
119868 1205734is the contact rate between 119867
119868
120588120577
1205731HS
1205732HS
1205733HE
1205734HI1205735VS
120575120582
120574VI
HS HE
VI
HI
VS
Figure 2 Model transfer diagram for onchocerciasis
and 119881119878 1205735is the contact rate between 119881
119878and 119881
119868 120574 is the
transmission rate from 119881119868to119867119878 120588 and 120575 are the proportions
of IvermectinMectizan treatment that cured the patient 120577is the rate of movement of humans from the infective 119867
119868
compartment into the human susceptible 119867119878compartment
120582 is the rate ofmovement of human from the human infective119867119868compartment into the latent (incubation) period 119867
119864
compartment
3 Model Diagram
Figure 2 shows the model diagram which consists of fivesets of ordinary differential equations highlighting the rate ofchange with respect to time of human susceptible compart-ment 119867
119878 human latent (incubation) period compartment
119867119864 human infective compartment 119867
119868 vector susceptible
compartment 119881119878 and the vector infective compartment 119881
119868
Abstract and Applied Analysis 3
4 Mathematical Formulation of the Model
From the model transfer diagram (Figure 2) we have thefollowing set of equations
119889119867119878
119889119905= 120588120577 + 120574119881
119868minus (1205731+ 1205732)119867119878 (1)
119889119867119864
119889119905= 120575120582 + 120573
1119867119878minus 1205733119867119864 (2)
119889119867119868
119889119905= 1205732119867119878+ 1205733119867119864minus 120588120577 minus 120575120582 minus 120573
4119867119868 (3)
119889119881119878
119889119905= 1205734119867119868minus 1205735119881119878 (4)
119889119881119868
119889119905= 1205735119881119878minus 120574119881119868 (5)
To find the steady state we assume that the compartments119881119878
119881119868119867119864119867119878 and119867
119868do not depend on time Therefore
119889119867119878
119889119905=119889119867119864
119889119905=119889119867119868
119889119905=119889119881119878
119889119905=119889119881119868
119889119905= 0 (6)
We now solve the differential equations 119889119867119878119889119905 = 119889119867
119864119889119905 =
119889119867119868119889119905 = 119889119881
119878119889119905 = 119889119881
119868119889119905 = 0 to find the value for119867
119878119867119864
119867119868 119881119878 and 119881
119868
From (4)1205734119867119868minus 1205735119881119878= 0
119867119868=1205735
1205734
119881119878
119867119868= 119860119881119878
(7)
From (5)1205735119881119878minus 120574119881119868= 0
119881119868=1205735119881119878
120574= 119861119881119878
(8)
From (1)120588120577 + 120574119881
119868minus (1205731+ 1205732)119867119878= 0
(1205731+ 1205732)119867119878= 120588120577 + 120574119881
119868
(9)
But
119881119868=1205735119881119878
120574
(1205731+ 1205732)119867119878= 120588120577 +
1205741205735119881119878
120574
119867119878=120588120577 + 120573
5119881119878
1205731+ 1205732
= 119862119881119878
(10)
From (2)120575120582 + 120573
1119867119878minus 1205733119867119864= 0
1205733119867119864= 120575120582 + 120573
1119867119878
but 119867119878=120588120577 + 120573
5119881119878
1205731+ 1205732
1205733119867119864= 120575120582 + 120573
1(120588120577 + 120573
5119881119878
1205731+ 1205732
)
119867119864=120575120582 (1205731+ 1205732) + 1205731120588120577 + 120573
11205735119881119878
1205733
= 119863119881119878
(11)
Lastly from (3)
1205732119867119878+ 1205733119867119864minus 120588120577 minus 120575120582 minus 120573
4119867119868= 0
1205732119867119878+ 1205733119867119864minus 120588120577 minus 120575120582 minus 120573
4119867119868= 0
1205732119862119881119878+ 1205733119863119881119878minus 120588120577 minus 120575120582 minus 120573
4119860119881119878= 0
(12)
We have that
119881119878=
120588120577 + 120575120582
1205732119862 + 120573
3119863 minus 120573
4119860 (13)
provided that 1205732119862 + 120573
3119863 gt 120573
4119860
Therefore the disease free equilibrium (DFE) is given as
1198640= (
120588120577
1205731+ 1205732
0 0 0 0) (14)
and the endemic equilibrium is given as
119867119878=120588120577 + 120573
5
1205731+ 1205732
((120588120577 + 120575120582)
times (1205732(120588120577 + 120573
5
1205731+ 1205732
) + 1205733
times (120575120582 (1205731+ 1205732) + 1205731120588120577 + 120573
11205735
1205733
)
minus1205734(1205735
1205734
))
minus1
)
= ((120588120577 + 1205735) (120588120577 + 120575120582))
times (1205732(120588120577 + 120573
5) + (120573
1+ 1205732)
times [120575120582 (1205731+ 1205732) + 1205731120588120577 + 120573
11205735]
minus (1205731+ 1205732) 1205735)minus1
(15)
119867119864=120575120582 (1205731+ 1205732) + 1205731120588120577 + 120573
11205735
1205733
times ((120588120577 + 120575120582) (1205732(120588120577 + 120573
5
1205731+ 1205732
) + 120575120582 (1205731+ 1205732)
+ 1205731120588120577 + 120573
11205735minus 1205735)minus1
)
4 Abstract and Applied Analysis
= ([120575120582 (1205731+ 1205732) + 1205731120588120577 + 120573
11205735] (120588120577 + 120575120582))
times ((12057321205733) (120588120577 + 120573
5)
1205731+ 1205732
+ 120575120582 (1205731+ 1205732) 1205733
+ 12057311205733120588120577 + 120573
112057331205735minus 12057331205735)
minus1
(16)
119867119868=1205735
1205734
((120588120577 + 120575120582)
times (1205732(120588120577 + 120573
5
1205731+ 1205732
)
+ 1205733(120575120582 (1205731+ 1205732) + 1205731120588120577 + 120573
11205735
1205733
)
minus1205734(1205735
1205734
))
minus1
)
= (1205735(120588120577 + 120575120582)) (120573
21205734(120588120577 + 120573
5
1205731+ 1205732
) + 1205734120575120582 (1205731+ 1205732)
+ 12057311205734120588120577 + 120573
112057341205735minus 12057341205735)minus1
(17)
119881119878= ((120588120577 + 120575120582)
times (1205732(120588120577 + 120573
5
1205731+ 1205732
)
+ 1205733(120575120582 (1205731+ 1205732) + 1205731120588120577 + 120573
11205735
1205733
)
minus1205734(1205735
1205734
))
minus1
)
(18)
119881119868=1205735
120574((120588120577 + 120575120582)
times (1205732(120588120577 + 120573
5
1205731+ 1205732
)
+ 1205733(120575120582 (1205731+ 1205732) + 1205731120588120577 + 120573
11205735
1205733
)
minus1205734(1205735
1205734
))
minus1
)
= (1205735(120588120577 + 120575120582))
times (1205741205732(120588120577 + 120573
5
1205731+ 1205732
) + 120574120575120582 (1205731+ 1205732)
+ 1205741205731120588120577 + 120574120573
11205735minus 1205741205735)minus1
(19)
5 Stability Analysis
To assess the stability of the (DFE) we solve the model char-acteristics equation using the Jacobean matrix formulatedfrom (1) (2) (3) (4) and (5) above as follows
119869 = (
minus(1205731+ 1205732) 0 0 0 120574
1205731
minus1205733
0 0 01205732
1205733minus1205734
0 00 0 120573
4minus12057350
0 0 0 1205735minus120574
) (20)
Applying the eigenvalue
120582 det (119869 minus 120582119868) = 0 (21)
we have
det[[[[[[
[
minus (1205731+ 1205732) minus 120582 0 0 0 120574
1205731
minus1205733minus 120582 0 0 0
1205732
1205733
minus1205734minus 120582 0 0
0 0 1205734
minus1205735minus 120582 0
0 0 0 1205735
minus120574 minus 120582
]]]]]]
]
= 0
det[[[[[[
[
minus (1205731+ 1205732+ 120582) 0 0 0 120574
1205731
minus (1205733+ 120582) 0 0 0
1205732
1205733
minus (1205734+ 120582) 0 0
0 0 1205734
minus (1205735+ 120582) 0
0 0 0 1205735
minus (120574 + 120582)
]]]]]]
]
= 0
minus (1205731+ 1205732+ 120582)
[[[[
[
minus (1205733+ 120582) 0 0 0
1205733
minus (1205734+ 120582) 0 0
0 1205734
minus (1205735+ 120582) 0
0 0 1205735
minus (120574 + 120582)
]]]]
]
+ 120574[[[[
[
1205731minus (1205733+ 120582) 0 0
1205732
1205733
minus (1205734+ 120582) 0
0 0 1205734
minus (1205735+ 120582)
0 0 0 1205735
]]]]
]
= 0
Abstract and Applied Analysis 5
minus (1205731+ 1205732+ 120582)
[[
[
minus (1205733+ 120582)
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
minus (1205734+ 120582) 0 0
1205734
minus (1205735+ 120582) 0
0 1205735
minus (120574 + 120582)
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
]]
]
+ 120574[[
[
1205731
[[
[
1205733minus (1205734+ 120582) 0
0 1205734
minus (1205735+ 120582)
0 0 1205735
]]
]
+ (1205733+ 120582)
[[
[
1205732minus (1205734+ 120582) 0
0 1205734
minus (1205735+ 120582)
0 0 1205735
]]
]
]]
]
= 0
minus (1205731+ 1205732+ 120582) [minus (120573
3+ 120582) [minus (120573
4+ 120582)
100381610038161003816100381610038161003816100381610038161003816
minus (1205735+ 120582) 0
1205735
minus (120574 + 120582)
100381610038161003816100381610038161003816100381610038161003816]] + 120574120573
1120573312057341205735+ 120574 (120573
3+ 120582) 120573
212057341205735= 0
minus [(1205731+ 1205732+ 120582) (120573
3+ 120582) (120573
4+ 120582) (120573
5+ 120582) (120574 + 120582)] + 120574120573
1120573312057341205735+ 1205741205732120573312057341205735+ 120574120582120573
212057341205735= 0
(22)
Hence we obtain from minus(1205731+ 1205732+ 120582) = 0
1205821= minus (120573
1+ 1205732) (23)
From minus(1205733+ 120582) = 0
1205822= minus1205733 (24)
From minus(1205734+ 120582) = 0
1205823= minus1205734 (25)
From (1205735+ 120582) = 0
1205824= minus1205735 (26)
From (120574 + 120582) = 0
1205825= minus120574 (27)
Given that the results of 1205821 1205822 1205823 1205824 and 120582
5are all negative
and considering the principles of linearized stability whichsays that the systems are asymptotically stable if and only ifall the eigenvalues of the system have negative real parts wesay that the disease-free equilibrium (DFE) is asymptoticallystable
6 Application of the HomotopyDecomposition Method (HDM)
In this section we apply the HDM [15ndash17] to solve thesemathematical sets of equations representing the onchocerci-asis disease model as follows
119867119878(119905) minus 119867
119878(0) = int
119905
0
(120588120577 + 120574119881119868(120591) minus (120573
1+ 1205732)119867119878(120591)) 119889120591
119867119864(119905) minus 119867
119864(0) = int
119905
0
(120575120582 + 1205731119867119878(120591) minus 120573
3119867119864(120591)) 119889120591
119867119868(119905) minus 119867
119868(0)
= int119905
0
(1205732119867119878(120591) + 120573
3119867119864(120591) minus 120588120577 minus 120575120582 minus 120573
4119867119868(120591)) 119889120591
119881119878(119905) minus 119881
119878(0) = int
119905
0
(1205734119867119868(120591) minus 120573
5119881119878(120591)) 119889120591
119881119868(119905) minus 119881
119868(0) = int
119905
0
(1205735119881119878(120591) minus 120574119881
119868(120591)) 119889120591
(28)
We then assume that the solutions of the above integralequations can be put in the following form for 119875 isin (0 1)
119867119878(119905) =
infin
sum119899=0
119875119899119867119878119899(119905) 119867
119864(119905) =
infin
sum119899=0
119875119899119867119864119899(119905)
119867119868(119905) =
infin
sum119899=0
119875119899119867119868119899(119905) 119881
119878(119905) =
infin
sum119899=0
119875119899119881119878119899(119905)
119881119868(119905) =
infin
sum119899=0
119875119899119881119868119899(119905)
(29)
Replacing the above expressions in the integral equations weobtain the following expressions
infin
sum119899=0
119875119899119867119878119899(119905) minus 119867
119878(0)
= 119875int119905
0
[(120588120577 + 120574infin
sum119899=0
119875119899119881119868119899(120591)
6 Abstract and Applied Analysis
minus (1205731+ 1205732)infin
sum119899=0
119875119899119867119878119899(120591)) 119889120591]
infin
sum119899=0
119875119899119867119864119899(119905) minus 119867
119864(0)
= 119875int119905
0
[(120575120582 + 1205731
infin
sum119899=0
119875119899119867119878119899(120591)
minus 1205733
infin
sum119899=0
119875119899119867119864119899(120591)) 119889120591]
infin
sum119899=0
119875119899119867119868119899(119905) minus 119867
119868(0)
= 119875int119905
0
[(1205732
infin
sum119899=0
119875119899119867119878119899(120591) + 120573
3
infin
sum119899=0
119875119899119867119864119899(120591)
minus 120588120577 minus 120575120582 minus 1205734
infin
sum119899=0
119875119899119867119868119899(120591)) 119889120591]
infin
sum119899=0
119875119899119881119878119899(119905) minus 119881
119878(0)
= 119875int119905
0
[(1205734
infin
sum119899=0
119875119899119867119868119899(120591) minus 120573
5
infin
sum119899=0
119875119899119881119878119899(120591)) 119889120591]
infin
sum119899=0
119875119899119881119868119899(119905) minus 119881
119868(0)
= 119875int119905
0
[(1205735
infin
sum119899=0
119875119899119881119878119899(120591) minus 120574
infin
sum119899=0
119875119899119881119868119899(120591)) 119889120591]
(30)
We compared the terms of the same power of 119875 and obtainedthe following
1198750 1198671198780(119905) = 119867
119878(0)
1198750 1198671198640(119905) = 119867
119864(0)
1198750 1198671198680(119905) = 119867
119868(0)
1198750 1198811198780(119905) = 119881
119878(0)
1198750 1198811198680(119905) = 119881
119868(0)
1198751 1198671198781(119905) = int
119905
0
(120588120577 + 1205741198811198680(120591) minus (120573
1+ 1205732)1198671198780(120591)) 119889120591
1198751 1198671198641(119905) = int
119905
0
(120575120582 + 12057311198671198780(120591) minus 120573
31198671198640(120591)) 119889120591
1198751 1198671198681(119905)
= int119905
0
(12057321198671198780(120591)+120573
31198671198640(120591)minus120588120577minus 120575120582 minus120573
41198671198680(120591)) 119889120591
1198751 1198811198781(119905) = int
119905
0
(12057341198671198680(120591) minus 120573
51198811198780(120591)) 119889120591
1198751 1198811198681(119905) = int
119905
0
(12057351198811198780(120591) minus 120574119881
1198680(120591)) 119889120591
(31)
Continuing for 119899 ge 2 we have
119867119878119899(119905) = int
119905
0
(120574119881119868(119899minus1)
(120591) minus (1205731+ 1205732)119867119878(119899minus1)
(120591)) 119889120591
119867119864119899(119905) = int
119905
0
(1205731119867119878(119899minus1)
(120591) minus 1205733119867119864(119899minus1)
(120591)) 119889120591
119867119868119899(119905) = int
119905
0
(1205732119867119878(119899minus1)
(120591) + 1205733119867119864(119899minus1)
(120591)
minus 1205734119867119868(119899minus1)
(120591)) 119889120591
119881119878119899(119905) = int
119905
0
(1205734119867119868(119899minus1)
(120591) minus 1205735119881119878(119899minus1)
(120591)) 119889120591
119881119868119899(119905) = int
119905
0
(1205735119881119878(119899minus1)
(120591) minus 120574119881119868(119899minus1)
(120591)) 119889120591
(32)
We now integrate the above equations to obtain the following
1198671198680(119905) = 0 119867
1198640(119905) = 0 119867
1198780 [119905] = 119873 1198811198780 [119905] = 119872
1198811198680 [119905] = 0 119867
1198681(119905) = 119905 (minus120575120582 minus 120577120588 + 119873120573
2)
1198671198781 [119905] = 119905 (120577120588 minus 119873 (1205731 + 1205732))
1198671198641(119905) = 119905 (120575120582 + 119873120573
1)
1198811198681 [119905] = 1198721199051205735 119881
1198781 [119905] = minus1198721199051205735
1198671198682(119905) =
1
21199052120577120588120573
2minus1
2119873119905212057311205732minus1
2119873119905212057322+1
21199052120575120582120573
3
+1
2119873119905212057311205733+1
21199052120575120582120573
4+1
21199052120577120588120573
4minus1
2119873119905212057321205734
1198671198642(119905) =
1
21199052120577120588120573
1minus1
2119873119905212057321minus1
2119873119905212057311205732
minus1
21199052120575120582120573
3minus1
2119873119905212057311205733
1198671198782 [119905] = minus
1
21199052120577120588120573
1+1
2119873119905212057321minus1
21199052120577120588120573
2
+ 119873119905212057311205732+1
2119873119905212057322+1
21198721199052120574120573
5
1198811198782 [119905] = minus
1
21199052120575120582120573
4minus1
21199052120577120588120573
4
+1
2119873119905212057321205734+1
211987211990521205732
5
1198811198682 [119905] = minus
1
21198721199052120574120573
5minus1
211987211990521205732
5
1198671198683(119905) = minus
1
61199053120577120588120573
11205732+1
61198731199053120573211205732minus1
611990531205771205881205732
2
+1
31198731199053120573112057322+1
6119873119905312057332+1
61199053120577120588120573
11205733
minus1
61198731199053120573211205733minus1
61198731199053120573112057321205733
minus1
611990531205751205821205732
3
minus1
61198731199053120573112057323minus1
61199053120577120588120573
21205734
+1
61198731199053120573112057321205734+1
61198731199053120573221205734
minus1
61199053120575120582120573
31205734minus1
61198731199053120573112057331205734
Abstract and Applied Analysis 7
minus1
611990531205751205821205732
4minus1
611990531205771205881205732
4+1
61198731199053120573212057324
+1
61198721199053120574120573
21205735
1198671198643(119905) = minus
1
611990531205771205881205732
1+1
6119873119905312057331minus1
61199053120577120588120573
11205732
+1
31198731199053120573211205732+1
61198731199053120573112057322
minus1
61199053120577120588120573
11205733+1
61198731199053120573211205733
+1
61198731199053120573112057321205733+1
611990531205751205821205732
3
+1
61198731199053120573112057323+1
61198721199053120574120573
11205735
1198671198783 [119905] =
1
611990531205771205881205732
1minus1
6119873119905312057331+1
31199053120577120588120573
11205732
minus1
21198731199053120573211205732+1
611990531205771205881205732
2minus1
21198731199053120573112057322
minus1
6119873119905312057332minus1
611987211990531205742120573
5
minus1
61198721199053120574120573
11205735
minus1
61198721199053120574120573
21205735minus1
611987211990531205741205732
5
1198811198783 [119905] =
1
61199053120577120588120573
21205734minus1
61198731199053120573112057321205734minus1
61198731199053120573221205734
+1
61199053120575120582120573
31205734+1
61198731199053120573112057331205734+1
611990531205751205821205732
4
+1
611990531205771205881205732
4minus1
61198731199053120573212057324+1
61199053120575120582120573
41205735
+1
61199053120577120588120573
41205735minus1
61198731199053120573212057341205735minus1
611987211990531205733
5
1198811198683 [119905] =
1
611987211990531205742120573
5minus1
61199053120575120582120573
41205735minus1
61199053120577120588120573
41205735
+1
61198731199053120573212057341205735+1
611987211990531205741205732
5+1
611987211990531205733
5
1198671198784 [119905] = minus
1
2411990541205771205881205733
1+1
24119873119905412057341minus1
811990541205771205881205732
11205732
+1
61198731199054120573311205732minus1
81199054120577120588120573
112057322+1
411987311990541205732112057322
minus1
2411990541205771205881205733
2+1
61198731199054120573112057332+1
24119873119905412057342
+1
2411987211990541205743120573
5+1
2411987211990541205742120573
11205735
+1
2411987211990541205741205732
11205735+1
2411987211990541205742120573
21205735
+1
121198721199054120574120573
112057321205735+1
2411987211990541205741205732
21205735
minus1
241199054120574120575120582120573
41205735minus1
241199054120574120577120588120573
41205735
+1
241198731199054120574120573
212057341205735+1
24119872119905412057421205732
5
+1
241198721199054120574120573
112057325+1
241198721199054120574120573
212057325+1
2411987211990541205741205733
5
1198671198644(119905) =
1
2411990541205771205881205733
1minus1
24119873119905412057341+1
1211990541205771205881205732
11205732
minus1
81198731199054120573311205732+1
241199054120577120588120573
112057322minus1
811987311990541205732112057322
minus1
241198731199054120573112057332+1
2411990541205771205881205732
11205733minus1
241198731199054120573311205733
+1
241199054120577120588120573
112057321205733minus1
1211987311990541205732112057321205733minus1
2411987311990541205731120573221205733
+1
241199054120577120588120573
112057323minus1
2411987311990541205732112057323minus1
2411987311990541205731120573212057323
minus1
2411990541205751205821205733
3minus1
241198731199054120573112057333minus1
2411987211990541205742120573
11205735
minus1
2411987211990541205741205732
11205735minus1
241198721199054120574120573
112057321205735
minus1
241198721199054120574120573
112057331205735minus1
241198721199054120574120573
112057325
1198671198684(119905) =
1
2411990541205771205881205732
11205732minus1
241198731199054120573311205732+1
121199054120577120588120573
112057322
minus1
811987311990541205732112057322+1
2411990541205771205881205733
2minus1
81198731199054120573112057332
minus1
24119873119905412057342minus1
2411990541205771205881205732
11205733+1
241198731199054120573311205733
minus1
241199054120577120588120573
112057321205733+1
1211987311990541205732112057321205733
+1
2411987311990541205731120573221205733
minus1
241199054120577120588120573
112057323+1
2411987311990541205732112057323+1
2411987311990541205731120573212057323
+1
2411990541205751205821205733
3+1
241198731199054120573112057333+1
241199054120577120588120573
112057321205734
minus1
2411987311990541205732112057321205734+1
2411990541205771205881205732
21205734minus1
1211987311990541205731120573221205734
minus1
241198731199054120573321205734minus1
241199054120577120588120573
112057331205734
+1
2411987311990541205732112057331205734+1
2411987311990541205731120573212057331205734
+1
2411990541205751205821205732
31205734+1
2411987311990541205731120573231205734
+1
241199054120577120588120573
212057324minus1
2411987311990541205731120573212057324minus1
2411987311990541205732212057324
+1
241199054120575120582120573
312057324+1
2411987311990541205731120573312057324+1
2411990541205751205821205733
4
+1
2411990541205771205881205733
4minus1
241198731199054120573212057334minus1
2411987211990541205742120573
21205735
minus1
241198721199054120574120573
112057321205735minus1
2411987211990541205741205732
21205735
+1
241198721199054120574120573
112057331205735
minus1
241198721199054120574120573
212057341205735minus1
241198721199054120574120573
212057325
1198811198784 [119905] = minus
1
241199054120577120588120573
112057321205734+1
2411987311990541205732112057321205734
minus1
2411990541205771205881205732
21205734+1
1211987311990541205731120573221205734
8 Abstract and Applied Analysis
+1
241198731199054120573321205734+1
241199054120577120588120573
112057331205734
minus1
2411987311990541205732112057331205734minus1
2411987311990541205731120573212057331205734
minus1
2411990541205751205821205732
31205734
minus1
2411987311990541205731120573231205734minus1
241199054120577120588120573
212057324
+1
2411987311990541205731120573212057324+1
2411987311990541205732212057324
minus1
241199054120575120582120573
312057324minus1
2411987311990541205731120573312057324
minus1
2411990541205751205821205733
4minus1
2411990541205771205881205733
4
+1
241198731199054120573212057334
+1
241198721199054120574120573
212057341205735minus1
241199054120577120588120573
212057341205735
+1
2411987311990541205731120573212057341205735+1
2411987311990541205732212057341205735
minus1
241199054120575120582120573
312057341205735minus1
2411987311990541205731120573312057341205735
minus1
2411990541205751205821205732
41205735minus1
2411990541205771205881205732
41205735+1
2411987311990541205732120573241205735
minus1
241199054120575120582120573
412057325minus1
241199054120577120588120573
412057325
+1
2411987311990541205732120573412057325+1
2411987211990541205734
5
1198811198684 [119905] = minus
1
2411987211990541205743120573
5+1
241199054120574120575120582120573
41205735+1
241199054120574120577120588120573
41205735
minus1
241198731199054120574120573
212057341205735+1
241199054120577120588120573
212057341205735
minus1
2411987311990541205731120573212057341205735minus1
2411987311990541205732212057341205735
+1
241199054120575120582120573
312057341205735+1
2411987311990541205731120573312057341205735
+1
2411990541205751205821205732
41205735+1
2411990541205771205881205732
41205735
minus1
2411987311990541205732120573241205735minus1
24119872119905412057421205732
5
+1
241199054120575120582120573
412057325+1
241199054120577120588120573
412057325
minus1
2411987311990541205732120573412057325minus1
2411987211990541205741205733
5minus1
2411987211990541205734
5
1198671198785 [119905] =
1
12011990551205771205881205734
1minus
1
120119873119905512057351+1
3011990551205771205881205733
11205732
minus1
241198731199055120573411205732+1
2011990551205771205881205732
112057322minus1
1211987311990551205733112057322
+1
301199055120577120588120573
112057332minus1
1211987311990551205732112057332+
1
12011990551205771205881205734
2
minus1
241198731199055120573112057342minus
1
120119873119905512057352minus
1
12011987211990551205744120573
5
minus1
12011987211990551205743120573
11205735minus
1
120119872119905512057421205732
11205735
minus1
12011987211990551205741205733
11205735minus
1
12011987211990551205743120573
21205735
minus1
6011987211990551205742120573
112057321205735minus1
4011987211990551205741205732
112057321205735
minus1
120119872119905512057421205732
21205735minus1
401198721199055120574120573
1120573221205735
minus1
12011987211990551205741205733
21205735+
1
12011990551205742120575120582120573
41205735
+1
12011990551205742120577120588120573
41205735+
1
1201199055120574120575120582120573
112057341205735
+1
1201199055120574120577120588120573
112057341205735minus
1
12011987311990551205742120573
212057341205735
+1
1201199055120574120575120582120573
212057341205735+1
601199055120574120577120588120573
212057341205735
minus1
601198731199055120574120573
1120573212057341205735minus1
6011987311990551205741205732
212057341205735
+1
1201199055120574120575120582120573
312057341205735+
1
1201198731199055120574120573
1120573312057341205735
+1
12011990551205741205751205821205732
41205735+
1
12011990551205741205771205881205732
41205735
minus1
1201198731199055120574120573
2120573241205735minus
1
120119872119905512057431205732
5
minus1
12011987211990551205742120573
112057325minus
1
12011987211990551205741205732
112057325
minus1
12011987211990551205742120573
212057325minus1
601198721199055120574120573
1120573212057325
minus1
12011987211990551205741205732
212057325+
1
1201199055120574120575120582120573
412057325
+1
1201199055120574120577120588120573
412057325minus
1
1201198731199055120574120573
2120573412057325
minus1
120119872119905512057421205733
5minus
1
1201198721199055120574120573
112057335
minus1
1201198721199055120574120573
212057335minus
1
12011987211990551205741205734
5
1198671198645 [119905] = minus
1
12011990551205771205881205734
1+
1
120119873119905512057351
minus1
4011990551205771205881205733
11205732+1
301198731199055120573411205732
minus1
4011990551205771205881205732
112057322+1
2011987311990551205733112057322
minus1
1201199055120577120588120573
112057332+1
3011987311990551205732112057332
+1
1201198731199055120573112057342
minus1
12011990551205771205881205733
11205733+
1
1201198731199055120573411205733
minus1
6011990551205771205881205732
112057321205733+1
4011987311990551205733112057321205733
minus1
1201199055120577120588120573
1120573221205733+1
40119873119905512057321120573221205733
+1
12011987311990551205731120573321205733minus
1
12011990551205771205881205732
112057323
+1
12011987311990551205733112057323minus
1
1201199055120577120588120573
1120573212057323
Abstract and Applied Analysis 9
+1
60119873119905512057321120573212057323+
1
120119873119905512057311205732212057323
minus1
1201199055120577120588120573
112057333+
1
12011987311990551205732112057333
+1
12011987311990551205731120573212057333+
1
12011990551205751205821205734
3
+1
1201198731199055120573112057343+
1
12011987211990551205743120573
11205735
+1
120119872119905512057421205732
11205735+
1
12011987211990551205741205733
11205735
+1
12011987211990551205742120573
112057321205735+1
6011987211990551205741205732
112057321205735
+1
1201198721199055120574120573
1120573221205735+
1
12011987211990551205742120573
112057331205735
+1
12011987211990551205741205732
112057331205735+
1
1201198721199055120574120573
1120573212057331205735
+1
1201198721199055120574120573
1120573231205735minus
1
1201199055120574120575120582120573
112057341205735
minus1
1201199055120574120577120588120573
112057341205735+
1
1201198731199055120574120573
1120573212057341205735
+1
12011987211990551205742120573
112057325+
1
12011987211990551205741205732
112057325
+1
1201198721199055120574120573
1120573212057325+
1
1201198721199055120574120573
1120573312057325
+1
1201198721199055120574120573
112057335
1198671198685 [119905]= minus
1
12011990551205771205881205733
11205732+
1
1201198731199055120573411205732minus1
4011990551205771205881205732
112057322
+1
3011987311990551205733112057322minus1
401199055120577120588120573
112057332+1
2011987311990551205732112057332
minus1
12011990551205771205881205734
2+1
301198731199055120573112057342+
1
120119873119905512057352
+1
12011990551205771205881205733
11205733minus
1
1201198731199055120573411205733
+1
6011990551205771205881205732
112057321205733minus1
4011987311990551205733112057321205733
+1
1201199055120577120588120573
1120573221205733minus1
40119873119905512057321120573221205733
minus1
12011987311990551205731120573321205733+
1
12011990551205771205881205732
112057323
minus1
12011987311990551205733112057323+
1
1201199055120577120588120573
1120573212057323
minus1
60119873119905512057321120573212057323minus
1
120119873119905512057311205732212057323
+1
1201199055120577120588120573
112057333minus
1
12011987311990551205732112057333
minus1
12011987311990551205731120573212057333minus
1
12011990551205751205821205734
3
minus1
1201198731199055120573112057343minus
1
12011990551205771205881205732
112057321205734
+1
12011987311990551205733112057321205734minus1
601199055120577120588120573
1120573221205734
+1
40119873119905512057321120573221205734minus
1
12011990551205771205881205733
21205734
+1
4011987311990551205731120573321205734+
1
1201198731199055120573421205734
+1
12011990551205771205881205732
112057331205734minus
1
12011987311990551205733112057331205734
+1
1201199055120577120588120573
1120573212057331205734minus1
60119873119905512057321120573212057331205734
minus1
120119873119905512057311205732212057331205734+
1
1201199055120577120588120573
1120573231205734
minus1
120119873119905512057321120573231205734minus
1
120119873119905512057311205732120573231205734
minus1
12011990551205751205821205733
31205734minus
1
12011987311990551205731120573331205734
minus1
1201199055120577120588120573
1120573212057324+
1
120119873119905512057321120573212057324
minus1
12011990551205771205881205732
212057324+1
60119873119905512057311205732212057324
+1
12011987311990551205733212057324+
1
1201199055120577120588120573
1120573312057324
minus1
120119873119905512057321120573312057324minus
1
120119873119905512057311205732120573312057324
minus1
12011990551205751205821205732
312057324minus
1
120119873119905512057311205732312057324
minus1
1201199055120577120588120573
212057334+
1
12011987311990551205731120573212057334
+1
12011987311990551205732212057334minus
1
1201199055120575120582120573
312057334
minus1
12011987311990551205731120573312057334minus
1
12011990551205751205821205734
4
minus1
12011990551205771205881205734
4+
1
1201198731199055120573212057344
+1
12011987211990551205743120573
21205735+
1
12011987211990551205742120573
112057321205735
+1
12011987211990551205741205732
112057321205735+
1
120119872119905512057421205732
21205735
+1
601198721199055120574120573
1120573221205735+
1
12011987211990551205741205733
21205735
minus1
12011987211990551205742120573
112057331205735minus
1
12011987211990551205741205732
112057331205735
minus1
1201198721199055120574120573
1120573212057331205735minus
1
1201198721199055120574120573
1120573231205735
+1
12011987211990551205742120573
212057341205735minus
1
1201199055120574120575120582120573
212057341205735
minus1
1201199055120574120577120588120573
212057341205735+
1
1201198721199055120574120573
1120573212057341205735
+1
12011987211990551205741205732
212057341205735+
1
12011987311990551205741205732
212057341205735
minus1
1201198721199055120574120573
1120573312057341205735+
1
1201198721199055120574120573
2120573241205735
+1
12011987211990551205742120573
212057325+
1
1201198721199055120574120573
1120573212057325
+1
12011987211990551205741205732
212057325minus
1
1201198721199055120574120573
1120573312057325
+1
1201198721199055120574120573
2120573412057325+
1
1201198721199055120574120573
212057335
10 Abstract and Applied Analysis
1198811198785 [119905] =
1
12011990551205771205881205732
112057321205734minus
1
12011987311990551205733112057321205734
+1
601199055120577120588120573
1120573221205734minus1
40119873119905512057321120573221205734
+1
12011990551205771205881205733
21205734minus1
4011987311990551205731120573321205734
minus1
1201198731199055120573421205734minus
1
12011990551205771205881205732
112057331205734
+1
12011987311990551205733112057331205734minus
1
1201199055120577120588120573
1120573212057331205734
+1
60119873119905512057321120573212057331205734+
1
120119873119905512057311205732212057331205734
minus1
1201199055120577120588120573
1120573231205734+
1
120119873119905512057321120573231205734
+1
120119873119905512057311205732120573231205734+
1
12011990551205751205821205733
31205734
+1
12011987311990551205731120573331205734+
1
1201199055120577120588120573
1120573212057324
minus1
120119873119905512057321120573212057324+
1
12011990551205771205881205732
212057324
minus1
60119873119905512057311205732212057324minus
1
12011987311990551205733212057324
minus1
1201199055120577120588120573
1120573312057324+
1
120119873119905512057321120573312057324
+1
120119873119905512057311205732120573312057324+
1
12011990551205751205821205732
312057324
+1
120119873119905512057311205732312057324+
1
1201199055120577120588120573
212057334
minus1
12011987311990551205731120573212057334minus
1
12011987311990551205732212057334
+1
1201199055120575120582120573
312057334+
1
12011987311990551205731120573312057334
+1
12011990551205751205821205734
4+
1
12011990551205771205881205734
4minus
1
1201198731199055120573212057344
minus1
12011987211990551205742120573
212057341205735minus
1
1201198721199055120574120573
1120573212057341205735
+1
1201199055120577120588120573
1120573212057341205735minus
1
120119873119905512057321120573212057341205735
minus1
12011987211990551205741205732
212057341205735+
1
12011990551205771205881205732
212057341205735
minus1
60119873119905512057311205732212057341205735minus
1
12011987311990551205733212057341205735
+1
1201198721199055120574120573
1120573312057341205735minus
1
1201199055120577120588120573
1120573312057341205735
+1
120119873119905512057321120573312057341205735+
1
120119873119905512057311205732120573312057341205735
+1
12011990551205751205821205732
312057341205735+
1
120119873119905512057311205732312057341205735
minus1
1201198721199055120574120573
2120573241205735+
1
1201199055120577120588120573
2120573241205735
minus1
120119873119905512057311205732120573241205735minus
1
120119873119905512057322120573241205735
+1
1201199055120575120582120573
3120573241205735+
1
120119873119905512057311205733120573241205735
+1
12011990551205751205821205733
41205735+
1
12011990551205771205881205733
41205735
minus1
12011987311990551205732120573341205735minus1
601198721199055120574120573
2120573412057325
+1
1201199055120577120588120573
2120573412057325minus
1
120119873119905512057311205732120573412057325
minus1
120119873119905512057322120573412057325+
1
1201199055120575120582120573
3120573412057325
+1
120119873119905512057311205733120573412057325+
1
12011990551205751205821205732
412057325
+1
12011990551205771205881205732
412057325minus
1
120119873119905512057321205732412057325
+1
1201199055120575120582120573
412057335+
1
1201199055120577120588120573
412057335
minus1
12011987311990551205732120573412057335minus
1
12011987211990551205735
5
1198811198685 [119905] =
1
12011987211990551205744120573
5minus
1
12011990551205742120575120582120573
41205735
minus1
12011990551205742120577120588120573
41205735+
1
12011987311990551205742120573
212057341205735
minus1
1201199055120574120577120588120573
212057341205735+
1
1201198731199055120574120573
1120573212057341205735
minus1
1201199055120577120588120573
1120573212057341205735+
1
120119873119905512057321120573212057341205735
+1
12011987311990551205741205732
212057341205735minus
1
12011990551205771205881205732
212057341205735
+1
60119873119905512057311205732212057341205735+
1
12011987311990551205733212057341205735
minus1
1201199055120574120575120582120573
312057341205735minus
1
1201198731199055120574120573
1120573312057341205735
+1
1201199055120577120588120573
1120573312057341205735minus
1
120119873119905512057321120573312057341205735
minus1
120119873119905512057311205732120573312057341205735minus
1
12011990551205751205821205732
312057341205735
minus1
120119873119905512057311205732312057341205735minus
1
12011990551205741205751205821205732
41205735
minus1
12011990551205741205771205881205732
41205735+
1
1201198731199055120574120573
2120573241205735
minus1
1201199055120577120588120573
2120573241205735+
1
120119873119905512057311205732120573241205735
+1
120119873119905512057322120573241205735minus
1
1201199055120575120582120573
3120573241205735
minus1
120119873119905512057311205733120573241205735minus
1
12011990551205751205821205733
41205735
minus1
12011990551205771205881205733
41205735+
1
12011987311990551205732120573341205735
+1
120119872119905512057431205732
5minus
1
1201199055120574120575120582120573
412057325
minus1
1201199055120574120577120588120573
412057325+
1
1201198721199055120574120573
2120573412057325
+1
1201198731199055120574120573
2120573412057325minus
1
1201199055120577120588120573
2120573412057325
+1
120119873119905512057311205732120573412057325+
1
120119873119905512057322120573412057325
Abstract and Applied Analysis 11
minus1
1201199055120575120582120573
3120573412057325minus
1
120119873119905512057311205733120573412057325
minus1
12011990551205751205821205732
412057325minus
1
12011990551205771205881205732
412057325
+1
120119873119905512057321205732412057325+
1
120119872119905512057421205733
5
minus1
1201199055120575120582120573
412057335minus
1
1201199055120577120588120573
412057335
+1
12011987311990551205732120573412057335+
1
12011987211990551205741205734
5
+1
12011987211990551205735
5
(33)
In general by the repeated application of the above proce-dure we have that
119867119878119899=119905119899
1198991198861119899 119867
119864119899=119905119899
1198991198862119899 119867
119868119899=119905119899
1198991198863119899
119881119878119899=119905119899
1198991198864119899 119881
119868119899=119905119899
1198991198865119899
(34)
where 1198861119899 1198862119899 1198863119899 1198864119899 1198865119899
depend on fixed set of empiricalparameters Hence the approximate solution of the modelabove is given as
119867119878(119905) =
infin
sum119899=0
119905119899
1198991198861119899
119867119864(119905) =
infin
sum119899=0
119905119899
1198991198862119899
119867119868(119905) =
infin
sum119899=0
119905119899
1198991198863119899
119881119878(119905) =
infin
sum119899=0
119905119899
1198991198864119899
119881119868(119905) =
infin
sum119899=0
119905119899
1198991198865119899
(35)
We therefore use the following theoretical parameters 1205731=
05 1205732= 05 120573
3= 05 120573
4= 05 120573
5= 05 120574 = 1 120577 = 1
119873 = 10 119872 = 10 120588 = 1 120575 = 1 and 120582 = 1 to examinethe behavior of the five different compartmental numericalsolutions in the system Figure 3 shows the behavior of thenumerical solution
7 Conclusion
Onchocerciasis model was examined and the mathematicalformulation underpinning the spread and control of thisdisease has been proposed on one hand On the otherhand we made use of some new analytical methods calledthe homotopy decomposition method (HDM) to derive thesolution of the resulting set of (1) to (5)The numerical resultswere presented to test the efficiency and the accuracy of themethod used The mathematical analysis of the resulting setof (1) to (5) above revealed that the stability analysis of the
05 10 15 20Time (per year)
2
4
6
8
10
Appr
oxim
ate s
olut
ions VS
HS
VI
HI
HE
Figure 3 Numerical solution of the model
five roots of the characteristics equations has all negativereal part In particular given that the stability analysis showsthat the DFE is asymptotically stable it would be possiblethat the parasite Onchocerca volvulus be eradicated in a finitetime On the other hand the numerical solution confirmedthat as the number of humans and vectors in the susceptiblecompartments reduces the number of infective humansand vectors in the infectious compartment increases Thenumerical solutions in both cases displayed the biologicalbehaviour of the real world situation Therefore in orderto build a healthy and a sustainable population treatmentwith IvermectinMectizan drugs and a more hygienic envi-ronment that does not allow the breeding and survival ofblackflies could lead to a total eradication of the disease
Conflict of Interests
The authors declare that they have no conflict of interestsregarding the publication of this paper
References
[1] J H F Remme The Global Burden of Onchocerciasis in 1990World Health Organization Geneva Switzerland 2004 httpwwwwhointhealthinfoglobal burden diseaseOnchocercia-sis201990pdf
[2] WorldHealthOrganization ldquoPrevention of blindness and visualimpairment priority eye diseasesrdquo 2014 httpwwwwhointblindnesscausespriorityenindex3html
[3] World Health Organization Certification of Elimination ofHuman Onchocerciasis Criteria and Procedures World HealthOrganizationGeneva Switzerland 2001 httpwhqlibdocwhointhq2001WHO CDS CPE CEE 200118bpdf
[4] K A Lindblade B Arana G Zea-Flores et al ldquoEliminationof Onchocerca volvulus transmission in the Santa Rosa focusof Guatemalardquo The American Journal of Tropical Medicine andHygiene vol 77 pp 334ndash341 2007
[5] J C Anosike and C O E Onwuliri ldquoStudies on filariasis inBauchi State Nigeria 1 Endemicity of human onchocerciasisin Ningi Local Government Areardquo Annals of Tropical Medicineand Parasitology vol 89 no 1 pp 31ndash38 1995
[6] WorldHealthOrganizationWeekly Epidemiological Records 71WHO Geneva Switzerland 1993
12 Abstract and Applied Analysis
[7] WorldHealthOrganization httpwwwwhointneglected dis-easesen
[8] Sightsaver httpwwwsightsaversorgabout uspublicationsonchocerciasis
[9] E C Uttah ldquoOnchocerciasis in the Upper Ima River BasinrdquoIranian Journal of Parasitology vol 5 no 2 pp 33ndash41 2010
[10] J C Anosike and C O E Onwuliri ldquoStudies on filariasis inBauchi State Nigeriamdash1 endemicity of human onchocerciasisin Ningi Local Government Areardquo Annals of Tropical Medicineand Parasitology vol 89 no 1 pp 31ndash38 1995
[11] L D Edungbola and S O Asaolu ldquoParasitologic survey ofonchocerciasis (river blindness) in Babana district Kwara StateNigeriardquo American Journal of Tropical Medicine and Hygienevol 33 no 6 pp 1147ndash1154 1984
[12] O C Nwaorgu A Ohaegbula I E Onweluzo et al ldquoResultsof a large scale onchocercosis survey in Enugu State NigeriardquoJournal of Helminthology vol 68 no 2 pp 155ndash159 1994
[13] F O Akinbo and C E Okaka ldquoPrevalence and socio-economiceffects of onchocerciasis in Ovia North East Local GovernmentArea of Edo State Nigeriardquo Rivista Di Parasitologie vol 22 no3 pp 215ndash221 2005
[14] M D Wogu and C E Okaka ldquoPrevalence and socio-economic effects of onchocerciasis in Okpuje Owan WestLocal Government Area Edo State Nigeriardquo InternationalJournal of Biomedical and Health Sciences vol 4 no 3 2008httpwwwasopahorg
[15] S C O Noutchie ldquoHomotopy Decomposition analysis of atuberculosis modelrdquo Life Science Journal vol 10 no 3 pp 459ndash465 2013
[16] A Atangana and A Secer ldquoThe time-fractional coupled-Korteweg-de-Vries equationsrdquo Abstract and Applied Analysisvol 2013 Article ID 947986 8 pages 2013
[17] A Atangana and N Bildik ldquoApproximate solution of tubercu-losis disease population dynamics modelrdquo Abstract and AppliedAnalysis vol 2013 Article ID 759801 8 pages 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 3
4 Mathematical Formulation of the Model
From the model transfer diagram (Figure 2) we have thefollowing set of equations
119889119867119878
119889119905= 120588120577 + 120574119881
119868minus (1205731+ 1205732)119867119878 (1)
119889119867119864
119889119905= 120575120582 + 120573
1119867119878minus 1205733119867119864 (2)
119889119867119868
119889119905= 1205732119867119878+ 1205733119867119864minus 120588120577 minus 120575120582 minus 120573
4119867119868 (3)
119889119881119878
119889119905= 1205734119867119868minus 1205735119881119878 (4)
119889119881119868
119889119905= 1205735119881119878minus 120574119881119868 (5)
To find the steady state we assume that the compartments119881119878
119881119868119867119864119867119878 and119867
119868do not depend on time Therefore
119889119867119878
119889119905=119889119867119864
119889119905=119889119867119868
119889119905=119889119881119878
119889119905=119889119881119868
119889119905= 0 (6)
We now solve the differential equations 119889119867119878119889119905 = 119889119867
119864119889119905 =
119889119867119868119889119905 = 119889119881
119878119889119905 = 119889119881
119868119889119905 = 0 to find the value for119867
119878119867119864
119867119868 119881119878 and 119881
119868
From (4)1205734119867119868minus 1205735119881119878= 0
119867119868=1205735
1205734
119881119878
119867119868= 119860119881119878
(7)
From (5)1205735119881119878minus 120574119881119868= 0
119881119868=1205735119881119878
120574= 119861119881119878
(8)
From (1)120588120577 + 120574119881
119868minus (1205731+ 1205732)119867119878= 0
(1205731+ 1205732)119867119878= 120588120577 + 120574119881
119868
(9)
But
119881119868=1205735119881119878
120574
(1205731+ 1205732)119867119878= 120588120577 +
1205741205735119881119878
120574
119867119878=120588120577 + 120573
5119881119878
1205731+ 1205732
= 119862119881119878
(10)
From (2)120575120582 + 120573
1119867119878minus 1205733119867119864= 0
1205733119867119864= 120575120582 + 120573
1119867119878
but 119867119878=120588120577 + 120573
5119881119878
1205731+ 1205732
1205733119867119864= 120575120582 + 120573
1(120588120577 + 120573
5119881119878
1205731+ 1205732
)
119867119864=120575120582 (1205731+ 1205732) + 1205731120588120577 + 120573
11205735119881119878
1205733
= 119863119881119878
(11)
Lastly from (3)
1205732119867119878+ 1205733119867119864minus 120588120577 minus 120575120582 minus 120573
4119867119868= 0
1205732119867119878+ 1205733119867119864minus 120588120577 minus 120575120582 minus 120573
4119867119868= 0
1205732119862119881119878+ 1205733119863119881119878minus 120588120577 minus 120575120582 minus 120573
4119860119881119878= 0
(12)
We have that
119881119878=
120588120577 + 120575120582
1205732119862 + 120573
3119863 minus 120573
4119860 (13)
provided that 1205732119862 + 120573
3119863 gt 120573
4119860
Therefore the disease free equilibrium (DFE) is given as
1198640= (
120588120577
1205731+ 1205732
0 0 0 0) (14)
and the endemic equilibrium is given as
119867119878=120588120577 + 120573
5
1205731+ 1205732
((120588120577 + 120575120582)
times (1205732(120588120577 + 120573
5
1205731+ 1205732
) + 1205733
times (120575120582 (1205731+ 1205732) + 1205731120588120577 + 120573
11205735
1205733
)
minus1205734(1205735
1205734
))
minus1
)
= ((120588120577 + 1205735) (120588120577 + 120575120582))
times (1205732(120588120577 + 120573
5) + (120573
1+ 1205732)
times [120575120582 (1205731+ 1205732) + 1205731120588120577 + 120573
11205735]
minus (1205731+ 1205732) 1205735)minus1
(15)
119867119864=120575120582 (1205731+ 1205732) + 1205731120588120577 + 120573
11205735
1205733
times ((120588120577 + 120575120582) (1205732(120588120577 + 120573
5
1205731+ 1205732
) + 120575120582 (1205731+ 1205732)
+ 1205731120588120577 + 120573
11205735minus 1205735)minus1
)
4 Abstract and Applied Analysis
= ([120575120582 (1205731+ 1205732) + 1205731120588120577 + 120573
11205735] (120588120577 + 120575120582))
times ((12057321205733) (120588120577 + 120573
5)
1205731+ 1205732
+ 120575120582 (1205731+ 1205732) 1205733
+ 12057311205733120588120577 + 120573
112057331205735minus 12057331205735)
minus1
(16)
119867119868=1205735
1205734
((120588120577 + 120575120582)
times (1205732(120588120577 + 120573
5
1205731+ 1205732
)
+ 1205733(120575120582 (1205731+ 1205732) + 1205731120588120577 + 120573
11205735
1205733
)
minus1205734(1205735
1205734
))
minus1
)
= (1205735(120588120577 + 120575120582)) (120573
21205734(120588120577 + 120573
5
1205731+ 1205732
) + 1205734120575120582 (1205731+ 1205732)
+ 12057311205734120588120577 + 120573
112057341205735minus 12057341205735)minus1
(17)
119881119878= ((120588120577 + 120575120582)
times (1205732(120588120577 + 120573
5
1205731+ 1205732
)
+ 1205733(120575120582 (1205731+ 1205732) + 1205731120588120577 + 120573
11205735
1205733
)
minus1205734(1205735
1205734
))
minus1
)
(18)
119881119868=1205735
120574((120588120577 + 120575120582)
times (1205732(120588120577 + 120573
5
1205731+ 1205732
)
+ 1205733(120575120582 (1205731+ 1205732) + 1205731120588120577 + 120573
11205735
1205733
)
minus1205734(1205735
1205734
))
minus1
)
= (1205735(120588120577 + 120575120582))
times (1205741205732(120588120577 + 120573
5
1205731+ 1205732
) + 120574120575120582 (1205731+ 1205732)
+ 1205741205731120588120577 + 120574120573
11205735minus 1205741205735)minus1
(19)
5 Stability Analysis
To assess the stability of the (DFE) we solve the model char-acteristics equation using the Jacobean matrix formulatedfrom (1) (2) (3) (4) and (5) above as follows
119869 = (
minus(1205731+ 1205732) 0 0 0 120574
1205731
minus1205733
0 0 01205732
1205733minus1205734
0 00 0 120573
4minus12057350
0 0 0 1205735minus120574
) (20)
Applying the eigenvalue
120582 det (119869 minus 120582119868) = 0 (21)
we have
det[[[[[[
[
minus (1205731+ 1205732) minus 120582 0 0 0 120574
1205731
minus1205733minus 120582 0 0 0
1205732
1205733
minus1205734minus 120582 0 0
0 0 1205734
minus1205735minus 120582 0
0 0 0 1205735
minus120574 minus 120582
]]]]]]
]
= 0
det[[[[[[
[
minus (1205731+ 1205732+ 120582) 0 0 0 120574
1205731
minus (1205733+ 120582) 0 0 0
1205732
1205733
minus (1205734+ 120582) 0 0
0 0 1205734
minus (1205735+ 120582) 0
0 0 0 1205735
minus (120574 + 120582)
]]]]]]
]
= 0
minus (1205731+ 1205732+ 120582)
[[[[
[
minus (1205733+ 120582) 0 0 0
1205733
minus (1205734+ 120582) 0 0
0 1205734
minus (1205735+ 120582) 0
0 0 1205735
minus (120574 + 120582)
]]]]
]
+ 120574[[[[
[
1205731minus (1205733+ 120582) 0 0
1205732
1205733
minus (1205734+ 120582) 0
0 0 1205734
minus (1205735+ 120582)
0 0 0 1205735
]]]]
]
= 0
Abstract and Applied Analysis 5
minus (1205731+ 1205732+ 120582)
[[
[
minus (1205733+ 120582)
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
minus (1205734+ 120582) 0 0
1205734
minus (1205735+ 120582) 0
0 1205735
minus (120574 + 120582)
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
]]
]
+ 120574[[
[
1205731
[[
[
1205733minus (1205734+ 120582) 0
0 1205734
minus (1205735+ 120582)
0 0 1205735
]]
]
+ (1205733+ 120582)
[[
[
1205732minus (1205734+ 120582) 0
0 1205734
minus (1205735+ 120582)
0 0 1205735
]]
]
]]
]
= 0
minus (1205731+ 1205732+ 120582) [minus (120573
3+ 120582) [minus (120573
4+ 120582)
100381610038161003816100381610038161003816100381610038161003816
minus (1205735+ 120582) 0
1205735
minus (120574 + 120582)
100381610038161003816100381610038161003816100381610038161003816]] + 120574120573
1120573312057341205735+ 120574 (120573
3+ 120582) 120573
212057341205735= 0
minus [(1205731+ 1205732+ 120582) (120573
3+ 120582) (120573
4+ 120582) (120573
5+ 120582) (120574 + 120582)] + 120574120573
1120573312057341205735+ 1205741205732120573312057341205735+ 120574120582120573
212057341205735= 0
(22)
Hence we obtain from minus(1205731+ 1205732+ 120582) = 0
1205821= minus (120573
1+ 1205732) (23)
From minus(1205733+ 120582) = 0
1205822= minus1205733 (24)
From minus(1205734+ 120582) = 0
1205823= minus1205734 (25)
From (1205735+ 120582) = 0
1205824= minus1205735 (26)
From (120574 + 120582) = 0
1205825= minus120574 (27)
Given that the results of 1205821 1205822 1205823 1205824 and 120582
5are all negative
and considering the principles of linearized stability whichsays that the systems are asymptotically stable if and only ifall the eigenvalues of the system have negative real parts wesay that the disease-free equilibrium (DFE) is asymptoticallystable
6 Application of the HomotopyDecomposition Method (HDM)
In this section we apply the HDM [15ndash17] to solve thesemathematical sets of equations representing the onchocerci-asis disease model as follows
119867119878(119905) minus 119867
119878(0) = int
119905
0
(120588120577 + 120574119881119868(120591) minus (120573
1+ 1205732)119867119878(120591)) 119889120591
119867119864(119905) minus 119867
119864(0) = int
119905
0
(120575120582 + 1205731119867119878(120591) minus 120573
3119867119864(120591)) 119889120591
119867119868(119905) minus 119867
119868(0)
= int119905
0
(1205732119867119878(120591) + 120573
3119867119864(120591) minus 120588120577 minus 120575120582 minus 120573
4119867119868(120591)) 119889120591
119881119878(119905) minus 119881
119878(0) = int
119905
0
(1205734119867119868(120591) minus 120573
5119881119878(120591)) 119889120591
119881119868(119905) minus 119881
119868(0) = int
119905
0
(1205735119881119878(120591) minus 120574119881
119868(120591)) 119889120591
(28)
We then assume that the solutions of the above integralequations can be put in the following form for 119875 isin (0 1)
119867119878(119905) =
infin
sum119899=0
119875119899119867119878119899(119905) 119867
119864(119905) =
infin
sum119899=0
119875119899119867119864119899(119905)
119867119868(119905) =
infin
sum119899=0
119875119899119867119868119899(119905) 119881
119878(119905) =
infin
sum119899=0
119875119899119881119878119899(119905)
119881119868(119905) =
infin
sum119899=0
119875119899119881119868119899(119905)
(29)
Replacing the above expressions in the integral equations weobtain the following expressions
infin
sum119899=0
119875119899119867119878119899(119905) minus 119867
119878(0)
= 119875int119905
0
[(120588120577 + 120574infin
sum119899=0
119875119899119881119868119899(120591)
6 Abstract and Applied Analysis
minus (1205731+ 1205732)infin
sum119899=0
119875119899119867119878119899(120591)) 119889120591]
infin
sum119899=0
119875119899119867119864119899(119905) minus 119867
119864(0)
= 119875int119905
0
[(120575120582 + 1205731
infin
sum119899=0
119875119899119867119878119899(120591)
minus 1205733
infin
sum119899=0
119875119899119867119864119899(120591)) 119889120591]
infin
sum119899=0
119875119899119867119868119899(119905) minus 119867
119868(0)
= 119875int119905
0
[(1205732
infin
sum119899=0
119875119899119867119878119899(120591) + 120573
3
infin
sum119899=0
119875119899119867119864119899(120591)
minus 120588120577 minus 120575120582 minus 1205734
infin
sum119899=0
119875119899119867119868119899(120591)) 119889120591]
infin
sum119899=0
119875119899119881119878119899(119905) minus 119881
119878(0)
= 119875int119905
0
[(1205734
infin
sum119899=0
119875119899119867119868119899(120591) minus 120573
5
infin
sum119899=0
119875119899119881119878119899(120591)) 119889120591]
infin
sum119899=0
119875119899119881119868119899(119905) minus 119881
119868(0)
= 119875int119905
0
[(1205735
infin
sum119899=0
119875119899119881119878119899(120591) minus 120574
infin
sum119899=0
119875119899119881119868119899(120591)) 119889120591]
(30)
We compared the terms of the same power of 119875 and obtainedthe following
1198750 1198671198780(119905) = 119867
119878(0)
1198750 1198671198640(119905) = 119867
119864(0)
1198750 1198671198680(119905) = 119867
119868(0)
1198750 1198811198780(119905) = 119881
119878(0)
1198750 1198811198680(119905) = 119881
119868(0)
1198751 1198671198781(119905) = int
119905
0
(120588120577 + 1205741198811198680(120591) minus (120573
1+ 1205732)1198671198780(120591)) 119889120591
1198751 1198671198641(119905) = int
119905
0
(120575120582 + 12057311198671198780(120591) minus 120573
31198671198640(120591)) 119889120591
1198751 1198671198681(119905)
= int119905
0
(12057321198671198780(120591)+120573
31198671198640(120591)minus120588120577minus 120575120582 minus120573
41198671198680(120591)) 119889120591
1198751 1198811198781(119905) = int
119905
0
(12057341198671198680(120591) minus 120573
51198811198780(120591)) 119889120591
1198751 1198811198681(119905) = int
119905
0
(12057351198811198780(120591) minus 120574119881
1198680(120591)) 119889120591
(31)
Continuing for 119899 ge 2 we have
119867119878119899(119905) = int
119905
0
(120574119881119868(119899minus1)
(120591) minus (1205731+ 1205732)119867119878(119899minus1)
(120591)) 119889120591
119867119864119899(119905) = int
119905
0
(1205731119867119878(119899minus1)
(120591) minus 1205733119867119864(119899minus1)
(120591)) 119889120591
119867119868119899(119905) = int
119905
0
(1205732119867119878(119899minus1)
(120591) + 1205733119867119864(119899minus1)
(120591)
minus 1205734119867119868(119899minus1)
(120591)) 119889120591
119881119878119899(119905) = int
119905
0
(1205734119867119868(119899minus1)
(120591) minus 1205735119881119878(119899minus1)
(120591)) 119889120591
119881119868119899(119905) = int
119905
0
(1205735119881119878(119899minus1)
(120591) minus 120574119881119868(119899minus1)
(120591)) 119889120591
(32)
We now integrate the above equations to obtain the following
1198671198680(119905) = 0 119867
1198640(119905) = 0 119867
1198780 [119905] = 119873 1198811198780 [119905] = 119872
1198811198680 [119905] = 0 119867
1198681(119905) = 119905 (minus120575120582 minus 120577120588 + 119873120573
2)
1198671198781 [119905] = 119905 (120577120588 minus 119873 (1205731 + 1205732))
1198671198641(119905) = 119905 (120575120582 + 119873120573
1)
1198811198681 [119905] = 1198721199051205735 119881
1198781 [119905] = minus1198721199051205735
1198671198682(119905) =
1
21199052120577120588120573
2minus1
2119873119905212057311205732minus1
2119873119905212057322+1
21199052120575120582120573
3
+1
2119873119905212057311205733+1
21199052120575120582120573
4+1
21199052120577120588120573
4minus1
2119873119905212057321205734
1198671198642(119905) =
1
21199052120577120588120573
1minus1
2119873119905212057321minus1
2119873119905212057311205732
minus1
21199052120575120582120573
3minus1
2119873119905212057311205733
1198671198782 [119905] = minus
1
21199052120577120588120573
1+1
2119873119905212057321minus1
21199052120577120588120573
2
+ 119873119905212057311205732+1
2119873119905212057322+1
21198721199052120574120573
5
1198811198782 [119905] = minus
1
21199052120575120582120573
4minus1
21199052120577120588120573
4
+1
2119873119905212057321205734+1
211987211990521205732
5
1198811198682 [119905] = minus
1
21198721199052120574120573
5minus1
211987211990521205732
5
1198671198683(119905) = minus
1
61199053120577120588120573
11205732+1
61198731199053120573211205732minus1
611990531205771205881205732
2
+1
31198731199053120573112057322+1
6119873119905312057332+1
61199053120577120588120573
11205733
minus1
61198731199053120573211205733minus1
61198731199053120573112057321205733
minus1
611990531205751205821205732
3
minus1
61198731199053120573112057323minus1
61199053120577120588120573
21205734
+1
61198731199053120573112057321205734+1
61198731199053120573221205734
minus1
61199053120575120582120573
31205734minus1
61198731199053120573112057331205734
Abstract and Applied Analysis 7
minus1
611990531205751205821205732
4minus1
611990531205771205881205732
4+1
61198731199053120573212057324
+1
61198721199053120574120573
21205735
1198671198643(119905) = minus
1
611990531205771205881205732
1+1
6119873119905312057331minus1
61199053120577120588120573
11205732
+1
31198731199053120573211205732+1
61198731199053120573112057322
minus1
61199053120577120588120573
11205733+1
61198731199053120573211205733
+1
61198731199053120573112057321205733+1
611990531205751205821205732
3
+1
61198731199053120573112057323+1
61198721199053120574120573
11205735
1198671198783 [119905] =
1
611990531205771205881205732
1minus1
6119873119905312057331+1
31199053120577120588120573
11205732
minus1
21198731199053120573211205732+1
611990531205771205881205732
2minus1
21198731199053120573112057322
minus1
6119873119905312057332minus1
611987211990531205742120573
5
minus1
61198721199053120574120573
11205735
minus1
61198721199053120574120573
21205735minus1
611987211990531205741205732
5
1198811198783 [119905] =
1
61199053120577120588120573
21205734minus1
61198731199053120573112057321205734minus1
61198731199053120573221205734
+1
61199053120575120582120573
31205734+1
61198731199053120573112057331205734+1
611990531205751205821205732
4
+1
611990531205771205881205732
4minus1
61198731199053120573212057324+1
61199053120575120582120573
41205735
+1
61199053120577120588120573
41205735minus1
61198731199053120573212057341205735minus1
611987211990531205733
5
1198811198683 [119905] =
1
611987211990531205742120573
5minus1
61199053120575120582120573
41205735minus1
61199053120577120588120573
41205735
+1
61198731199053120573212057341205735+1
611987211990531205741205732
5+1
611987211990531205733
5
1198671198784 [119905] = minus
1
2411990541205771205881205733
1+1
24119873119905412057341minus1
811990541205771205881205732
11205732
+1
61198731199054120573311205732minus1
81199054120577120588120573
112057322+1
411987311990541205732112057322
minus1
2411990541205771205881205733
2+1
61198731199054120573112057332+1
24119873119905412057342
+1
2411987211990541205743120573
5+1
2411987211990541205742120573
11205735
+1
2411987211990541205741205732
11205735+1
2411987211990541205742120573
21205735
+1
121198721199054120574120573
112057321205735+1
2411987211990541205741205732
21205735
minus1
241199054120574120575120582120573
41205735minus1
241199054120574120577120588120573
41205735
+1
241198731199054120574120573
212057341205735+1
24119872119905412057421205732
5
+1
241198721199054120574120573
112057325+1
241198721199054120574120573
212057325+1
2411987211990541205741205733
5
1198671198644(119905) =
1
2411990541205771205881205733
1minus1
24119873119905412057341+1
1211990541205771205881205732
11205732
minus1
81198731199054120573311205732+1
241199054120577120588120573
112057322minus1
811987311990541205732112057322
minus1
241198731199054120573112057332+1
2411990541205771205881205732
11205733minus1
241198731199054120573311205733
+1
241199054120577120588120573
112057321205733minus1
1211987311990541205732112057321205733minus1
2411987311990541205731120573221205733
+1
241199054120577120588120573
112057323minus1
2411987311990541205732112057323minus1
2411987311990541205731120573212057323
minus1
2411990541205751205821205733
3minus1
241198731199054120573112057333minus1
2411987211990541205742120573
11205735
minus1
2411987211990541205741205732
11205735minus1
241198721199054120574120573
112057321205735
minus1
241198721199054120574120573
112057331205735minus1
241198721199054120574120573
112057325
1198671198684(119905) =
1
2411990541205771205881205732
11205732minus1
241198731199054120573311205732+1
121199054120577120588120573
112057322
minus1
811987311990541205732112057322+1
2411990541205771205881205733
2minus1
81198731199054120573112057332
minus1
24119873119905412057342minus1
2411990541205771205881205732
11205733+1
241198731199054120573311205733
minus1
241199054120577120588120573
112057321205733+1
1211987311990541205732112057321205733
+1
2411987311990541205731120573221205733
minus1
241199054120577120588120573
112057323+1
2411987311990541205732112057323+1
2411987311990541205731120573212057323
+1
2411990541205751205821205733
3+1
241198731199054120573112057333+1
241199054120577120588120573
112057321205734
minus1
2411987311990541205732112057321205734+1
2411990541205771205881205732
21205734minus1
1211987311990541205731120573221205734
minus1
241198731199054120573321205734minus1
241199054120577120588120573
112057331205734
+1
2411987311990541205732112057331205734+1
2411987311990541205731120573212057331205734
+1
2411990541205751205821205732
31205734+1
2411987311990541205731120573231205734
+1
241199054120577120588120573
212057324minus1
2411987311990541205731120573212057324minus1
2411987311990541205732212057324
+1
241199054120575120582120573
312057324+1
2411987311990541205731120573312057324+1
2411990541205751205821205733
4
+1
2411990541205771205881205733
4minus1
241198731199054120573212057334minus1
2411987211990541205742120573
21205735
minus1
241198721199054120574120573
112057321205735minus1
2411987211990541205741205732
21205735
+1
241198721199054120574120573
112057331205735
minus1
241198721199054120574120573
212057341205735minus1
241198721199054120574120573
212057325
1198811198784 [119905] = minus
1
241199054120577120588120573
112057321205734+1
2411987311990541205732112057321205734
minus1
2411990541205771205881205732
21205734+1
1211987311990541205731120573221205734
8 Abstract and Applied Analysis
+1
241198731199054120573321205734+1
241199054120577120588120573
112057331205734
minus1
2411987311990541205732112057331205734minus1
2411987311990541205731120573212057331205734
minus1
2411990541205751205821205732
31205734
minus1
2411987311990541205731120573231205734minus1
241199054120577120588120573
212057324
+1
2411987311990541205731120573212057324+1
2411987311990541205732212057324
minus1
241199054120575120582120573
312057324minus1
2411987311990541205731120573312057324
minus1
2411990541205751205821205733
4minus1
2411990541205771205881205733
4
+1
241198731199054120573212057334
+1
241198721199054120574120573
212057341205735minus1
241199054120577120588120573
212057341205735
+1
2411987311990541205731120573212057341205735+1
2411987311990541205732212057341205735
minus1
241199054120575120582120573
312057341205735minus1
2411987311990541205731120573312057341205735
minus1
2411990541205751205821205732
41205735minus1
2411990541205771205881205732
41205735+1
2411987311990541205732120573241205735
minus1
241199054120575120582120573
412057325minus1
241199054120577120588120573
412057325
+1
2411987311990541205732120573412057325+1
2411987211990541205734
5
1198811198684 [119905] = minus
1
2411987211990541205743120573
5+1
241199054120574120575120582120573
41205735+1
241199054120574120577120588120573
41205735
minus1
241198731199054120574120573
212057341205735+1
241199054120577120588120573
212057341205735
minus1
2411987311990541205731120573212057341205735minus1
2411987311990541205732212057341205735
+1
241199054120575120582120573
312057341205735+1
2411987311990541205731120573312057341205735
+1
2411990541205751205821205732
41205735+1
2411990541205771205881205732
41205735
minus1
2411987311990541205732120573241205735minus1
24119872119905412057421205732
5
+1
241199054120575120582120573
412057325+1
241199054120577120588120573
412057325
minus1
2411987311990541205732120573412057325minus1
2411987211990541205741205733
5minus1
2411987211990541205734
5
1198671198785 [119905] =
1
12011990551205771205881205734
1minus
1
120119873119905512057351+1
3011990551205771205881205733
11205732
minus1
241198731199055120573411205732+1
2011990551205771205881205732
112057322minus1
1211987311990551205733112057322
+1
301199055120577120588120573
112057332minus1
1211987311990551205732112057332+
1
12011990551205771205881205734
2
minus1
241198731199055120573112057342minus
1
120119873119905512057352minus
1
12011987211990551205744120573
5
minus1
12011987211990551205743120573
11205735minus
1
120119872119905512057421205732
11205735
minus1
12011987211990551205741205733
11205735minus
1
12011987211990551205743120573
21205735
minus1
6011987211990551205742120573
112057321205735minus1
4011987211990551205741205732
112057321205735
minus1
120119872119905512057421205732
21205735minus1
401198721199055120574120573
1120573221205735
minus1
12011987211990551205741205733
21205735+
1
12011990551205742120575120582120573
41205735
+1
12011990551205742120577120588120573
41205735+
1
1201199055120574120575120582120573
112057341205735
+1
1201199055120574120577120588120573
112057341205735minus
1
12011987311990551205742120573
212057341205735
+1
1201199055120574120575120582120573
212057341205735+1
601199055120574120577120588120573
212057341205735
minus1
601198731199055120574120573
1120573212057341205735minus1
6011987311990551205741205732
212057341205735
+1
1201199055120574120575120582120573
312057341205735+
1
1201198731199055120574120573
1120573312057341205735
+1
12011990551205741205751205821205732
41205735+
1
12011990551205741205771205881205732
41205735
minus1
1201198731199055120574120573
2120573241205735minus
1
120119872119905512057431205732
5
minus1
12011987211990551205742120573
112057325minus
1
12011987211990551205741205732
112057325
minus1
12011987211990551205742120573
212057325minus1
601198721199055120574120573
1120573212057325
minus1
12011987211990551205741205732
212057325+
1
1201199055120574120575120582120573
412057325
+1
1201199055120574120577120588120573
412057325minus
1
1201198731199055120574120573
2120573412057325
minus1
120119872119905512057421205733
5minus
1
1201198721199055120574120573
112057335
minus1
1201198721199055120574120573
212057335minus
1
12011987211990551205741205734
5
1198671198645 [119905] = minus
1
12011990551205771205881205734
1+
1
120119873119905512057351
minus1
4011990551205771205881205733
11205732+1
301198731199055120573411205732
minus1
4011990551205771205881205732
112057322+1
2011987311990551205733112057322
minus1
1201199055120577120588120573
112057332+1
3011987311990551205732112057332
+1
1201198731199055120573112057342
minus1
12011990551205771205881205733
11205733+
1
1201198731199055120573411205733
minus1
6011990551205771205881205732
112057321205733+1
4011987311990551205733112057321205733
minus1
1201199055120577120588120573
1120573221205733+1
40119873119905512057321120573221205733
+1
12011987311990551205731120573321205733minus
1
12011990551205771205881205732
112057323
+1
12011987311990551205733112057323minus
1
1201199055120577120588120573
1120573212057323
Abstract and Applied Analysis 9
+1
60119873119905512057321120573212057323+
1
120119873119905512057311205732212057323
minus1
1201199055120577120588120573
112057333+
1
12011987311990551205732112057333
+1
12011987311990551205731120573212057333+
1
12011990551205751205821205734
3
+1
1201198731199055120573112057343+
1
12011987211990551205743120573
11205735
+1
120119872119905512057421205732
11205735+
1
12011987211990551205741205733
11205735
+1
12011987211990551205742120573
112057321205735+1
6011987211990551205741205732
112057321205735
+1
1201198721199055120574120573
1120573221205735+
1
12011987211990551205742120573
112057331205735
+1
12011987211990551205741205732
112057331205735+
1
1201198721199055120574120573
1120573212057331205735
+1
1201198721199055120574120573
1120573231205735minus
1
1201199055120574120575120582120573
112057341205735
minus1
1201199055120574120577120588120573
112057341205735+
1
1201198731199055120574120573
1120573212057341205735
+1
12011987211990551205742120573
112057325+
1
12011987211990551205741205732
112057325
+1
1201198721199055120574120573
1120573212057325+
1
1201198721199055120574120573
1120573312057325
+1
1201198721199055120574120573
112057335
1198671198685 [119905]= minus
1
12011990551205771205881205733
11205732+
1
1201198731199055120573411205732minus1
4011990551205771205881205732
112057322
+1
3011987311990551205733112057322minus1
401199055120577120588120573
112057332+1
2011987311990551205732112057332
minus1
12011990551205771205881205734
2+1
301198731199055120573112057342+
1
120119873119905512057352
+1
12011990551205771205881205733
11205733minus
1
1201198731199055120573411205733
+1
6011990551205771205881205732
112057321205733minus1
4011987311990551205733112057321205733
+1
1201199055120577120588120573
1120573221205733minus1
40119873119905512057321120573221205733
minus1
12011987311990551205731120573321205733+
1
12011990551205771205881205732
112057323
minus1
12011987311990551205733112057323+
1
1201199055120577120588120573
1120573212057323
minus1
60119873119905512057321120573212057323minus
1
120119873119905512057311205732212057323
+1
1201199055120577120588120573
112057333minus
1
12011987311990551205732112057333
minus1
12011987311990551205731120573212057333minus
1
12011990551205751205821205734
3
minus1
1201198731199055120573112057343minus
1
12011990551205771205881205732
112057321205734
+1
12011987311990551205733112057321205734minus1
601199055120577120588120573
1120573221205734
+1
40119873119905512057321120573221205734minus
1
12011990551205771205881205733
21205734
+1
4011987311990551205731120573321205734+
1
1201198731199055120573421205734
+1
12011990551205771205881205732
112057331205734minus
1
12011987311990551205733112057331205734
+1
1201199055120577120588120573
1120573212057331205734minus1
60119873119905512057321120573212057331205734
minus1
120119873119905512057311205732212057331205734+
1
1201199055120577120588120573
1120573231205734
minus1
120119873119905512057321120573231205734minus
1
120119873119905512057311205732120573231205734
minus1
12011990551205751205821205733
31205734minus
1
12011987311990551205731120573331205734
minus1
1201199055120577120588120573
1120573212057324+
1
120119873119905512057321120573212057324
minus1
12011990551205771205881205732
212057324+1
60119873119905512057311205732212057324
+1
12011987311990551205733212057324+
1
1201199055120577120588120573
1120573312057324
minus1
120119873119905512057321120573312057324minus
1
120119873119905512057311205732120573312057324
minus1
12011990551205751205821205732
312057324minus
1
120119873119905512057311205732312057324
minus1
1201199055120577120588120573
212057334+
1
12011987311990551205731120573212057334
+1
12011987311990551205732212057334minus
1
1201199055120575120582120573
312057334
minus1
12011987311990551205731120573312057334minus
1
12011990551205751205821205734
4
minus1
12011990551205771205881205734
4+
1
1201198731199055120573212057344
+1
12011987211990551205743120573
21205735+
1
12011987211990551205742120573
112057321205735
+1
12011987211990551205741205732
112057321205735+
1
120119872119905512057421205732
21205735
+1
601198721199055120574120573
1120573221205735+
1
12011987211990551205741205733
21205735
minus1
12011987211990551205742120573
112057331205735minus
1
12011987211990551205741205732
112057331205735
minus1
1201198721199055120574120573
1120573212057331205735minus
1
1201198721199055120574120573
1120573231205735
+1
12011987211990551205742120573
212057341205735minus
1
1201199055120574120575120582120573
212057341205735
minus1
1201199055120574120577120588120573
212057341205735+
1
1201198721199055120574120573
1120573212057341205735
+1
12011987211990551205741205732
212057341205735+
1
12011987311990551205741205732
212057341205735
minus1
1201198721199055120574120573
1120573312057341205735+
1
1201198721199055120574120573
2120573241205735
+1
12011987211990551205742120573
212057325+
1
1201198721199055120574120573
1120573212057325
+1
12011987211990551205741205732
212057325minus
1
1201198721199055120574120573
1120573312057325
+1
1201198721199055120574120573
2120573412057325+
1
1201198721199055120574120573
212057335
10 Abstract and Applied Analysis
1198811198785 [119905] =
1
12011990551205771205881205732
112057321205734minus
1
12011987311990551205733112057321205734
+1
601199055120577120588120573
1120573221205734minus1
40119873119905512057321120573221205734
+1
12011990551205771205881205733
21205734minus1
4011987311990551205731120573321205734
minus1
1201198731199055120573421205734minus
1
12011990551205771205881205732
112057331205734
+1
12011987311990551205733112057331205734minus
1
1201199055120577120588120573
1120573212057331205734
+1
60119873119905512057321120573212057331205734+
1
120119873119905512057311205732212057331205734
minus1
1201199055120577120588120573
1120573231205734+
1
120119873119905512057321120573231205734
+1
120119873119905512057311205732120573231205734+
1
12011990551205751205821205733
31205734
+1
12011987311990551205731120573331205734+
1
1201199055120577120588120573
1120573212057324
minus1
120119873119905512057321120573212057324+
1
12011990551205771205881205732
212057324
minus1
60119873119905512057311205732212057324minus
1
12011987311990551205733212057324
minus1
1201199055120577120588120573
1120573312057324+
1
120119873119905512057321120573312057324
+1
120119873119905512057311205732120573312057324+
1
12011990551205751205821205732
312057324
+1
120119873119905512057311205732312057324+
1
1201199055120577120588120573
212057334
minus1
12011987311990551205731120573212057334minus
1
12011987311990551205732212057334
+1
1201199055120575120582120573
312057334+
1
12011987311990551205731120573312057334
+1
12011990551205751205821205734
4+
1
12011990551205771205881205734
4minus
1
1201198731199055120573212057344
minus1
12011987211990551205742120573
212057341205735minus
1
1201198721199055120574120573
1120573212057341205735
+1
1201199055120577120588120573
1120573212057341205735minus
1
120119873119905512057321120573212057341205735
minus1
12011987211990551205741205732
212057341205735+
1
12011990551205771205881205732
212057341205735
minus1
60119873119905512057311205732212057341205735minus
1
12011987311990551205733212057341205735
+1
1201198721199055120574120573
1120573312057341205735minus
1
1201199055120577120588120573
1120573312057341205735
+1
120119873119905512057321120573312057341205735+
1
120119873119905512057311205732120573312057341205735
+1
12011990551205751205821205732
312057341205735+
1
120119873119905512057311205732312057341205735
minus1
1201198721199055120574120573
2120573241205735+
1
1201199055120577120588120573
2120573241205735
minus1
120119873119905512057311205732120573241205735minus
1
120119873119905512057322120573241205735
+1
1201199055120575120582120573
3120573241205735+
1
120119873119905512057311205733120573241205735
+1
12011990551205751205821205733
41205735+
1
12011990551205771205881205733
41205735
minus1
12011987311990551205732120573341205735minus1
601198721199055120574120573
2120573412057325
+1
1201199055120577120588120573
2120573412057325minus
1
120119873119905512057311205732120573412057325
minus1
120119873119905512057322120573412057325+
1
1201199055120575120582120573
3120573412057325
+1
120119873119905512057311205733120573412057325+
1
12011990551205751205821205732
412057325
+1
12011990551205771205881205732
412057325minus
1
120119873119905512057321205732412057325
+1
1201199055120575120582120573
412057335+
1
1201199055120577120588120573
412057335
minus1
12011987311990551205732120573412057335minus
1
12011987211990551205735
5
1198811198685 [119905] =
1
12011987211990551205744120573
5minus
1
12011990551205742120575120582120573
41205735
minus1
12011990551205742120577120588120573
41205735+
1
12011987311990551205742120573
212057341205735
minus1
1201199055120574120577120588120573
212057341205735+
1
1201198731199055120574120573
1120573212057341205735
minus1
1201199055120577120588120573
1120573212057341205735+
1
120119873119905512057321120573212057341205735
+1
12011987311990551205741205732
212057341205735minus
1
12011990551205771205881205732
212057341205735
+1
60119873119905512057311205732212057341205735+
1
12011987311990551205733212057341205735
minus1
1201199055120574120575120582120573
312057341205735minus
1
1201198731199055120574120573
1120573312057341205735
+1
1201199055120577120588120573
1120573312057341205735minus
1
120119873119905512057321120573312057341205735
minus1
120119873119905512057311205732120573312057341205735minus
1
12011990551205751205821205732
312057341205735
minus1
120119873119905512057311205732312057341205735minus
1
12011990551205741205751205821205732
41205735
minus1
12011990551205741205771205881205732
41205735+
1
1201198731199055120574120573
2120573241205735
minus1
1201199055120577120588120573
2120573241205735+
1
120119873119905512057311205732120573241205735
+1
120119873119905512057322120573241205735minus
1
1201199055120575120582120573
3120573241205735
minus1
120119873119905512057311205733120573241205735minus
1
12011990551205751205821205733
41205735
minus1
12011990551205771205881205733
41205735+
1
12011987311990551205732120573341205735
+1
120119872119905512057431205732
5minus
1
1201199055120574120575120582120573
412057325
minus1
1201199055120574120577120588120573
412057325+
1
1201198721199055120574120573
2120573412057325
+1
1201198731199055120574120573
2120573412057325minus
1
1201199055120577120588120573
2120573412057325
+1
120119873119905512057311205732120573412057325+
1
120119873119905512057322120573412057325
Abstract and Applied Analysis 11
minus1
1201199055120575120582120573
3120573412057325minus
1
120119873119905512057311205733120573412057325
minus1
12011990551205751205821205732
412057325minus
1
12011990551205771205881205732
412057325
+1
120119873119905512057321205732412057325+
1
120119872119905512057421205733
5
minus1
1201199055120575120582120573
412057335minus
1
1201199055120577120588120573
412057335
+1
12011987311990551205732120573412057335+
1
12011987211990551205741205734
5
+1
12011987211990551205735
5
(33)
In general by the repeated application of the above proce-dure we have that
119867119878119899=119905119899
1198991198861119899 119867
119864119899=119905119899
1198991198862119899 119867
119868119899=119905119899
1198991198863119899
119881119878119899=119905119899
1198991198864119899 119881
119868119899=119905119899
1198991198865119899
(34)
where 1198861119899 1198862119899 1198863119899 1198864119899 1198865119899
depend on fixed set of empiricalparameters Hence the approximate solution of the modelabove is given as
119867119878(119905) =
infin
sum119899=0
119905119899
1198991198861119899
119867119864(119905) =
infin
sum119899=0
119905119899
1198991198862119899
119867119868(119905) =
infin
sum119899=0
119905119899
1198991198863119899
119881119878(119905) =
infin
sum119899=0
119905119899
1198991198864119899
119881119868(119905) =
infin
sum119899=0
119905119899
1198991198865119899
(35)
We therefore use the following theoretical parameters 1205731=
05 1205732= 05 120573
3= 05 120573
4= 05 120573
5= 05 120574 = 1 120577 = 1
119873 = 10 119872 = 10 120588 = 1 120575 = 1 and 120582 = 1 to examinethe behavior of the five different compartmental numericalsolutions in the system Figure 3 shows the behavior of thenumerical solution
7 Conclusion
Onchocerciasis model was examined and the mathematicalformulation underpinning the spread and control of thisdisease has been proposed on one hand On the otherhand we made use of some new analytical methods calledthe homotopy decomposition method (HDM) to derive thesolution of the resulting set of (1) to (5)The numerical resultswere presented to test the efficiency and the accuracy of themethod used The mathematical analysis of the resulting setof (1) to (5) above revealed that the stability analysis of the
05 10 15 20Time (per year)
2
4
6
8
10
Appr
oxim
ate s
olut
ions VS
HS
VI
HI
HE
Figure 3 Numerical solution of the model
five roots of the characteristics equations has all negativereal part In particular given that the stability analysis showsthat the DFE is asymptotically stable it would be possiblethat the parasite Onchocerca volvulus be eradicated in a finitetime On the other hand the numerical solution confirmedthat as the number of humans and vectors in the susceptiblecompartments reduces the number of infective humansand vectors in the infectious compartment increases Thenumerical solutions in both cases displayed the biologicalbehaviour of the real world situation Therefore in orderto build a healthy and a sustainable population treatmentwith IvermectinMectizan drugs and a more hygienic envi-ronment that does not allow the breeding and survival ofblackflies could lead to a total eradication of the disease
Conflict of Interests
The authors declare that they have no conflict of interestsregarding the publication of this paper
References
[1] J H F Remme The Global Burden of Onchocerciasis in 1990World Health Organization Geneva Switzerland 2004 httpwwwwhointhealthinfoglobal burden diseaseOnchocercia-sis201990pdf
[2] WorldHealthOrganization ldquoPrevention of blindness and visualimpairment priority eye diseasesrdquo 2014 httpwwwwhointblindnesscausespriorityenindex3html
[3] World Health Organization Certification of Elimination ofHuman Onchocerciasis Criteria and Procedures World HealthOrganizationGeneva Switzerland 2001 httpwhqlibdocwhointhq2001WHO CDS CPE CEE 200118bpdf
[4] K A Lindblade B Arana G Zea-Flores et al ldquoEliminationof Onchocerca volvulus transmission in the Santa Rosa focusof Guatemalardquo The American Journal of Tropical Medicine andHygiene vol 77 pp 334ndash341 2007
[5] J C Anosike and C O E Onwuliri ldquoStudies on filariasis inBauchi State Nigeria 1 Endemicity of human onchocerciasisin Ningi Local Government Areardquo Annals of Tropical Medicineand Parasitology vol 89 no 1 pp 31ndash38 1995
[6] WorldHealthOrganizationWeekly Epidemiological Records 71WHO Geneva Switzerland 1993
12 Abstract and Applied Analysis
[7] WorldHealthOrganization httpwwwwhointneglected dis-easesen
[8] Sightsaver httpwwwsightsaversorgabout uspublicationsonchocerciasis
[9] E C Uttah ldquoOnchocerciasis in the Upper Ima River BasinrdquoIranian Journal of Parasitology vol 5 no 2 pp 33ndash41 2010
[10] J C Anosike and C O E Onwuliri ldquoStudies on filariasis inBauchi State Nigeriamdash1 endemicity of human onchocerciasisin Ningi Local Government Areardquo Annals of Tropical Medicineand Parasitology vol 89 no 1 pp 31ndash38 1995
[11] L D Edungbola and S O Asaolu ldquoParasitologic survey ofonchocerciasis (river blindness) in Babana district Kwara StateNigeriardquo American Journal of Tropical Medicine and Hygienevol 33 no 6 pp 1147ndash1154 1984
[12] O C Nwaorgu A Ohaegbula I E Onweluzo et al ldquoResultsof a large scale onchocercosis survey in Enugu State NigeriardquoJournal of Helminthology vol 68 no 2 pp 155ndash159 1994
[13] F O Akinbo and C E Okaka ldquoPrevalence and socio-economiceffects of onchocerciasis in Ovia North East Local GovernmentArea of Edo State Nigeriardquo Rivista Di Parasitologie vol 22 no3 pp 215ndash221 2005
[14] M D Wogu and C E Okaka ldquoPrevalence and socio-economic effects of onchocerciasis in Okpuje Owan WestLocal Government Area Edo State Nigeriardquo InternationalJournal of Biomedical and Health Sciences vol 4 no 3 2008httpwwwasopahorg
[15] S C O Noutchie ldquoHomotopy Decomposition analysis of atuberculosis modelrdquo Life Science Journal vol 10 no 3 pp 459ndash465 2013
[16] A Atangana and A Secer ldquoThe time-fractional coupled-Korteweg-de-Vries equationsrdquo Abstract and Applied Analysisvol 2013 Article ID 947986 8 pages 2013
[17] A Atangana and N Bildik ldquoApproximate solution of tubercu-losis disease population dynamics modelrdquo Abstract and AppliedAnalysis vol 2013 Article ID 759801 8 pages 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Abstract and Applied Analysis
= ([120575120582 (1205731+ 1205732) + 1205731120588120577 + 120573
11205735] (120588120577 + 120575120582))
times ((12057321205733) (120588120577 + 120573
5)
1205731+ 1205732
+ 120575120582 (1205731+ 1205732) 1205733
+ 12057311205733120588120577 + 120573
112057331205735minus 12057331205735)
minus1
(16)
119867119868=1205735
1205734
((120588120577 + 120575120582)
times (1205732(120588120577 + 120573
5
1205731+ 1205732
)
+ 1205733(120575120582 (1205731+ 1205732) + 1205731120588120577 + 120573
11205735
1205733
)
minus1205734(1205735
1205734
))
minus1
)
= (1205735(120588120577 + 120575120582)) (120573
21205734(120588120577 + 120573
5
1205731+ 1205732
) + 1205734120575120582 (1205731+ 1205732)
+ 12057311205734120588120577 + 120573
112057341205735minus 12057341205735)minus1
(17)
119881119878= ((120588120577 + 120575120582)
times (1205732(120588120577 + 120573
5
1205731+ 1205732
)
+ 1205733(120575120582 (1205731+ 1205732) + 1205731120588120577 + 120573
11205735
1205733
)
minus1205734(1205735
1205734
))
minus1
)
(18)
119881119868=1205735
120574((120588120577 + 120575120582)
times (1205732(120588120577 + 120573
5
1205731+ 1205732
)
+ 1205733(120575120582 (1205731+ 1205732) + 1205731120588120577 + 120573
11205735
1205733
)
minus1205734(1205735
1205734
))
minus1
)
= (1205735(120588120577 + 120575120582))
times (1205741205732(120588120577 + 120573
5
1205731+ 1205732
) + 120574120575120582 (1205731+ 1205732)
+ 1205741205731120588120577 + 120574120573
11205735minus 1205741205735)minus1
(19)
5 Stability Analysis
To assess the stability of the (DFE) we solve the model char-acteristics equation using the Jacobean matrix formulatedfrom (1) (2) (3) (4) and (5) above as follows
119869 = (
minus(1205731+ 1205732) 0 0 0 120574
1205731
minus1205733
0 0 01205732
1205733minus1205734
0 00 0 120573
4minus12057350
0 0 0 1205735minus120574
) (20)
Applying the eigenvalue
120582 det (119869 minus 120582119868) = 0 (21)
we have
det[[[[[[
[
minus (1205731+ 1205732) minus 120582 0 0 0 120574
1205731
minus1205733minus 120582 0 0 0
1205732
1205733
minus1205734minus 120582 0 0
0 0 1205734
minus1205735minus 120582 0
0 0 0 1205735
minus120574 minus 120582
]]]]]]
]
= 0
det[[[[[[
[
minus (1205731+ 1205732+ 120582) 0 0 0 120574
1205731
minus (1205733+ 120582) 0 0 0
1205732
1205733
minus (1205734+ 120582) 0 0
0 0 1205734
minus (1205735+ 120582) 0
0 0 0 1205735
minus (120574 + 120582)
]]]]]]
]
= 0
minus (1205731+ 1205732+ 120582)
[[[[
[
minus (1205733+ 120582) 0 0 0
1205733
minus (1205734+ 120582) 0 0
0 1205734
minus (1205735+ 120582) 0
0 0 1205735
minus (120574 + 120582)
]]]]
]
+ 120574[[[[
[
1205731minus (1205733+ 120582) 0 0
1205732
1205733
minus (1205734+ 120582) 0
0 0 1205734
minus (1205735+ 120582)
0 0 0 1205735
]]]]
]
= 0
Abstract and Applied Analysis 5
minus (1205731+ 1205732+ 120582)
[[
[
minus (1205733+ 120582)
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
minus (1205734+ 120582) 0 0
1205734
minus (1205735+ 120582) 0
0 1205735
minus (120574 + 120582)
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
]]
]
+ 120574[[
[
1205731
[[
[
1205733minus (1205734+ 120582) 0
0 1205734
minus (1205735+ 120582)
0 0 1205735
]]
]
+ (1205733+ 120582)
[[
[
1205732minus (1205734+ 120582) 0
0 1205734
minus (1205735+ 120582)
0 0 1205735
]]
]
]]
]
= 0
minus (1205731+ 1205732+ 120582) [minus (120573
3+ 120582) [minus (120573
4+ 120582)
100381610038161003816100381610038161003816100381610038161003816
minus (1205735+ 120582) 0
1205735
minus (120574 + 120582)
100381610038161003816100381610038161003816100381610038161003816]] + 120574120573
1120573312057341205735+ 120574 (120573
3+ 120582) 120573
212057341205735= 0
minus [(1205731+ 1205732+ 120582) (120573
3+ 120582) (120573
4+ 120582) (120573
5+ 120582) (120574 + 120582)] + 120574120573
1120573312057341205735+ 1205741205732120573312057341205735+ 120574120582120573
212057341205735= 0
(22)
Hence we obtain from minus(1205731+ 1205732+ 120582) = 0
1205821= minus (120573
1+ 1205732) (23)
From minus(1205733+ 120582) = 0
1205822= minus1205733 (24)
From minus(1205734+ 120582) = 0
1205823= minus1205734 (25)
From (1205735+ 120582) = 0
1205824= minus1205735 (26)
From (120574 + 120582) = 0
1205825= minus120574 (27)
Given that the results of 1205821 1205822 1205823 1205824 and 120582
5are all negative
and considering the principles of linearized stability whichsays that the systems are asymptotically stable if and only ifall the eigenvalues of the system have negative real parts wesay that the disease-free equilibrium (DFE) is asymptoticallystable
6 Application of the HomotopyDecomposition Method (HDM)
In this section we apply the HDM [15ndash17] to solve thesemathematical sets of equations representing the onchocerci-asis disease model as follows
119867119878(119905) minus 119867
119878(0) = int
119905
0
(120588120577 + 120574119881119868(120591) minus (120573
1+ 1205732)119867119878(120591)) 119889120591
119867119864(119905) minus 119867
119864(0) = int
119905
0
(120575120582 + 1205731119867119878(120591) minus 120573
3119867119864(120591)) 119889120591
119867119868(119905) minus 119867
119868(0)
= int119905
0
(1205732119867119878(120591) + 120573
3119867119864(120591) minus 120588120577 minus 120575120582 minus 120573
4119867119868(120591)) 119889120591
119881119878(119905) minus 119881
119878(0) = int
119905
0
(1205734119867119868(120591) minus 120573
5119881119878(120591)) 119889120591
119881119868(119905) minus 119881
119868(0) = int
119905
0
(1205735119881119878(120591) minus 120574119881
119868(120591)) 119889120591
(28)
We then assume that the solutions of the above integralequations can be put in the following form for 119875 isin (0 1)
119867119878(119905) =
infin
sum119899=0
119875119899119867119878119899(119905) 119867
119864(119905) =
infin
sum119899=0
119875119899119867119864119899(119905)
119867119868(119905) =
infin
sum119899=0
119875119899119867119868119899(119905) 119881
119878(119905) =
infin
sum119899=0
119875119899119881119878119899(119905)
119881119868(119905) =
infin
sum119899=0
119875119899119881119868119899(119905)
(29)
Replacing the above expressions in the integral equations weobtain the following expressions
infin
sum119899=0
119875119899119867119878119899(119905) minus 119867
119878(0)
= 119875int119905
0
[(120588120577 + 120574infin
sum119899=0
119875119899119881119868119899(120591)
6 Abstract and Applied Analysis
minus (1205731+ 1205732)infin
sum119899=0
119875119899119867119878119899(120591)) 119889120591]
infin
sum119899=0
119875119899119867119864119899(119905) minus 119867
119864(0)
= 119875int119905
0
[(120575120582 + 1205731
infin
sum119899=0
119875119899119867119878119899(120591)
minus 1205733
infin
sum119899=0
119875119899119867119864119899(120591)) 119889120591]
infin
sum119899=0
119875119899119867119868119899(119905) minus 119867
119868(0)
= 119875int119905
0
[(1205732
infin
sum119899=0
119875119899119867119878119899(120591) + 120573
3
infin
sum119899=0
119875119899119867119864119899(120591)
minus 120588120577 minus 120575120582 minus 1205734
infin
sum119899=0
119875119899119867119868119899(120591)) 119889120591]
infin
sum119899=0
119875119899119881119878119899(119905) minus 119881
119878(0)
= 119875int119905
0
[(1205734
infin
sum119899=0
119875119899119867119868119899(120591) minus 120573
5
infin
sum119899=0
119875119899119881119878119899(120591)) 119889120591]
infin
sum119899=0
119875119899119881119868119899(119905) minus 119881
119868(0)
= 119875int119905
0
[(1205735
infin
sum119899=0
119875119899119881119878119899(120591) minus 120574
infin
sum119899=0
119875119899119881119868119899(120591)) 119889120591]
(30)
We compared the terms of the same power of 119875 and obtainedthe following
1198750 1198671198780(119905) = 119867
119878(0)
1198750 1198671198640(119905) = 119867
119864(0)
1198750 1198671198680(119905) = 119867
119868(0)
1198750 1198811198780(119905) = 119881
119878(0)
1198750 1198811198680(119905) = 119881
119868(0)
1198751 1198671198781(119905) = int
119905
0
(120588120577 + 1205741198811198680(120591) minus (120573
1+ 1205732)1198671198780(120591)) 119889120591
1198751 1198671198641(119905) = int
119905
0
(120575120582 + 12057311198671198780(120591) minus 120573
31198671198640(120591)) 119889120591
1198751 1198671198681(119905)
= int119905
0
(12057321198671198780(120591)+120573
31198671198640(120591)minus120588120577minus 120575120582 minus120573
41198671198680(120591)) 119889120591
1198751 1198811198781(119905) = int
119905
0
(12057341198671198680(120591) minus 120573
51198811198780(120591)) 119889120591
1198751 1198811198681(119905) = int
119905
0
(12057351198811198780(120591) minus 120574119881
1198680(120591)) 119889120591
(31)
Continuing for 119899 ge 2 we have
119867119878119899(119905) = int
119905
0
(120574119881119868(119899minus1)
(120591) minus (1205731+ 1205732)119867119878(119899minus1)
(120591)) 119889120591
119867119864119899(119905) = int
119905
0
(1205731119867119878(119899minus1)
(120591) minus 1205733119867119864(119899minus1)
(120591)) 119889120591
119867119868119899(119905) = int
119905
0
(1205732119867119878(119899minus1)
(120591) + 1205733119867119864(119899minus1)
(120591)
minus 1205734119867119868(119899minus1)
(120591)) 119889120591
119881119878119899(119905) = int
119905
0
(1205734119867119868(119899minus1)
(120591) minus 1205735119881119878(119899minus1)
(120591)) 119889120591
119881119868119899(119905) = int
119905
0
(1205735119881119878(119899minus1)
(120591) minus 120574119881119868(119899minus1)
(120591)) 119889120591
(32)
We now integrate the above equations to obtain the following
1198671198680(119905) = 0 119867
1198640(119905) = 0 119867
1198780 [119905] = 119873 1198811198780 [119905] = 119872
1198811198680 [119905] = 0 119867
1198681(119905) = 119905 (minus120575120582 minus 120577120588 + 119873120573
2)
1198671198781 [119905] = 119905 (120577120588 minus 119873 (1205731 + 1205732))
1198671198641(119905) = 119905 (120575120582 + 119873120573
1)
1198811198681 [119905] = 1198721199051205735 119881
1198781 [119905] = minus1198721199051205735
1198671198682(119905) =
1
21199052120577120588120573
2minus1
2119873119905212057311205732minus1
2119873119905212057322+1
21199052120575120582120573
3
+1
2119873119905212057311205733+1
21199052120575120582120573
4+1
21199052120577120588120573
4minus1
2119873119905212057321205734
1198671198642(119905) =
1
21199052120577120588120573
1minus1
2119873119905212057321minus1
2119873119905212057311205732
minus1
21199052120575120582120573
3minus1
2119873119905212057311205733
1198671198782 [119905] = minus
1
21199052120577120588120573
1+1
2119873119905212057321minus1
21199052120577120588120573
2
+ 119873119905212057311205732+1
2119873119905212057322+1
21198721199052120574120573
5
1198811198782 [119905] = minus
1
21199052120575120582120573
4minus1
21199052120577120588120573
4
+1
2119873119905212057321205734+1
211987211990521205732
5
1198811198682 [119905] = minus
1
21198721199052120574120573
5minus1
211987211990521205732
5
1198671198683(119905) = minus
1
61199053120577120588120573
11205732+1
61198731199053120573211205732minus1
611990531205771205881205732
2
+1
31198731199053120573112057322+1
6119873119905312057332+1
61199053120577120588120573
11205733
minus1
61198731199053120573211205733minus1
61198731199053120573112057321205733
minus1
611990531205751205821205732
3
minus1
61198731199053120573112057323minus1
61199053120577120588120573
21205734
+1
61198731199053120573112057321205734+1
61198731199053120573221205734
minus1
61199053120575120582120573
31205734minus1
61198731199053120573112057331205734
Abstract and Applied Analysis 7
minus1
611990531205751205821205732
4minus1
611990531205771205881205732
4+1
61198731199053120573212057324
+1
61198721199053120574120573
21205735
1198671198643(119905) = minus
1
611990531205771205881205732
1+1
6119873119905312057331minus1
61199053120577120588120573
11205732
+1
31198731199053120573211205732+1
61198731199053120573112057322
minus1
61199053120577120588120573
11205733+1
61198731199053120573211205733
+1
61198731199053120573112057321205733+1
611990531205751205821205732
3
+1
61198731199053120573112057323+1
61198721199053120574120573
11205735
1198671198783 [119905] =
1
611990531205771205881205732
1minus1
6119873119905312057331+1
31199053120577120588120573
11205732
minus1
21198731199053120573211205732+1
611990531205771205881205732
2minus1
21198731199053120573112057322
minus1
6119873119905312057332minus1
611987211990531205742120573
5
minus1
61198721199053120574120573
11205735
minus1
61198721199053120574120573
21205735minus1
611987211990531205741205732
5
1198811198783 [119905] =
1
61199053120577120588120573
21205734minus1
61198731199053120573112057321205734minus1
61198731199053120573221205734
+1
61199053120575120582120573
31205734+1
61198731199053120573112057331205734+1
611990531205751205821205732
4
+1
611990531205771205881205732
4minus1
61198731199053120573212057324+1
61199053120575120582120573
41205735
+1
61199053120577120588120573
41205735minus1
61198731199053120573212057341205735minus1
611987211990531205733
5
1198811198683 [119905] =
1
611987211990531205742120573
5minus1
61199053120575120582120573
41205735minus1
61199053120577120588120573
41205735
+1
61198731199053120573212057341205735+1
611987211990531205741205732
5+1
611987211990531205733
5
1198671198784 [119905] = minus
1
2411990541205771205881205733
1+1
24119873119905412057341minus1
811990541205771205881205732
11205732
+1
61198731199054120573311205732minus1
81199054120577120588120573
112057322+1
411987311990541205732112057322
minus1
2411990541205771205881205733
2+1
61198731199054120573112057332+1
24119873119905412057342
+1
2411987211990541205743120573
5+1
2411987211990541205742120573
11205735
+1
2411987211990541205741205732
11205735+1
2411987211990541205742120573
21205735
+1
121198721199054120574120573
112057321205735+1
2411987211990541205741205732
21205735
minus1
241199054120574120575120582120573
41205735minus1
241199054120574120577120588120573
41205735
+1
241198731199054120574120573
212057341205735+1
24119872119905412057421205732
5
+1
241198721199054120574120573
112057325+1
241198721199054120574120573
212057325+1
2411987211990541205741205733
5
1198671198644(119905) =
1
2411990541205771205881205733
1minus1
24119873119905412057341+1
1211990541205771205881205732
11205732
minus1
81198731199054120573311205732+1
241199054120577120588120573
112057322minus1
811987311990541205732112057322
minus1
241198731199054120573112057332+1
2411990541205771205881205732
11205733minus1
241198731199054120573311205733
+1
241199054120577120588120573
112057321205733minus1
1211987311990541205732112057321205733minus1
2411987311990541205731120573221205733
+1
241199054120577120588120573
112057323minus1
2411987311990541205732112057323minus1
2411987311990541205731120573212057323
minus1
2411990541205751205821205733
3minus1
241198731199054120573112057333minus1
2411987211990541205742120573
11205735
minus1
2411987211990541205741205732
11205735minus1
241198721199054120574120573
112057321205735
minus1
241198721199054120574120573
112057331205735minus1
241198721199054120574120573
112057325
1198671198684(119905) =
1
2411990541205771205881205732
11205732minus1
241198731199054120573311205732+1
121199054120577120588120573
112057322
minus1
811987311990541205732112057322+1
2411990541205771205881205733
2minus1
81198731199054120573112057332
minus1
24119873119905412057342minus1
2411990541205771205881205732
11205733+1
241198731199054120573311205733
minus1
241199054120577120588120573
112057321205733+1
1211987311990541205732112057321205733
+1
2411987311990541205731120573221205733
minus1
241199054120577120588120573
112057323+1
2411987311990541205732112057323+1
2411987311990541205731120573212057323
+1
2411990541205751205821205733
3+1
241198731199054120573112057333+1
241199054120577120588120573
112057321205734
minus1
2411987311990541205732112057321205734+1
2411990541205771205881205732
21205734minus1
1211987311990541205731120573221205734
minus1
241198731199054120573321205734minus1
241199054120577120588120573
112057331205734
+1
2411987311990541205732112057331205734+1
2411987311990541205731120573212057331205734
+1
2411990541205751205821205732
31205734+1
2411987311990541205731120573231205734
+1
241199054120577120588120573
212057324minus1
2411987311990541205731120573212057324minus1
2411987311990541205732212057324
+1
241199054120575120582120573
312057324+1
2411987311990541205731120573312057324+1
2411990541205751205821205733
4
+1
2411990541205771205881205733
4minus1
241198731199054120573212057334minus1
2411987211990541205742120573
21205735
minus1
241198721199054120574120573
112057321205735minus1
2411987211990541205741205732
21205735
+1
241198721199054120574120573
112057331205735
minus1
241198721199054120574120573
212057341205735minus1
241198721199054120574120573
212057325
1198811198784 [119905] = minus
1
241199054120577120588120573
112057321205734+1
2411987311990541205732112057321205734
minus1
2411990541205771205881205732
21205734+1
1211987311990541205731120573221205734
8 Abstract and Applied Analysis
+1
241198731199054120573321205734+1
241199054120577120588120573
112057331205734
minus1
2411987311990541205732112057331205734minus1
2411987311990541205731120573212057331205734
minus1
2411990541205751205821205732
31205734
minus1
2411987311990541205731120573231205734minus1
241199054120577120588120573
212057324
+1
2411987311990541205731120573212057324+1
2411987311990541205732212057324
minus1
241199054120575120582120573
312057324minus1
2411987311990541205731120573312057324
minus1
2411990541205751205821205733
4minus1
2411990541205771205881205733
4
+1
241198731199054120573212057334
+1
241198721199054120574120573
212057341205735minus1
241199054120577120588120573
212057341205735
+1
2411987311990541205731120573212057341205735+1
2411987311990541205732212057341205735
minus1
241199054120575120582120573
312057341205735minus1
2411987311990541205731120573312057341205735
minus1
2411990541205751205821205732
41205735minus1
2411990541205771205881205732
41205735+1
2411987311990541205732120573241205735
minus1
241199054120575120582120573
412057325minus1
241199054120577120588120573
412057325
+1
2411987311990541205732120573412057325+1
2411987211990541205734
5
1198811198684 [119905] = minus
1
2411987211990541205743120573
5+1
241199054120574120575120582120573
41205735+1
241199054120574120577120588120573
41205735
minus1
241198731199054120574120573
212057341205735+1
241199054120577120588120573
212057341205735
minus1
2411987311990541205731120573212057341205735minus1
2411987311990541205732212057341205735
+1
241199054120575120582120573
312057341205735+1
2411987311990541205731120573312057341205735
+1
2411990541205751205821205732
41205735+1
2411990541205771205881205732
41205735
minus1
2411987311990541205732120573241205735minus1
24119872119905412057421205732
5
+1
241199054120575120582120573
412057325+1
241199054120577120588120573
412057325
minus1
2411987311990541205732120573412057325minus1
2411987211990541205741205733
5minus1
2411987211990541205734
5
1198671198785 [119905] =
1
12011990551205771205881205734
1minus
1
120119873119905512057351+1
3011990551205771205881205733
11205732
minus1
241198731199055120573411205732+1
2011990551205771205881205732
112057322minus1
1211987311990551205733112057322
+1
301199055120577120588120573
112057332minus1
1211987311990551205732112057332+
1
12011990551205771205881205734
2
minus1
241198731199055120573112057342minus
1
120119873119905512057352minus
1
12011987211990551205744120573
5
minus1
12011987211990551205743120573
11205735minus
1
120119872119905512057421205732
11205735
minus1
12011987211990551205741205733
11205735minus
1
12011987211990551205743120573
21205735
minus1
6011987211990551205742120573
112057321205735minus1
4011987211990551205741205732
112057321205735
minus1
120119872119905512057421205732
21205735minus1
401198721199055120574120573
1120573221205735
minus1
12011987211990551205741205733
21205735+
1
12011990551205742120575120582120573
41205735
+1
12011990551205742120577120588120573
41205735+
1
1201199055120574120575120582120573
112057341205735
+1
1201199055120574120577120588120573
112057341205735minus
1
12011987311990551205742120573
212057341205735
+1
1201199055120574120575120582120573
212057341205735+1
601199055120574120577120588120573
212057341205735
minus1
601198731199055120574120573
1120573212057341205735minus1
6011987311990551205741205732
212057341205735
+1
1201199055120574120575120582120573
312057341205735+
1
1201198731199055120574120573
1120573312057341205735
+1
12011990551205741205751205821205732
41205735+
1
12011990551205741205771205881205732
41205735
minus1
1201198731199055120574120573
2120573241205735minus
1
120119872119905512057431205732
5
minus1
12011987211990551205742120573
112057325minus
1
12011987211990551205741205732
112057325
minus1
12011987211990551205742120573
212057325minus1
601198721199055120574120573
1120573212057325
minus1
12011987211990551205741205732
212057325+
1
1201199055120574120575120582120573
412057325
+1
1201199055120574120577120588120573
412057325minus
1
1201198731199055120574120573
2120573412057325
minus1
120119872119905512057421205733
5minus
1
1201198721199055120574120573
112057335
minus1
1201198721199055120574120573
212057335minus
1
12011987211990551205741205734
5
1198671198645 [119905] = minus
1
12011990551205771205881205734
1+
1
120119873119905512057351
minus1
4011990551205771205881205733
11205732+1
301198731199055120573411205732
minus1
4011990551205771205881205732
112057322+1
2011987311990551205733112057322
minus1
1201199055120577120588120573
112057332+1
3011987311990551205732112057332
+1
1201198731199055120573112057342
minus1
12011990551205771205881205733
11205733+
1
1201198731199055120573411205733
minus1
6011990551205771205881205732
112057321205733+1
4011987311990551205733112057321205733
minus1
1201199055120577120588120573
1120573221205733+1
40119873119905512057321120573221205733
+1
12011987311990551205731120573321205733minus
1
12011990551205771205881205732
112057323
+1
12011987311990551205733112057323minus
1
1201199055120577120588120573
1120573212057323
Abstract and Applied Analysis 9
+1
60119873119905512057321120573212057323+
1
120119873119905512057311205732212057323
minus1
1201199055120577120588120573
112057333+
1
12011987311990551205732112057333
+1
12011987311990551205731120573212057333+
1
12011990551205751205821205734
3
+1
1201198731199055120573112057343+
1
12011987211990551205743120573
11205735
+1
120119872119905512057421205732
11205735+
1
12011987211990551205741205733
11205735
+1
12011987211990551205742120573
112057321205735+1
6011987211990551205741205732
112057321205735
+1
1201198721199055120574120573
1120573221205735+
1
12011987211990551205742120573
112057331205735
+1
12011987211990551205741205732
112057331205735+
1
1201198721199055120574120573
1120573212057331205735
+1
1201198721199055120574120573
1120573231205735minus
1
1201199055120574120575120582120573
112057341205735
minus1
1201199055120574120577120588120573
112057341205735+
1
1201198731199055120574120573
1120573212057341205735
+1
12011987211990551205742120573
112057325+
1
12011987211990551205741205732
112057325
+1
1201198721199055120574120573
1120573212057325+
1
1201198721199055120574120573
1120573312057325
+1
1201198721199055120574120573
112057335
1198671198685 [119905]= minus
1
12011990551205771205881205733
11205732+
1
1201198731199055120573411205732minus1
4011990551205771205881205732
112057322
+1
3011987311990551205733112057322minus1
401199055120577120588120573
112057332+1
2011987311990551205732112057332
minus1
12011990551205771205881205734
2+1
301198731199055120573112057342+
1
120119873119905512057352
+1
12011990551205771205881205733
11205733minus
1
1201198731199055120573411205733
+1
6011990551205771205881205732
112057321205733minus1
4011987311990551205733112057321205733
+1
1201199055120577120588120573
1120573221205733minus1
40119873119905512057321120573221205733
minus1
12011987311990551205731120573321205733+
1
12011990551205771205881205732
112057323
minus1
12011987311990551205733112057323+
1
1201199055120577120588120573
1120573212057323
minus1
60119873119905512057321120573212057323minus
1
120119873119905512057311205732212057323
+1
1201199055120577120588120573
112057333minus
1
12011987311990551205732112057333
minus1
12011987311990551205731120573212057333minus
1
12011990551205751205821205734
3
minus1
1201198731199055120573112057343minus
1
12011990551205771205881205732
112057321205734
+1
12011987311990551205733112057321205734minus1
601199055120577120588120573
1120573221205734
+1
40119873119905512057321120573221205734minus
1
12011990551205771205881205733
21205734
+1
4011987311990551205731120573321205734+
1
1201198731199055120573421205734
+1
12011990551205771205881205732
112057331205734minus
1
12011987311990551205733112057331205734
+1
1201199055120577120588120573
1120573212057331205734minus1
60119873119905512057321120573212057331205734
minus1
120119873119905512057311205732212057331205734+
1
1201199055120577120588120573
1120573231205734
minus1
120119873119905512057321120573231205734minus
1
120119873119905512057311205732120573231205734
minus1
12011990551205751205821205733
31205734minus
1
12011987311990551205731120573331205734
minus1
1201199055120577120588120573
1120573212057324+
1
120119873119905512057321120573212057324
minus1
12011990551205771205881205732
212057324+1
60119873119905512057311205732212057324
+1
12011987311990551205733212057324+
1
1201199055120577120588120573
1120573312057324
minus1
120119873119905512057321120573312057324minus
1
120119873119905512057311205732120573312057324
minus1
12011990551205751205821205732
312057324minus
1
120119873119905512057311205732312057324
minus1
1201199055120577120588120573
212057334+
1
12011987311990551205731120573212057334
+1
12011987311990551205732212057334minus
1
1201199055120575120582120573
312057334
minus1
12011987311990551205731120573312057334minus
1
12011990551205751205821205734
4
minus1
12011990551205771205881205734
4+
1
1201198731199055120573212057344
+1
12011987211990551205743120573
21205735+
1
12011987211990551205742120573
112057321205735
+1
12011987211990551205741205732
112057321205735+
1
120119872119905512057421205732
21205735
+1
601198721199055120574120573
1120573221205735+
1
12011987211990551205741205733
21205735
minus1
12011987211990551205742120573
112057331205735minus
1
12011987211990551205741205732
112057331205735
minus1
1201198721199055120574120573
1120573212057331205735minus
1
1201198721199055120574120573
1120573231205735
+1
12011987211990551205742120573
212057341205735minus
1
1201199055120574120575120582120573
212057341205735
minus1
1201199055120574120577120588120573
212057341205735+
1
1201198721199055120574120573
1120573212057341205735
+1
12011987211990551205741205732
212057341205735+
1
12011987311990551205741205732
212057341205735
minus1
1201198721199055120574120573
1120573312057341205735+
1
1201198721199055120574120573
2120573241205735
+1
12011987211990551205742120573
212057325+
1
1201198721199055120574120573
1120573212057325
+1
12011987211990551205741205732
212057325minus
1
1201198721199055120574120573
1120573312057325
+1
1201198721199055120574120573
2120573412057325+
1
1201198721199055120574120573
212057335
10 Abstract and Applied Analysis
1198811198785 [119905] =
1
12011990551205771205881205732
112057321205734minus
1
12011987311990551205733112057321205734
+1
601199055120577120588120573
1120573221205734minus1
40119873119905512057321120573221205734
+1
12011990551205771205881205733
21205734minus1
4011987311990551205731120573321205734
minus1
1201198731199055120573421205734minus
1
12011990551205771205881205732
112057331205734
+1
12011987311990551205733112057331205734minus
1
1201199055120577120588120573
1120573212057331205734
+1
60119873119905512057321120573212057331205734+
1
120119873119905512057311205732212057331205734
minus1
1201199055120577120588120573
1120573231205734+
1
120119873119905512057321120573231205734
+1
120119873119905512057311205732120573231205734+
1
12011990551205751205821205733
31205734
+1
12011987311990551205731120573331205734+
1
1201199055120577120588120573
1120573212057324
minus1
120119873119905512057321120573212057324+
1
12011990551205771205881205732
212057324
minus1
60119873119905512057311205732212057324minus
1
12011987311990551205733212057324
minus1
1201199055120577120588120573
1120573312057324+
1
120119873119905512057321120573312057324
+1
120119873119905512057311205732120573312057324+
1
12011990551205751205821205732
312057324
+1
120119873119905512057311205732312057324+
1
1201199055120577120588120573
212057334
minus1
12011987311990551205731120573212057334minus
1
12011987311990551205732212057334
+1
1201199055120575120582120573
312057334+
1
12011987311990551205731120573312057334
+1
12011990551205751205821205734
4+
1
12011990551205771205881205734
4minus
1
1201198731199055120573212057344
minus1
12011987211990551205742120573
212057341205735minus
1
1201198721199055120574120573
1120573212057341205735
+1
1201199055120577120588120573
1120573212057341205735minus
1
120119873119905512057321120573212057341205735
minus1
12011987211990551205741205732
212057341205735+
1
12011990551205771205881205732
212057341205735
minus1
60119873119905512057311205732212057341205735minus
1
12011987311990551205733212057341205735
+1
1201198721199055120574120573
1120573312057341205735minus
1
1201199055120577120588120573
1120573312057341205735
+1
120119873119905512057321120573312057341205735+
1
120119873119905512057311205732120573312057341205735
+1
12011990551205751205821205732
312057341205735+
1
120119873119905512057311205732312057341205735
minus1
1201198721199055120574120573
2120573241205735+
1
1201199055120577120588120573
2120573241205735
minus1
120119873119905512057311205732120573241205735minus
1
120119873119905512057322120573241205735
+1
1201199055120575120582120573
3120573241205735+
1
120119873119905512057311205733120573241205735
+1
12011990551205751205821205733
41205735+
1
12011990551205771205881205733
41205735
minus1
12011987311990551205732120573341205735minus1
601198721199055120574120573
2120573412057325
+1
1201199055120577120588120573
2120573412057325minus
1
120119873119905512057311205732120573412057325
minus1
120119873119905512057322120573412057325+
1
1201199055120575120582120573
3120573412057325
+1
120119873119905512057311205733120573412057325+
1
12011990551205751205821205732
412057325
+1
12011990551205771205881205732
412057325minus
1
120119873119905512057321205732412057325
+1
1201199055120575120582120573
412057335+
1
1201199055120577120588120573
412057335
minus1
12011987311990551205732120573412057335minus
1
12011987211990551205735
5
1198811198685 [119905] =
1
12011987211990551205744120573
5minus
1
12011990551205742120575120582120573
41205735
minus1
12011990551205742120577120588120573
41205735+
1
12011987311990551205742120573
212057341205735
minus1
1201199055120574120577120588120573
212057341205735+
1
1201198731199055120574120573
1120573212057341205735
minus1
1201199055120577120588120573
1120573212057341205735+
1
120119873119905512057321120573212057341205735
+1
12011987311990551205741205732
212057341205735minus
1
12011990551205771205881205732
212057341205735
+1
60119873119905512057311205732212057341205735+
1
12011987311990551205733212057341205735
minus1
1201199055120574120575120582120573
312057341205735minus
1
1201198731199055120574120573
1120573312057341205735
+1
1201199055120577120588120573
1120573312057341205735minus
1
120119873119905512057321120573312057341205735
minus1
120119873119905512057311205732120573312057341205735minus
1
12011990551205751205821205732
312057341205735
minus1
120119873119905512057311205732312057341205735minus
1
12011990551205741205751205821205732
41205735
minus1
12011990551205741205771205881205732
41205735+
1
1201198731199055120574120573
2120573241205735
minus1
1201199055120577120588120573
2120573241205735+
1
120119873119905512057311205732120573241205735
+1
120119873119905512057322120573241205735minus
1
1201199055120575120582120573
3120573241205735
minus1
120119873119905512057311205733120573241205735minus
1
12011990551205751205821205733
41205735
minus1
12011990551205771205881205733
41205735+
1
12011987311990551205732120573341205735
+1
120119872119905512057431205732
5minus
1
1201199055120574120575120582120573
412057325
minus1
1201199055120574120577120588120573
412057325+
1
1201198721199055120574120573
2120573412057325
+1
1201198731199055120574120573
2120573412057325minus
1
1201199055120577120588120573
2120573412057325
+1
120119873119905512057311205732120573412057325+
1
120119873119905512057322120573412057325
Abstract and Applied Analysis 11
minus1
1201199055120575120582120573
3120573412057325minus
1
120119873119905512057311205733120573412057325
minus1
12011990551205751205821205732
412057325minus
1
12011990551205771205881205732
412057325
+1
120119873119905512057321205732412057325+
1
120119872119905512057421205733
5
minus1
1201199055120575120582120573
412057335minus
1
1201199055120577120588120573
412057335
+1
12011987311990551205732120573412057335+
1
12011987211990551205741205734
5
+1
12011987211990551205735
5
(33)
In general by the repeated application of the above proce-dure we have that
119867119878119899=119905119899
1198991198861119899 119867
119864119899=119905119899
1198991198862119899 119867
119868119899=119905119899
1198991198863119899
119881119878119899=119905119899
1198991198864119899 119881
119868119899=119905119899
1198991198865119899
(34)
where 1198861119899 1198862119899 1198863119899 1198864119899 1198865119899
depend on fixed set of empiricalparameters Hence the approximate solution of the modelabove is given as
119867119878(119905) =
infin
sum119899=0
119905119899
1198991198861119899
119867119864(119905) =
infin
sum119899=0
119905119899
1198991198862119899
119867119868(119905) =
infin
sum119899=0
119905119899
1198991198863119899
119881119878(119905) =
infin
sum119899=0
119905119899
1198991198864119899
119881119868(119905) =
infin
sum119899=0
119905119899
1198991198865119899
(35)
We therefore use the following theoretical parameters 1205731=
05 1205732= 05 120573
3= 05 120573
4= 05 120573
5= 05 120574 = 1 120577 = 1
119873 = 10 119872 = 10 120588 = 1 120575 = 1 and 120582 = 1 to examinethe behavior of the five different compartmental numericalsolutions in the system Figure 3 shows the behavior of thenumerical solution
7 Conclusion
Onchocerciasis model was examined and the mathematicalformulation underpinning the spread and control of thisdisease has been proposed on one hand On the otherhand we made use of some new analytical methods calledthe homotopy decomposition method (HDM) to derive thesolution of the resulting set of (1) to (5)The numerical resultswere presented to test the efficiency and the accuracy of themethod used The mathematical analysis of the resulting setof (1) to (5) above revealed that the stability analysis of the
05 10 15 20Time (per year)
2
4
6
8
10
Appr
oxim
ate s
olut
ions VS
HS
VI
HI
HE
Figure 3 Numerical solution of the model
five roots of the characteristics equations has all negativereal part In particular given that the stability analysis showsthat the DFE is asymptotically stable it would be possiblethat the parasite Onchocerca volvulus be eradicated in a finitetime On the other hand the numerical solution confirmedthat as the number of humans and vectors in the susceptiblecompartments reduces the number of infective humansand vectors in the infectious compartment increases Thenumerical solutions in both cases displayed the biologicalbehaviour of the real world situation Therefore in orderto build a healthy and a sustainable population treatmentwith IvermectinMectizan drugs and a more hygienic envi-ronment that does not allow the breeding and survival ofblackflies could lead to a total eradication of the disease
Conflict of Interests
The authors declare that they have no conflict of interestsregarding the publication of this paper
References
[1] J H F Remme The Global Burden of Onchocerciasis in 1990World Health Organization Geneva Switzerland 2004 httpwwwwhointhealthinfoglobal burden diseaseOnchocercia-sis201990pdf
[2] WorldHealthOrganization ldquoPrevention of blindness and visualimpairment priority eye diseasesrdquo 2014 httpwwwwhointblindnesscausespriorityenindex3html
[3] World Health Organization Certification of Elimination ofHuman Onchocerciasis Criteria and Procedures World HealthOrganizationGeneva Switzerland 2001 httpwhqlibdocwhointhq2001WHO CDS CPE CEE 200118bpdf
[4] K A Lindblade B Arana G Zea-Flores et al ldquoEliminationof Onchocerca volvulus transmission in the Santa Rosa focusof Guatemalardquo The American Journal of Tropical Medicine andHygiene vol 77 pp 334ndash341 2007
[5] J C Anosike and C O E Onwuliri ldquoStudies on filariasis inBauchi State Nigeria 1 Endemicity of human onchocerciasisin Ningi Local Government Areardquo Annals of Tropical Medicineand Parasitology vol 89 no 1 pp 31ndash38 1995
[6] WorldHealthOrganizationWeekly Epidemiological Records 71WHO Geneva Switzerland 1993
12 Abstract and Applied Analysis
[7] WorldHealthOrganization httpwwwwhointneglected dis-easesen
[8] Sightsaver httpwwwsightsaversorgabout uspublicationsonchocerciasis
[9] E C Uttah ldquoOnchocerciasis in the Upper Ima River BasinrdquoIranian Journal of Parasitology vol 5 no 2 pp 33ndash41 2010
[10] J C Anosike and C O E Onwuliri ldquoStudies on filariasis inBauchi State Nigeriamdash1 endemicity of human onchocerciasisin Ningi Local Government Areardquo Annals of Tropical Medicineand Parasitology vol 89 no 1 pp 31ndash38 1995
[11] L D Edungbola and S O Asaolu ldquoParasitologic survey ofonchocerciasis (river blindness) in Babana district Kwara StateNigeriardquo American Journal of Tropical Medicine and Hygienevol 33 no 6 pp 1147ndash1154 1984
[12] O C Nwaorgu A Ohaegbula I E Onweluzo et al ldquoResultsof a large scale onchocercosis survey in Enugu State NigeriardquoJournal of Helminthology vol 68 no 2 pp 155ndash159 1994
[13] F O Akinbo and C E Okaka ldquoPrevalence and socio-economiceffects of onchocerciasis in Ovia North East Local GovernmentArea of Edo State Nigeriardquo Rivista Di Parasitologie vol 22 no3 pp 215ndash221 2005
[14] M D Wogu and C E Okaka ldquoPrevalence and socio-economic effects of onchocerciasis in Okpuje Owan WestLocal Government Area Edo State Nigeriardquo InternationalJournal of Biomedical and Health Sciences vol 4 no 3 2008httpwwwasopahorg
[15] S C O Noutchie ldquoHomotopy Decomposition analysis of atuberculosis modelrdquo Life Science Journal vol 10 no 3 pp 459ndash465 2013
[16] A Atangana and A Secer ldquoThe time-fractional coupled-Korteweg-de-Vries equationsrdquo Abstract and Applied Analysisvol 2013 Article ID 947986 8 pages 2013
[17] A Atangana and N Bildik ldquoApproximate solution of tubercu-losis disease population dynamics modelrdquo Abstract and AppliedAnalysis vol 2013 Article ID 759801 8 pages 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 5
minus (1205731+ 1205732+ 120582)
[[
[
minus (1205733+ 120582)
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
minus (1205734+ 120582) 0 0
1205734
minus (1205735+ 120582) 0
0 1205735
minus (120574 + 120582)
1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816
]]
]
+ 120574[[
[
1205731
[[
[
1205733minus (1205734+ 120582) 0
0 1205734
minus (1205735+ 120582)
0 0 1205735
]]
]
+ (1205733+ 120582)
[[
[
1205732minus (1205734+ 120582) 0
0 1205734
minus (1205735+ 120582)
0 0 1205735
]]
]
]]
]
= 0
minus (1205731+ 1205732+ 120582) [minus (120573
3+ 120582) [minus (120573
4+ 120582)
100381610038161003816100381610038161003816100381610038161003816
minus (1205735+ 120582) 0
1205735
minus (120574 + 120582)
100381610038161003816100381610038161003816100381610038161003816]] + 120574120573
1120573312057341205735+ 120574 (120573
3+ 120582) 120573
212057341205735= 0
minus [(1205731+ 1205732+ 120582) (120573
3+ 120582) (120573
4+ 120582) (120573
5+ 120582) (120574 + 120582)] + 120574120573
1120573312057341205735+ 1205741205732120573312057341205735+ 120574120582120573
212057341205735= 0
(22)
Hence we obtain from minus(1205731+ 1205732+ 120582) = 0
1205821= minus (120573
1+ 1205732) (23)
From minus(1205733+ 120582) = 0
1205822= minus1205733 (24)
From minus(1205734+ 120582) = 0
1205823= minus1205734 (25)
From (1205735+ 120582) = 0
1205824= minus1205735 (26)
From (120574 + 120582) = 0
1205825= minus120574 (27)
Given that the results of 1205821 1205822 1205823 1205824 and 120582
5are all negative
and considering the principles of linearized stability whichsays that the systems are asymptotically stable if and only ifall the eigenvalues of the system have negative real parts wesay that the disease-free equilibrium (DFE) is asymptoticallystable
6 Application of the HomotopyDecomposition Method (HDM)
In this section we apply the HDM [15ndash17] to solve thesemathematical sets of equations representing the onchocerci-asis disease model as follows
119867119878(119905) minus 119867
119878(0) = int
119905
0
(120588120577 + 120574119881119868(120591) minus (120573
1+ 1205732)119867119878(120591)) 119889120591
119867119864(119905) minus 119867
119864(0) = int
119905
0
(120575120582 + 1205731119867119878(120591) minus 120573
3119867119864(120591)) 119889120591
119867119868(119905) minus 119867
119868(0)
= int119905
0
(1205732119867119878(120591) + 120573
3119867119864(120591) minus 120588120577 minus 120575120582 minus 120573
4119867119868(120591)) 119889120591
119881119878(119905) minus 119881
119878(0) = int
119905
0
(1205734119867119868(120591) minus 120573
5119881119878(120591)) 119889120591
119881119868(119905) minus 119881
119868(0) = int
119905
0
(1205735119881119878(120591) minus 120574119881
119868(120591)) 119889120591
(28)
We then assume that the solutions of the above integralequations can be put in the following form for 119875 isin (0 1)
119867119878(119905) =
infin
sum119899=0
119875119899119867119878119899(119905) 119867
119864(119905) =
infin
sum119899=0
119875119899119867119864119899(119905)
119867119868(119905) =
infin
sum119899=0
119875119899119867119868119899(119905) 119881
119878(119905) =
infin
sum119899=0
119875119899119881119878119899(119905)
119881119868(119905) =
infin
sum119899=0
119875119899119881119868119899(119905)
(29)
Replacing the above expressions in the integral equations weobtain the following expressions
infin
sum119899=0
119875119899119867119878119899(119905) minus 119867
119878(0)
= 119875int119905
0
[(120588120577 + 120574infin
sum119899=0
119875119899119881119868119899(120591)
6 Abstract and Applied Analysis
minus (1205731+ 1205732)infin
sum119899=0
119875119899119867119878119899(120591)) 119889120591]
infin
sum119899=0
119875119899119867119864119899(119905) minus 119867
119864(0)
= 119875int119905
0
[(120575120582 + 1205731
infin
sum119899=0
119875119899119867119878119899(120591)
minus 1205733
infin
sum119899=0
119875119899119867119864119899(120591)) 119889120591]
infin
sum119899=0
119875119899119867119868119899(119905) minus 119867
119868(0)
= 119875int119905
0
[(1205732
infin
sum119899=0
119875119899119867119878119899(120591) + 120573
3
infin
sum119899=0
119875119899119867119864119899(120591)
minus 120588120577 minus 120575120582 minus 1205734
infin
sum119899=0
119875119899119867119868119899(120591)) 119889120591]
infin
sum119899=0
119875119899119881119878119899(119905) minus 119881
119878(0)
= 119875int119905
0
[(1205734
infin
sum119899=0
119875119899119867119868119899(120591) minus 120573
5
infin
sum119899=0
119875119899119881119878119899(120591)) 119889120591]
infin
sum119899=0
119875119899119881119868119899(119905) minus 119881
119868(0)
= 119875int119905
0
[(1205735
infin
sum119899=0
119875119899119881119878119899(120591) minus 120574
infin
sum119899=0
119875119899119881119868119899(120591)) 119889120591]
(30)
We compared the terms of the same power of 119875 and obtainedthe following
1198750 1198671198780(119905) = 119867
119878(0)
1198750 1198671198640(119905) = 119867
119864(0)
1198750 1198671198680(119905) = 119867
119868(0)
1198750 1198811198780(119905) = 119881
119878(0)
1198750 1198811198680(119905) = 119881
119868(0)
1198751 1198671198781(119905) = int
119905
0
(120588120577 + 1205741198811198680(120591) minus (120573
1+ 1205732)1198671198780(120591)) 119889120591
1198751 1198671198641(119905) = int
119905
0
(120575120582 + 12057311198671198780(120591) minus 120573
31198671198640(120591)) 119889120591
1198751 1198671198681(119905)
= int119905
0
(12057321198671198780(120591)+120573
31198671198640(120591)minus120588120577minus 120575120582 minus120573
41198671198680(120591)) 119889120591
1198751 1198811198781(119905) = int
119905
0
(12057341198671198680(120591) minus 120573
51198811198780(120591)) 119889120591
1198751 1198811198681(119905) = int
119905
0
(12057351198811198780(120591) minus 120574119881
1198680(120591)) 119889120591
(31)
Continuing for 119899 ge 2 we have
119867119878119899(119905) = int
119905
0
(120574119881119868(119899minus1)
(120591) minus (1205731+ 1205732)119867119878(119899minus1)
(120591)) 119889120591
119867119864119899(119905) = int
119905
0
(1205731119867119878(119899minus1)
(120591) minus 1205733119867119864(119899minus1)
(120591)) 119889120591
119867119868119899(119905) = int
119905
0
(1205732119867119878(119899minus1)
(120591) + 1205733119867119864(119899minus1)
(120591)
minus 1205734119867119868(119899minus1)
(120591)) 119889120591
119881119878119899(119905) = int
119905
0
(1205734119867119868(119899minus1)
(120591) minus 1205735119881119878(119899minus1)
(120591)) 119889120591
119881119868119899(119905) = int
119905
0
(1205735119881119878(119899minus1)
(120591) minus 120574119881119868(119899minus1)
(120591)) 119889120591
(32)
We now integrate the above equations to obtain the following
1198671198680(119905) = 0 119867
1198640(119905) = 0 119867
1198780 [119905] = 119873 1198811198780 [119905] = 119872
1198811198680 [119905] = 0 119867
1198681(119905) = 119905 (minus120575120582 minus 120577120588 + 119873120573
2)
1198671198781 [119905] = 119905 (120577120588 minus 119873 (1205731 + 1205732))
1198671198641(119905) = 119905 (120575120582 + 119873120573
1)
1198811198681 [119905] = 1198721199051205735 119881
1198781 [119905] = minus1198721199051205735
1198671198682(119905) =
1
21199052120577120588120573
2minus1
2119873119905212057311205732minus1
2119873119905212057322+1
21199052120575120582120573
3
+1
2119873119905212057311205733+1
21199052120575120582120573
4+1
21199052120577120588120573
4minus1
2119873119905212057321205734
1198671198642(119905) =
1
21199052120577120588120573
1minus1
2119873119905212057321minus1
2119873119905212057311205732
minus1
21199052120575120582120573
3minus1
2119873119905212057311205733
1198671198782 [119905] = minus
1
21199052120577120588120573
1+1
2119873119905212057321minus1
21199052120577120588120573
2
+ 119873119905212057311205732+1
2119873119905212057322+1
21198721199052120574120573
5
1198811198782 [119905] = minus
1
21199052120575120582120573
4minus1
21199052120577120588120573
4
+1
2119873119905212057321205734+1
211987211990521205732
5
1198811198682 [119905] = minus
1
21198721199052120574120573
5minus1
211987211990521205732
5
1198671198683(119905) = minus
1
61199053120577120588120573
11205732+1
61198731199053120573211205732minus1
611990531205771205881205732
2
+1
31198731199053120573112057322+1
6119873119905312057332+1
61199053120577120588120573
11205733
minus1
61198731199053120573211205733minus1
61198731199053120573112057321205733
minus1
611990531205751205821205732
3
minus1
61198731199053120573112057323minus1
61199053120577120588120573
21205734
+1
61198731199053120573112057321205734+1
61198731199053120573221205734
minus1
61199053120575120582120573
31205734minus1
61198731199053120573112057331205734
Abstract and Applied Analysis 7
minus1
611990531205751205821205732
4minus1
611990531205771205881205732
4+1
61198731199053120573212057324
+1
61198721199053120574120573
21205735
1198671198643(119905) = minus
1
611990531205771205881205732
1+1
6119873119905312057331minus1
61199053120577120588120573
11205732
+1
31198731199053120573211205732+1
61198731199053120573112057322
minus1
61199053120577120588120573
11205733+1
61198731199053120573211205733
+1
61198731199053120573112057321205733+1
611990531205751205821205732
3
+1
61198731199053120573112057323+1
61198721199053120574120573
11205735
1198671198783 [119905] =
1
611990531205771205881205732
1minus1
6119873119905312057331+1
31199053120577120588120573
11205732
minus1
21198731199053120573211205732+1
611990531205771205881205732
2minus1
21198731199053120573112057322
minus1
6119873119905312057332minus1
611987211990531205742120573
5
minus1
61198721199053120574120573
11205735
minus1
61198721199053120574120573
21205735minus1
611987211990531205741205732
5
1198811198783 [119905] =
1
61199053120577120588120573
21205734minus1
61198731199053120573112057321205734minus1
61198731199053120573221205734
+1
61199053120575120582120573
31205734+1
61198731199053120573112057331205734+1
611990531205751205821205732
4
+1
611990531205771205881205732
4minus1
61198731199053120573212057324+1
61199053120575120582120573
41205735
+1
61199053120577120588120573
41205735minus1
61198731199053120573212057341205735minus1
611987211990531205733
5
1198811198683 [119905] =
1
611987211990531205742120573
5minus1
61199053120575120582120573
41205735minus1
61199053120577120588120573
41205735
+1
61198731199053120573212057341205735+1
611987211990531205741205732
5+1
611987211990531205733
5
1198671198784 [119905] = minus
1
2411990541205771205881205733
1+1
24119873119905412057341minus1
811990541205771205881205732
11205732
+1
61198731199054120573311205732minus1
81199054120577120588120573
112057322+1
411987311990541205732112057322
minus1
2411990541205771205881205733
2+1
61198731199054120573112057332+1
24119873119905412057342
+1
2411987211990541205743120573
5+1
2411987211990541205742120573
11205735
+1
2411987211990541205741205732
11205735+1
2411987211990541205742120573
21205735
+1
121198721199054120574120573
112057321205735+1
2411987211990541205741205732
21205735
minus1
241199054120574120575120582120573
41205735minus1
241199054120574120577120588120573
41205735
+1
241198731199054120574120573
212057341205735+1
24119872119905412057421205732
5
+1
241198721199054120574120573
112057325+1
241198721199054120574120573
212057325+1
2411987211990541205741205733
5
1198671198644(119905) =
1
2411990541205771205881205733
1minus1
24119873119905412057341+1
1211990541205771205881205732
11205732
minus1
81198731199054120573311205732+1
241199054120577120588120573
112057322minus1
811987311990541205732112057322
minus1
241198731199054120573112057332+1
2411990541205771205881205732
11205733minus1
241198731199054120573311205733
+1
241199054120577120588120573
112057321205733minus1
1211987311990541205732112057321205733minus1
2411987311990541205731120573221205733
+1
241199054120577120588120573
112057323minus1
2411987311990541205732112057323minus1
2411987311990541205731120573212057323
minus1
2411990541205751205821205733
3minus1
241198731199054120573112057333minus1
2411987211990541205742120573
11205735
minus1
2411987211990541205741205732
11205735minus1
241198721199054120574120573
112057321205735
minus1
241198721199054120574120573
112057331205735minus1
241198721199054120574120573
112057325
1198671198684(119905) =
1
2411990541205771205881205732
11205732minus1
241198731199054120573311205732+1
121199054120577120588120573
112057322
minus1
811987311990541205732112057322+1
2411990541205771205881205733
2minus1
81198731199054120573112057332
minus1
24119873119905412057342minus1
2411990541205771205881205732
11205733+1
241198731199054120573311205733
minus1
241199054120577120588120573
112057321205733+1
1211987311990541205732112057321205733
+1
2411987311990541205731120573221205733
minus1
241199054120577120588120573
112057323+1
2411987311990541205732112057323+1
2411987311990541205731120573212057323
+1
2411990541205751205821205733
3+1
241198731199054120573112057333+1
241199054120577120588120573
112057321205734
minus1
2411987311990541205732112057321205734+1
2411990541205771205881205732
21205734minus1
1211987311990541205731120573221205734
minus1
241198731199054120573321205734minus1
241199054120577120588120573
112057331205734
+1
2411987311990541205732112057331205734+1
2411987311990541205731120573212057331205734
+1
2411990541205751205821205732
31205734+1
2411987311990541205731120573231205734
+1
241199054120577120588120573
212057324minus1
2411987311990541205731120573212057324minus1
2411987311990541205732212057324
+1
241199054120575120582120573
312057324+1
2411987311990541205731120573312057324+1
2411990541205751205821205733
4
+1
2411990541205771205881205733
4minus1
241198731199054120573212057334minus1
2411987211990541205742120573
21205735
minus1
241198721199054120574120573
112057321205735minus1
2411987211990541205741205732
21205735
+1
241198721199054120574120573
112057331205735
minus1
241198721199054120574120573
212057341205735minus1
241198721199054120574120573
212057325
1198811198784 [119905] = minus
1
241199054120577120588120573
112057321205734+1
2411987311990541205732112057321205734
minus1
2411990541205771205881205732
21205734+1
1211987311990541205731120573221205734
8 Abstract and Applied Analysis
+1
241198731199054120573321205734+1
241199054120577120588120573
112057331205734
minus1
2411987311990541205732112057331205734minus1
2411987311990541205731120573212057331205734
minus1
2411990541205751205821205732
31205734
minus1
2411987311990541205731120573231205734minus1
241199054120577120588120573
212057324
+1
2411987311990541205731120573212057324+1
2411987311990541205732212057324
minus1
241199054120575120582120573
312057324minus1
2411987311990541205731120573312057324
minus1
2411990541205751205821205733
4minus1
2411990541205771205881205733
4
+1
241198731199054120573212057334
+1
241198721199054120574120573
212057341205735minus1
241199054120577120588120573
212057341205735
+1
2411987311990541205731120573212057341205735+1
2411987311990541205732212057341205735
minus1
241199054120575120582120573
312057341205735minus1
2411987311990541205731120573312057341205735
minus1
2411990541205751205821205732
41205735minus1
2411990541205771205881205732
41205735+1
2411987311990541205732120573241205735
minus1
241199054120575120582120573
412057325minus1
241199054120577120588120573
412057325
+1
2411987311990541205732120573412057325+1
2411987211990541205734
5
1198811198684 [119905] = minus
1
2411987211990541205743120573
5+1
241199054120574120575120582120573
41205735+1
241199054120574120577120588120573
41205735
minus1
241198731199054120574120573
212057341205735+1
241199054120577120588120573
212057341205735
minus1
2411987311990541205731120573212057341205735minus1
2411987311990541205732212057341205735
+1
241199054120575120582120573
312057341205735+1
2411987311990541205731120573312057341205735
+1
2411990541205751205821205732
41205735+1
2411990541205771205881205732
41205735
minus1
2411987311990541205732120573241205735minus1
24119872119905412057421205732
5
+1
241199054120575120582120573
412057325+1
241199054120577120588120573
412057325
minus1
2411987311990541205732120573412057325minus1
2411987211990541205741205733
5minus1
2411987211990541205734
5
1198671198785 [119905] =
1
12011990551205771205881205734
1minus
1
120119873119905512057351+1
3011990551205771205881205733
11205732
minus1
241198731199055120573411205732+1
2011990551205771205881205732
112057322minus1
1211987311990551205733112057322
+1
301199055120577120588120573
112057332minus1
1211987311990551205732112057332+
1
12011990551205771205881205734
2
minus1
241198731199055120573112057342minus
1
120119873119905512057352minus
1
12011987211990551205744120573
5
minus1
12011987211990551205743120573
11205735minus
1
120119872119905512057421205732
11205735
minus1
12011987211990551205741205733
11205735minus
1
12011987211990551205743120573
21205735
minus1
6011987211990551205742120573
112057321205735minus1
4011987211990551205741205732
112057321205735
minus1
120119872119905512057421205732
21205735minus1
401198721199055120574120573
1120573221205735
minus1
12011987211990551205741205733
21205735+
1
12011990551205742120575120582120573
41205735
+1
12011990551205742120577120588120573
41205735+
1
1201199055120574120575120582120573
112057341205735
+1
1201199055120574120577120588120573
112057341205735minus
1
12011987311990551205742120573
212057341205735
+1
1201199055120574120575120582120573
212057341205735+1
601199055120574120577120588120573
212057341205735
minus1
601198731199055120574120573
1120573212057341205735minus1
6011987311990551205741205732
212057341205735
+1
1201199055120574120575120582120573
312057341205735+
1
1201198731199055120574120573
1120573312057341205735
+1
12011990551205741205751205821205732
41205735+
1
12011990551205741205771205881205732
41205735
minus1
1201198731199055120574120573
2120573241205735minus
1
120119872119905512057431205732
5
minus1
12011987211990551205742120573
112057325minus
1
12011987211990551205741205732
112057325
minus1
12011987211990551205742120573
212057325minus1
601198721199055120574120573
1120573212057325
minus1
12011987211990551205741205732
212057325+
1
1201199055120574120575120582120573
412057325
+1
1201199055120574120577120588120573
412057325minus
1
1201198731199055120574120573
2120573412057325
minus1
120119872119905512057421205733
5minus
1
1201198721199055120574120573
112057335
minus1
1201198721199055120574120573
212057335minus
1
12011987211990551205741205734
5
1198671198645 [119905] = minus
1
12011990551205771205881205734
1+
1
120119873119905512057351
minus1
4011990551205771205881205733
11205732+1
301198731199055120573411205732
minus1
4011990551205771205881205732
112057322+1
2011987311990551205733112057322
minus1
1201199055120577120588120573
112057332+1
3011987311990551205732112057332
+1
1201198731199055120573112057342
minus1
12011990551205771205881205733
11205733+
1
1201198731199055120573411205733
minus1
6011990551205771205881205732
112057321205733+1
4011987311990551205733112057321205733
minus1
1201199055120577120588120573
1120573221205733+1
40119873119905512057321120573221205733
+1
12011987311990551205731120573321205733minus
1
12011990551205771205881205732
112057323
+1
12011987311990551205733112057323minus
1
1201199055120577120588120573
1120573212057323
Abstract and Applied Analysis 9
+1
60119873119905512057321120573212057323+
1
120119873119905512057311205732212057323
minus1
1201199055120577120588120573
112057333+
1
12011987311990551205732112057333
+1
12011987311990551205731120573212057333+
1
12011990551205751205821205734
3
+1
1201198731199055120573112057343+
1
12011987211990551205743120573
11205735
+1
120119872119905512057421205732
11205735+
1
12011987211990551205741205733
11205735
+1
12011987211990551205742120573
112057321205735+1
6011987211990551205741205732
112057321205735
+1
1201198721199055120574120573
1120573221205735+
1
12011987211990551205742120573
112057331205735
+1
12011987211990551205741205732
112057331205735+
1
1201198721199055120574120573
1120573212057331205735
+1
1201198721199055120574120573
1120573231205735minus
1
1201199055120574120575120582120573
112057341205735
minus1
1201199055120574120577120588120573
112057341205735+
1
1201198731199055120574120573
1120573212057341205735
+1
12011987211990551205742120573
112057325+
1
12011987211990551205741205732
112057325
+1
1201198721199055120574120573
1120573212057325+
1
1201198721199055120574120573
1120573312057325
+1
1201198721199055120574120573
112057335
1198671198685 [119905]= minus
1
12011990551205771205881205733
11205732+
1
1201198731199055120573411205732minus1
4011990551205771205881205732
112057322
+1
3011987311990551205733112057322minus1
401199055120577120588120573
112057332+1
2011987311990551205732112057332
minus1
12011990551205771205881205734
2+1
301198731199055120573112057342+
1
120119873119905512057352
+1
12011990551205771205881205733
11205733minus
1
1201198731199055120573411205733
+1
6011990551205771205881205732
112057321205733minus1
4011987311990551205733112057321205733
+1
1201199055120577120588120573
1120573221205733minus1
40119873119905512057321120573221205733
minus1
12011987311990551205731120573321205733+
1
12011990551205771205881205732
112057323
minus1
12011987311990551205733112057323+
1
1201199055120577120588120573
1120573212057323
minus1
60119873119905512057321120573212057323minus
1
120119873119905512057311205732212057323
+1
1201199055120577120588120573
112057333minus
1
12011987311990551205732112057333
minus1
12011987311990551205731120573212057333minus
1
12011990551205751205821205734
3
minus1
1201198731199055120573112057343minus
1
12011990551205771205881205732
112057321205734
+1
12011987311990551205733112057321205734minus1
601199055120577120588120573
1120573221205734
+1
40119873119905512057321120573221205734minus
1
12011990551205771205881205733
21205734
+1
4011987311990551205731120573321205734+
1
1201198731199055120573421205734
+1
12011990551205771205881205732
112057331205734minus
1
12011987311990551205733112057331205734
+1
1201199055120577120588120573
1120573212057331205734minus1
60119873119905512057321120573212057331205734
minus1
120119873119905512057311205732212057331205734+
1
1201199055120577120588120573
1120573231205734
minus1
120119873119905512057321120573231205734minus
1
120119873119905512057311205732120573231205734
minus1
12011990551205751205821205733
31205734minus
1
12011987311990551205731120573331205734
minus1
1201199055120577120588120573
1120573212057324+
1
120119873119905512057321120573212057324
minus1
12011990551205771205881205732
212057324+1
60119873119905512057311205732212057324
+1
12011987311990551205733212057324+
1
1201199055120577120588120573
1120573312057324
minus1
120119873119905512057321120573312057324minus
1
120119873119905512057311205732120573312057324
minus1
12011990551205751205821205732
312057324minus
1
120119873119905512057311205732312057324
minus1
1201199055120577120588120573
212057334+
1
12011987311990551205731120573212057334
+1
12011987311990551205732212057334minus
1
1201199055120575120582120573
312057334
minus1
12011987311990551205731120573312057334minus
1
12011990551205751205821205734
4
minus1
12011990551205771205881205734
4+
1
1201198731199055120573212057344
+1
12011987211990551205743120573
21205735+
1
12011987211990551205742120573
112057321205735
+1
12011987211990551205741205732
112057321205735+
1
120119872119905512057421205732
21205735
+1
601198721199055120574120573
1120573221205735+
1
12011987211990551205741205733
21205735
minus1
12011987211990551205742120573
112057331205735minus
1
12011987211990551205741205732
112057331205735
minus1
1201198721199055120574120573
1120573212057331205735minus
1
1201198721199055120574120573
1120573231205735
+1
12011987211990551205742120573
212057341205735minus
1
1201199055120574120575120582120573
212057341205735
minus1
1201199055120574120577120588120573
212057341205735+
1
1201198721199055120574120573
1120573212057341205735
+1
12011987211990551205741205732
212057341205735+
1
12011987311990551205741205732
212057341205735
minus1
1201198721199055120574120573
1120573312057341205735+
1
1201198721199055120574120573
2120573241205735
+1
12011987211990551205742120573
212057325+
1
1201198721199055120574120573
1120573212057325
+1
12011987211990551205741205732
212057325minus
1
1201198721199055120574120573
1120573312057325
+1
1201198721199055120574120573
2120573412057325+
1
1201198721199055120574120573
212057335
10 Abstract and Applied Analysis
1198811198785 [119905] =
1
12011990551205771205881205732
112057321205734minus
1
12011987311990551205733112057321205734
+1
601199055120577120588120573
1120573221205734minus1
40119873119905512057321120573221205734
+1
12011990551205771205881205733
21205734minus1
4011987311990551205731120573321205734
minus1
1201198731199055120573421205734minus
1
12011990551205771205881205732
112057331205734
+1
12011987311990551205733112057331205734minus
1
1201199055120577120588120573
1120573212057331205734
+1
60119873119905512057321120573212057331205734+
1
120119873119905512057311205732212057331205734
minus1
1201199055120577120588120573
1120573231205734+
1
120119873119905512057321120573231205734
+1
120119873119905512057311205732120573231205734+
1
12011990551205751205821205733
31205734
+1
12011987311990551205731120573331205734+
1
1201199055120577120588120573
1120573212057324
minus1
120119873119905512057321120573212057324+
1
12011990551205771205881205732
212057324
minus1
60119873119905512057311205732212057324minus
1
12011987311990551205733212057324
minus1
1201199055120577120588120573
1120573312057324+
1
120119873119905512057321120573312057324
+1
120119873119905512057311205732120573312057324+
1
12011990551205751205821205732
312057324
+1
120119873119905512057311205732312057324+
1
1201199055120577120588120573
212057334
minus1
12011987311990551205731120573212057334minus
1
12011987311990551205732212057334
+1
1201199055120575120582120573
312057334+
1
12011987311990551205731120573312057334
+1
12011990551205751205821205734
4+
1
12011990551205771205881205734
4minus
1
1201198731199055120573212057344
minus1
12011987211990551205742120573
212057341205735minus
1
1201198721199055120574120573
1120573212057341205735
+1
1201199055120577120588120573
1120573212057341205735minus
1
120119873119905512057321120573212057341205735
minus1
12011987211990551205741205732
212057341205735+
1
12011990551205771205881205732
212057341205735
minus1
60119873119905512057311205732212057341205735minus
1
12011987311990551205733212057341205735
+1
1201198721199055120574120573
1120573312057341205735minus
1
1201199055120577120588120573
1120573312057341205735
+1
120119873119905512057321120573312057341205735+
1
120119873119905512057311205732120573312057341205735
+1
12011990551205751205821205732
312057341205735+
1
120119873119905512057311205732312057341205735
minus1
1201198721199055120574120573
2120573241205735+
1
1201199055120577120588120573
2120573241205735
minus1
120119873119905512057311205732120573241205735minus
1
120119873119905512057322120573241205735
+1
1201199055120575120582120573
3120573241205735+
1
120119873119905512057311205733120573241205735
+1
12011990551205751205821205733
41205735+
1
12011990551205771205881205733
41205735
minus1
12011987311990551205732120573341205735minus1
601198721199055120574120573
2120573412057325
+1
1201199055120577120588120573
2120573412057325minus
1
120119873119905512057311205732120573412057325
minus1
120119873119905512057322120573412057325+
1
1201199055120575120582120573
3120573412057325
+1
120119873119905512057311205733120573412057325+
1
12011990551205751205821205732
412057325
+1
12011990551205771205881205732
412057325minus
1
120119873119905512057321205732412057325
+1
1201199055120575120582120573
412057335+
1
1201199055120577120588120573
412057335
minus1
12011987311990551205732120573412057335minus
1
12011987211990551205735
5
1198811198685 [119905] =
1
12011987211990551205744120573
5minus
1
12011990551205742120575120582120573
41205735
minus1
12011990551205742120577120588120573
41205735+
1
12011987311990551205742120573
212057341205735
minus1
1201199055120574120577120588120573
212057341205735+
1
1201198731199055120574120573
1120573212057341205735
minus1
1201199055120577120588120573
1120573212057341205735+
1
120119873119905512057321120573212057341205735
+1
12011987311990551205741205732
212057341205735minus
1
12011990551205771205881205732
212057341205735
+1
60119873119905512057311205732212057341205735+
1
12011987311990551205733212057341205735
minus1
1201199055120574120575120582120573
312057341205735minus
1
1201198731199055120574120573
1120573312057341205735
+1
1201199055120577120588120573
1120573312057341205735minus
1
120119873119905512057321120573312057341205735
minus1
120119873119905512057311205732120573312057341205735minus
1
12011990551205751205821205732
312057341205735
minus1
120119873119905512057311205732312057341205735minus
1
12011990551205741205751205821205732
41205735
minus1
12011990551205741205771205881205732
41205735+
1
1201198731199055120574120573
2120573241205735
minus1
1201199055120577120588120573
2120573241205735+
1
120119873119905512057311205732120573241205735
+1
120119873119905512057322120573241205735minus
1
1201199055120575120582120573
3120573241205735
minus1
120119873119905512057311205733120573241205735minus
1
12011990551205751205821205733
41205735
minus1
12011990551205771205881205733
41205735+
1
12011987311990551205732120573341205735
+1
120119872119905512057431205732
5minus
1
1201199055120574120575120582120573
412057325
minus1
1201199055120574120577120588120573
412057325+
1
1201198721199055120574120573
2120573412057325
+1
1201198731199055120574120573
2120573412057325minus
1
1201199055120577120588120573
2120573412057325
+1
120119873119905512057311205732120573412057325+
1
120119873119905512057322120573412057325
Abstract and Applied Analysis 11
minus1
1201199055120575120582120573
3120573412057325minus
1
120119873119905512057311205733120573412057325
minus1
12011990551205751205821205732
412057325minus
1
12011990551205771205881205732
412057325
+1
120119873119905512057321205732412057325+
1
120119872119905512057421205733
5
minus1
1201199055120575120582120573
412057335minus
1
1201199055120577120588120573
412057335
+1
12011987311990551205732120573412057335+
1
12011987211990551205741205734
5
+1
12011987211990551205735
5
(33)
In general by the repeated application of the above proce-dure we have that
119867119878119899=119905119899
1198991198861119899 119867
119864119899=119905119899
1198991198862119899 119867
119868119899=119905119899
1198991198863119899
119881119878119899=119905119899
1198991198864119899 119881
119868119899=119905119899
1198991198865119899
(34)
where 1198861119899 1198862119899 1198863119899 1198864119899 1198865119899
depend on fixed set of empiricalparameters Hence the approximate solution of the modelabove is given as
119867119878(119905) =
infin
sum119899=0
119905119899
1198991198861119899
119867119864(119905) =
infin
sum119899=0
119905119899
1198991198862119899
119867119868(119905) =
infin
sum119899=0
119905119899
1198991198863119899
119881119878(119905) =
infin
sum119899=0
119905119899
1198991198864119899
119881119868(119905) =
infin
sum119899=0
119905119899
1198991198865119899
(35)
We therefore use the following theoretical parameters 1205731=
05 1205732= 05 120573
3= 05 120573
4= 05 120573
5= 05 120574 = 1 120577 = 1
119873 = 10 119872 = 10 120588 = 1 120575 = 1 and 120582 = 1 to examinethe behavior of the five different compartmental numericalsolutions in the system Figure 3 shows the behavior of thenumerical solution
7 Conclusion
Onchocerciasis model was examined and the mathematicalformulation underpinning the spread and control of thisdisease has been proposed on one hand On the otherhand we made use of some new analytical methods calledthe homotopy decomposition method (HDM) to derive thesolution of the resulting set of (1) to (5)The numerical resultswere presented to test the efficiency and the accuracy of themethod used The mathematical analysis of the resulting setof (1) to (5) above revealed that the stability analysis of the
05 10 15 20Time (per year)
2
4
6
8
10
Appr
oxim
ate s
olut
ions VS
HS
VI
HI
HE
Figure 3 Numerical solution of the model
five roots of the characteristics equations has all negativereal part In particular given that the stability analysis showsthat the DFE is asymptotically stable it would be possiblethat the parasite Onchocerca volvulus be eradicated in a finitetime On the other hand the numerical solution confirmedthat as the number of humans and vectors in the susceptiblecompartments reduces the number of infective humansand vectors in the infectious compartment increases Thenumerical solutions in both cases displayed the biologicalbehaviour of the real world situation Therefore in orderto build a healthy and a sustainable population treatmentwith IvermectinMectizan drugs and a more hygienic envi-ronment that does not allow the breeding and survival ofblackflies could lead to a total eradication of the disease
Conflict of Interests
The authors declare that they have no conflict of interestsregarding the publication of this paper
References
[1] J H F Remme The Global Burden of Onchocerciasis in 1990World Health Organization Geneva Switzerland 2004 httpwwwwhointhealthinfoglobal burden diseaseOnchocercia-sis201990pdf
[2] WorldHealthOrganization ldquoPrevention of blindness and visualimpairment priority eye diseasesrdquo 2014 httpwwwwhointblindnesscausespriorityenindex3html
[3] World Health Organization Certification of Elimination ofHuman Onchocerciasis Criteria and Procedures World HealthOrganizationGeneva Switzerland 2001 httpwhqlibdocwhointhq2001WHO CDS CPE CEE 200118bpdf
[4] K A Lindblade B Arana G Zea-Flores et al ldquoEliminationof Onchocerca volvulus transmission in the Santa Rosa focusof Guatemalardquo The American Journal of Tropical Medicine andHygiene vol 77 pp 334ndash341 2007
[5] J C Anosike and C O E Onwuliri ldquoStudies on filariasis inBauchi State Nigeria 1 Endemicity of human onchocerciasisin Ningi Local Government Areardquo Annals of Tropical Medicineand Parasitology vol 89 no 1 pp 31ndash38 1995
[6] WorldHealthOrganizationWeekly Epidemiological Records 71WHO Geneva Switzerland 1993
12 Abstract and Applied Analysis
[7] WorldHealthOrganization httpwwwwhointneglected dis-easesen
[8] Sightsaver httpwwwsightsaversorgabout uspublicationsonchocerciasis
[9] E C Uttah ldquoOnchocerciasis in the Upper Ima River BasinrdquoIranian Journal of Parasitology vol 5 no 2 pp 33ndash41 2010
[10] J C Anosike and C O E Onwuliri ldquoStudies on filariasis inBauchi State Nigeriamdash1 endemicity of human onchocerciasisin Ningi Local Government Areardquo Annals of Tropical Medicineand Parasitology vol 89 no 1 pp 31ndash38 1995
[11] L D Edungbola and S O Asaolu ldquoParasitologic survey ofonchocerciasis (river blindness) in Babana district Kwara StateNigeriardquo American Journal of Tropical Medicine and Hygienevol 33 no 6 pp 1147ndash1154 1984
[12] O C Nwaorgu A Ohaegbula I E Onweluzo et al ldquoResultsof a large scale onchocercosis survey in Enugu State NigeriardquoJournal of Helminthology vol 68 no 2 pp 155ndash159 1994
[13] F O Akinbo and C E Okaka ldquoPrevalence and socio-economiceffects of onchocerciasis in Ovia North East Local GovernmentArea of Edo State Nigeriardquo Rivista Di Parasitologie vol 22 no3 pp 215ndash221 2005
[14] M D Wogu and C E Okaka ldquoPrevalence and socio-economic effects of onchocerciasis in Okpuje Owan WestLocal Government Area Edo State Nigeriardquo InternationalJournal of Biomedical and Health Sciences vol 4 no 3 2008httpwwwasopahorg
[15] S C O Noutchie ldquoHomotopy Decomposition analysis of atuberculosis modelrdquo Life Science Journal vol 10 no 3 pp 459ndash465 2013
[16] A Atangana and A Secer ldquoThe time-fractional coupled-Korteweg-de-Vries equationsrdquo Abstract and Applied Analysisvol 2013 Article ID 947986 8 pages 2013
[17] A Atangana and N Bildik ldquoApproximate solution of tubercu-losis disease population dynamics modelrdquo Abstract and AppliedAnalysis vol 2013 Article ID 759801 8 pages 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Abstract and Applied Analysis
minus (1205731+ 1205732)infin
sum119899=0
119875119899119867119878119899(120591)) 119889120591]
infin
sum119899=0
119875119899119867119864119899(119905) minus 119867
119864(0)
= 119875int119905
0
[(120575120582 + 1205731
infin
sum119899=0
119875119899119867119878119899(120591)
minus 1205733
infin
sum119899=0
119875119899119867119864119899(120591)) 119889120591]
infin
sum119899=0
119875119899119867119868119899(119905) minus 119867
119868(0)
= 119875int119905
0
[(1205732
infin
sum119899=0
119875119899119867119878119899(120591) + 120573
3
infin
sum119899=0
119875119899119867119864119899(120591)
minus 120588120577 minus 120575120582 minus 1205734
infin
sum119899=0
119875119899119867119868119899(120591)) 119889120591]
infin
sum119899=0
119875119899119881119878119899(119905) minus 119881
119878(0)
= 119875int119905
0
[(1205734
infin
sum119899=0
119875119899119867119868119899(120591) minus 120573
5
infin
sum119899=0
119875119899119881119878119899(120591)) 119889120591]
infin
sum119899=0
119875119899119881119868119899(119905) minus 119881
119868(0)
= 119875int119905
0
[(1205735
infin
sum119899=0
119875119899119881119878119899(120591) minus 120574
infin
sum119899=0
119875119899119881119868119899(120591)) 119889120591]
(30)
We compared the terms of the same power of 119875 and obtainedthe following
1198750 1198671198780(119905) = 119867
119878(0)
1198750 1198671198640(119905) = 119867
119864(0)
1198750 1198671198680(119905) = 119867
119868(0)
1198750 1198811198780(119905) = 119881
119878(0)
1198750 1198811198680(119905) = 119881
119868(0)
1198751 1198671198781(119905) = int
119905
0
(120588120577 + 1205741198811198680(120591) minus (120573
1+ 1205732)1198671198780(120591)) 119889120591
1198751 1198671198641(119905) = int
119905
0
(120575120582 + 12057311198671198780(120591) minus 120573
31198671198640(120591)) 119889120591
1198751 1198671198681(119905)
= int119905
0
(12057321198671198780(120591)+120573
31198671198640(120591)minus120588120577minus 120575120582 minus120573
41198671198680(120591)) 119889120591
1198751 1198811198781(119905) = int
119905
0
(12057341198671198680(120591) minus 120573
51198811198780(120591)) 119889120591
1198751 1198811198681(119905) = int
119905
0
(12057351198811198780(120591) minus 120574119881
1198680(120591)) 119889120591
(31)
Continuing for 119899 ge 2 we have
119867119878119899(119905) = int
119905
0
(120574119881119868(119899minus1)
(120591) minus (1205731+ 1205732)119867119878(119899minus1)
(120591)) 119889120591
119867119864119899(119905) = int
119905
0
(1205731119867119878(119899minus1)
(120591) minus 1205733119867119864(119899minus1)
(120591)) 119889120591
119867119868119899(119905) = int
119905
0
(1205732119867119878(119899minus1)
(120591) + 1205733119867119864(119899minus1)
(120591)
minus 1205734119867119868(119899minus1)
(120591)) 119889120591
119881119878119899(119905) = int
119905
0
(1205734119867119868(119899minus1)
(120591) minus 1205735119881119878(119899minus1)
(120591)) 119889120591
119881119868119899(119905) = int
119905
0
(1205735119881119878(119899minus1)
(120591) minus 120574119881119868(119899minus1)
(120591)) 119889120591
(32)
We now integrate the above equations to obtain the following
1198671198680(119905) = 0 119867
1198640(119905) = 0 119867
1198780 [119905] = 119873 1198811198780 [119905] = 119872
1198811198680 [119905] = 0 119867
1198681(119905) = 119905 (minus120575120582 minus 120577120588 + 119873120573
2)
1198671198781 [119905] = 119905 (120577120588 minus 119873 (1205731 + 1205732))
1198671198641(119905) = 119905 (120575120582 + 119873120573
1)
1198811198681 [119905] = 1198721199051205735 119881
1198781 [119905] = minus1198721199051205735
1198671198682(119905) =
1
21199052120577120588120573
2minus1
2119873119905212057311205732minus1
2119873119905212057322+1
21199052120575120582120573
3
+1
2119873119905212057311205733+1
21199052120575120582120573
4+1
21199052120577120588120573
4minus1
2119873119905212057321205734
1198671198642(119905) =
1
21199052120577120588120573
1minus1
2119873119905212057321minus1
2119873119905212057311205732
minus1
21199052120575120582120573
3minus1
2119873119905212057311205733
1198671198782 [119905] = minus
1
21199052120577120588120573
1+1
2119873119905212057321minus1
21199052120577120588120573
2
+ 119873119905212057311205732+1
2119873119905212057322+1
21198721199052120574120573
5
1198811198782 [119905] = minus
1
21199052120575120582120573
4minus1
21199052120577120588120573
4
+1
2119873119905212057321205734+1
211987211990521205732
5
1198811198682 [119905] = minus
1
21198721199052120574120573
5minus1
211987211990521205732
5
1198671198683(119905) = minus
1
61199053120577120588120573
11205732+1
61198731199053120573211205732minus1
611990531205771205881205732
2
+1
31198731199053120573112057322+1
6119873119905312057332+1
61199053120577120588120573
11205733
minus1
61198731199053120573211205733minus1
61198731199053120573112057321205733
minus1
611990531205751205821205732
3
minus1
61198731199053120573112057323minus1
61199053120577120588120573
21205734
+1
61198731199053120573112057321205734+1
61198731199053120573221205734
minus1
61199053120575120582120573
31205734minus1
61198731199053120573112057331205734
Abstract and Applied Analysis 7
minus1
611990531205751205821205732
4minus1
611990531205771205881205732
4+1
61198731199053120573212057324
+1
61198721199053120574120573
21205735
1198671198643(119905) = minus
1
611990531205771205881205732
1+1
6119873119905312057331minus1
61199053120577120588120573
11205732
+1
31198731199053120573211205732+1
61198731199053120573112057322
minus1
61199053120577120588120573
11205733+1
61198731199053120573211205733
+1
61198731199053120573112057321205733+1
611990531205751205821205732
3
+1
61198731199053120573112057323+1
61198721199053120574120573
11205735
1198671198783 [119905] =
1
611990531205771205881205732
1minus1
6119873119905312057331+1
31199053120577120588120573
11205732
minus1
21198731199053120573211205732+1
611990531205771205881205732
2minus1
21198731199053120573112057322
minus1
6119873119905312057332minus1
611987211990531205742120573
5
minus1
61198721199053120574120573
11205735
minus1
61198721199053120574120573
21205735minus1
611987211990531205741205732
5
1198811198783 [119905] =
1
61199053120577120588120573
21205734minus1
61198731199053120573112057321205734minus1
61198731199053120573221205734
+1
61199053120575120582120573
31205734+1
61198731199053120573112057331205734+1
611990531205751205821205732
4
+1
611990531205771205881205732
4minus1
61198731199053120573212057324+1
61199053120575120582120573
41205735
+1
61199053120577120588120573
41205735minus1
61198731199053120573212057341205735minus1
611987211990531205733
5
1198811198683 [119905] =
1
611987211990531205742120573
5minus1
61199053120575120582120573
41205735minus1
61199053120577120588120573
41205735
+1
61198731199053120573212057341205735+1
611987211990531205741205732
5+1
611987211990531205733
5
1198671198784 [119905] = minus
1
2411990541205771205881205733
1+1
24119873119905412057341minus1
811990541205771205881205732
11205732
+1
61198731199054120573311205732minus1
81199054120577120588120573
112057322+1
411987311990541205732112057322
minus1
2411990541205771205881205733
2+1
61198731199054120573112057332+1
24119873119905412057342
+1
2411987211990541205743120573
5+1
2411987211990541205742120573
11205735
+1
2411987211990541205741205732
11205735+1
2411987211990541205742120573
21205735
+1
121198721199054120574120573
112057321205735+1
2411987211990541205741205732
21205735
minus1
241199054120574120575120582120573
41205735minus1
241199054120574120577120588120573
41205735
+1
241198731199054120574120573
212057341205735+1
24119872119905412057421205732
5
+1
241198721199054120574120573
112057325+1
241198721199054120574120573
212057325+1
2411987211990541205741205733
5
1198671198644(119905) =
1
2411990541205771205881205733
1minus1
24119873119905412057341+1
1211990541205771205881205732
11205732
minus1
81198731199054120573311205732+1
241199054120577120588120573
112057322minus1
811987311990541205732112057322
minus1
241198731199054120573112057332+1
2411990541205771205881205732
11205733minus1
241198731199054120573311205733
+1
241199054120577120588120573
112057321205733minus1
1211987311990541205732112057321205733minus1
2411987311990541205731120573221205733
+1
241199054120577120588120573
112057323minus1
2411987311990541205732112057323minus1
2411987311990541205731120573212057323
minus1
2411990541205751205821205733
3minus1
241198731199054120573112057333minus1
2411987211990541205742120573
11205735
minus1
2411987211990541205741205732
11205735minus1
241198721199054120574120573
112057321205735
minus1
241198721199054120574120573
112057331205735minus1
241198721199054120574120573
112057325
1198671198684(119905) =
1
2411990541205771205881205732
11205732minus1
241198731199054120573311205732+1
121199054120577120588120573
112057322
minus1
811987311990541205732112057322+1
2411990541205771205881205733
2minus1
81198731199054120573112057332
minus1
24119873119905412057342minus1
2411990541205771205881205732
11205733+1
241198731199054120573311205733
minus1
241199054120577120588120573
112057321205733+1
1211987311990541205732112057321205733
+1
2411987311990541205731120573221205733
minus1
241199054120577120588120573
112057323+1
2411987311990541205732112057323+1
2411987311990541205731120573212057323
+1
2411990541205751205821205733
3+1
241198731199054120573112057333+1
241199054120577120588120573
112057321205734
minus1
2411987311990541205732112057321205734+1
2411990541205771205881205732
21205734minus1
1211987311990541205731120573221205734
minus1
241198731199054120573321205734minus1
241199054120577120588120573
112057331205734
+1
2411987311990541205732112057331205734+1
2411987311990541205731120573212057331205734
+1
2411990541205751205821205732
31205734+1
2411987311990541205731120573231205734
+1
241199054120577120588120573
212057324minus1
2411987311990541205731120573212057324minus1
2411987311990541205732212057324
+1
241199054120575120582120573
312057324+1
2411987311990541205731120573312057324+1
2411990541205751205821205733
4
+1
2411990541205771205881205733
4minus1
241198731199054120573212057334minus1
2411987211990541205742120573
21205735
minus1
241198721199054120574120573
112057321205735minus1
2411987211990541205741205732
21205735
+1
241198721199054120574120573
112057331205735
minus1
241198721199054120574120573
212057341205735minus1
241198721199054120574120573
212057325
1198811198784 [119905] = minus
1
241199054120577120588120573
112057321205734+1
2411987311990541205732112057321205734
minus1
2411990541205771205881205732
21205734+1
1211987311990541205731120573221205734
8 Abstract and Applied Analysis
+1
241198731199054120573321205734+1
241199054120577120588120573
112057331205734
minus1
2411987311990541205732112057331205734minus1
2411987311990541205731120573212057331205734
minus1
2411990541205751205821205732
31205734
minus1
2411987311990541205731120573231205734minus1
241199054120577120588120573
212057324
+1
2411987311990541205731120573212057324+1
2411987311990541205732212057324
minus1
241199054120575120582120573
312057324minus1
2411987311990541205731120573312057324
minus1
2411990541205751205821205733
4minus1
2411990541205771205881205733
4
+1
241198731199054120573212057334
+1
241198721199054120574120573
212057341205735minus1
241199054120577120588120573
212057341205735
+1
2411987311990541205731120573212057341205735+1
2411987311990541205732212057341205735
minus1
241199054120575120582120573
312057341205735minus1
2411987311990541205731120573312057341205735
minus1
2411990541205751205821205732
41205735minus1
2411990541205771205881205732
41205735+1
2411987311990541205732120573241205735
minus1
241199054120575120582120573
412057325minus1
241199054120577120588120573
412057325
+1
2411987311990541205732120573412057325+1
2411987211990541205734
5
1198811198684 [119905] = minus
1
2411987211990541205743120573
5+1
241199054120574120575120582120573
41205735+1
241199054120574120577120588120573
41205735
minus1
241198731199054120574120573
212057341205735+1
241199054120577120588120573
212057341205735
minus1
2411987311990541205731120573212057341205735minus1
2411987311990541205732212057341205735
+1
241199054120575120582120573
312057341205735+1
2411987311990541205731120573312057341205735
+1
2411990541205751205821205732
41205735+1
2411990541205771205881205732
41205735
minus1
2411987311990541205732120573241205735minus1
24119872119905412057421205732
5
+1
241199054120575120582120573
412057325+1
241199054120577120588120573
412057325
minus1
2411987311990541205732120573412057325minus1
2411987211990541205741205733
5minus1
2411987211990541205734
5
1198671198785 [119905] =
1
12011990551205771205881205734
1minus
1
120119873119905512057351+1
3011990551205771205881205733
11205732
minus1
241198731199055120573411205732+1
2011990551205771205881205732
112057322minus1
1211987311990551205733112057322
+1
301199055120577120588120573
112057332minus1
1211987311990551205732112057332+
1
12011990551205771205881205734
2
minus1
241198731199055120573112057342minus
1
120119873119905512057352minus
1
12011987211990551205744120573
5
minus1
12011987211990551205743120573
11205735minus
1
120119872119905512057421205732
11205735
minus1
12011987211990551205741205733
11205735minus
1
12011987211990551205743120573
21205735
minus1
6011987211990551205742120573
112057321205735minus1
4011987211990551205741205732
112057321205735
minus1
120119872119905512057421205732
21205735minus1
401198721199055120574120573
1120573221205735
minus1
12011987211990551205741205733
21205735+
1
12011990551205742120575120582120573
41205735
+1
12011990551205742120577120588120573
41205735+
1
1201199055120574120575120582120573
112057341205735
+1
1201199055120574120577120588120573
112057341205735minus
1
12011987311990551205742120573
212057341205735
+1
1201199055120574120575120582120573
212057341205735+1
601199055120574120577120588120573
212057341205735
minus1
601198731199055120574120573
1120573212057341205735minus1
6011987311990551205741205732
212057341205735
+1
1201199055120574120575120582120573
312057341205735+
1
1201198731199055120574120573
1120573312057341205735
+1
12011990551205741205751205821205732
41205735+
1
12011990551205741205771205881205732
41205735
minus1
1201198731199055120574120573
2120573241205735minus
1
120119872119905512057431205732
5
minus1
12011987211990551205742120573
112057325minus
1
12011987211990551205741205732
112057325
minus1
12011987211990551205742120573
212057325minus1
601198721199055120574120573
1120573212057325
minus1
12011987211990551205741205732
212057325+
1
1201199055120574120575120582120573
412057325
+1
1201199055120574120577120588120573
412057325minus
1
1201198731199055120574120573
2120573412057325
minus1
120119872119905512057421205733
5minus
1
1201198721199055120574120573
112057335
minus1
1201198721199055120574120573
212057335minus
1
12011987211990551205741205734
5
1198671198645 [119905] = minus
1
12011990551205771205881205734
1+
1
120119873119905512057351
minus1
4011990551205771205881205733
11205732+1
301198731199055120573411205732
minus1
4011990551205771205881205732
112057322+1
2011987311990551205733112057322
minus1
1201199055120577120588120573
112057332+1
3011987311990551205732112057332
+1
1201198731199055120573112057342
minus1
12011990551205771205881205733
11205733+
1
1201198731199055120573411205733
minus1
6011990551205771205881205732
112057321205733+1
4011987311990551205733112057321205733
minus1
1201199055120577120588120573
1120573221205733+1
40119873119905512057321120573221205733
+1
12011987311990551205731120573321205733minus
1
12011990551205771205881205732
112057323
+1
12011987311990551205733112057323minus
1
1201199055120577120588120573
1120573212057323
Abstract and Applied Analysis 9
+1
60119873119905512057321120573212057323+
1
120119873119905512057311205732212057323
minus1
1201199055120577120588120573
112057333+
1
12011987311990551205732112057333
+1
12011987311990551205731120573212057333+
1
12011990551205751205821205734
3
+1
1201198731199055120573112057343+
1
12011987211990551205743120573
11205735
+1
120119872119905512057421205732
11205735+
1
12011987211990551205741205733
11205735
+1
12011987211990551205742120573
112057321205735+1
6011987211990551205741205732
112057321205735
+1
1201198721199055120574120573
1120573221205735+
1
12011987211990551205742120573
112057331205735
+1
12011987211990551205741205732
112057331205735+
1
1201198721199055120574120573
1120573212057331205735
+1
1201198721199055120574120573
1120573231205735minus
1
1201199055120574120575120582120573
112057341205735
minus1
1201199055120574120577120588120573
112057341205735+
1
1201198731199055120574120573
1120573212057341205735
+1
12011987211990551205742120573
112057325+
1
12011987211990551205741205732
112057325
+1
1201198721199055120574120573
1120573212057325+
1
1201198721199055120574120573
1120573312057325
+1
1201198721199055120574120573
112057335
1198671198685 [119905]= minus
1
12011990551205771205881205733
11205732+
1
1201198731199055120573411205732minus1
4011990551205771205881205732
112057322
+1
3011987311990551205733112057322minus1
401199055120577120588120573
112057332+1
2011987311990551205732112057332
minus1
12011990551205771205881205734
2+1
301198731199055120573112057342+
1
120119873119905512057352
+1
12011990551205771205881205733
11205733minus
1
1201198731199055120573411205733
+1
6011990551205771205881205732
112057321205733minus1
4011987311990551205733112057321205733
+1
1201199055120577120588120573
1120573221205733minus1
40119873119905512057321120573221205733
minus1
12011987311990551205731120573321205733+
1
12011990551205771205881205732
112057323
minus1
12011987311990551205733112057323+
1
1201199055120577120588120573
1120573212057323
minus1
60119873119905512057321120573212057323minus
1
120119873119905512057311205732212057323
+1
1201199055120577120588120573
112057333minus
1
12011987311990551205732112057333
minus1
12011987311990551205731120573212057333minus
1
12011990551205751205821205734
3
minus1
1201198731199055120573112057343minus
1
12011990551205771205881205732
112057321205734
+1
12011987311990551205733112057321205734minus1
601199055120577120588120573
1120573221205734
+1
40119873119905512057321120573221205734minus
1
12011990551205771205881205733
21205734
+1
4011987311990551205731120573321205734+
1
1201198731199055120573421205734
+1
12011990551205771205881205732
112057331205734minus
1
12011987311990551205733112057331205734
+1
1201199055120577120588120573
1120573212057331205734minus1
60119873119905512057321120573212057331205734
minus1
120119873119905512057311205732212057331205734+
1
1201199055120577120588120573
1120573231205734
minus1
120119873119905512057321120573231205734minus
1
120119873119905512057311205732120573231205734
minus1
12011990551205751205821205733
31205734minus
1
12011987311990551205731120573331205734
minus1
1201199055120577120588120573
1120573212057324+
1
120119873119905512057321120573212057324
minus1
12011990551205771205881205732
212057324+1
60119873119905512057311205732212057324
+1
12011987311990551205733212057324+
1
1201199055120577120588120573
1120573312057324
minus1
120119873119905512057321120573312057324minus
1
120119873119905512057311205732120573312057324
minus1
12011990551205751205821205732
312057324minus
1
120119873119905512057311205732312057324
minus1
1201199055120577120588120573
212057334+
1
12011987311990551205731120573212057334
+1
12011987311990551205732212057334minus
1
1201199055120575120582120573
312057334
minus1
12011987311990551205731120573312057334minus
1
12011990551205751205821205734
4
minus1
12011990551205771205881205734
4+
1
1201198731199055120573212057344
+1
12011987211990551205743120573
21205735+
1
12011987211990551205742120573
112057321205735
+1
12011987211990551205741205732
112057321205735+
1
120119872119905512057421205732
21205735
+1
601198721199055120574120573
1120573221205735+
1
12011987211990551205741205733
21205735
minus1
12011987211990551205742120573
112057331205735minus
1
12011987211990551205741205732
112057331205735
minus1
1201198721199055120574120573
1120573212057331205735minus
1
1201198721199055120574120573
1120573231205735
+1
12011987211990551205742120573
212057341205735minus
1
1201199055120574120575120582120573
212057341205735
minus1
1201199055120574120577120588120573
212057341205735+
1
1201198721199055120574120573
1120573212057341205735
+1
12011987211990551205741205732
212057341205735+
1
12011987311990551205741205732
212057341205735
minus1
1201198721199055120574120573
1120573312057341205735+
1
1201198721199055120574120573
2120573241205735
+1
12011987211990551205742120573
212057325+
1
1201198721199055120574120573
1120573212057325
+1
12011987211990551205741205732
212057325minus
1
1201198721199055120574120573
1120573312057325
+1
1201198721199055120574120573
2120573412057325+
1
1201198721199055120574120573
212057335
10 Abstract and Applied Analysis
1198811198785 [119905] =
1
12011990551205771205881205732
112057321205734minus
1
12011987311990551205733112057321205734
+1
601199055120577120588120573
1120573221205734minus1
40119873119905512057321120573221205734
+1
12011990551205771205881205733
21205734minus1
4011987311990551205731120573321205734
minus1
1201198731199055120573421205734minus
1
12011990551205771205881205732
112057331205734
+1
12011987311990551205733112057331205734minus
1
1201199055120577120588120573
1120573212057331205734
+1
60119873119905512057321120573212057331205734+
1
120119873119905512057311205732212057331205734
minus1
1201199055120577120588120573
1120573231205734+
1
120119873119905512057321120573231205734
+1
120119873119905512057311205732120573231205734+
1
12011990551205751205821205733
31205734
+1
12011987311990551205731120573331205734+
1
1201199055120577120588120573
1120573212057324
minus1
120119873119905512057321120573212057324+
1
12011990551205771205881205732
212057324
minus1
60119873119905512057311205732212057324minus
1
12011987311990551205733212057324
minus1
1201199055120577120588120573
1120573312057324+
1
120119873119905512057321120573312057324
+1
120119873119905512057311205732120573312057324+
1
12011990551205751205821205732
312057324
+1
120119873119905512057311205732312057324+
1
1201199055120577120588120573
212057334
minus1
12011987311990551205731120573212057334minus
1
12011987311990551205732212057334
+1
1201199055120575120582120573
312057334+
1
12011987311990551205731120573312057334
+1
12011990551205751205821205734
4+
1
12011990551205771205881205734
4minus
1
1201198731199055120573212057344
minus1
12011987211990551205742120573
212057341205735minus
1
1201198721199055120574120573
1120573212057341205735
+1
1201199055120577120588120573
1120573212057341205735minus
1
120119873119905512057321120573212057341205735
minus1
12011987211990551205741205732
212057341205735+
1
12011990551205771205881205732
212057341205735
minus1
60119873119905512057311205732212057341205735minus
1
12011987311990551205733212057341205735
+1
1201198721199055120574120573
1120573312057341205735minus
1
1201199055120577120588120573
1120573312057341205735
+1
120119873119905512057321120573312057341205735+
1
120119873119905512057311205732120573312057341205735
+1
12011990551205751205821205732
312057341205735+
1
120119873119905512057311205732312057341205735
minus1
1201198721199055120574120573
2120573241205735+
1
1201199055120577120588120573
2120573241205735
minus1
120119873119905512057311205732120573241205735minus
1
120119873119905512057322120573241205735
+1
1201199055120575120582120573
3120573241205735+
1
120119873119905512057311205733120573241205735
+1
12011990551205751205821205733
41205735+
1
12011990551205771205881205733
41205735
minus1
12011987311990551205732120573341205735minus1
601198721199055120574120573
2120573412057325
+1
1201199055120577120588120573
2120573412057325minus
1
120119873119905512057311205732120573412057325
minus1
120119873119905512057322120573412057325+
1
1201199055120575120582120573
3120573412057325
+1
120119873119905512057311205733120573412057325+
1
12011990551205751205821205732
412057325
+1
12011990551205771205881205732
412057325minus
1
120119873119905512057321205732412057325
+1
1201199055120575120582120573
412057335+
1
1201199055120577120588120573
412057335
minus1
12011987311990551205732120573412057335minus
1
12011987211990551205735
5
1198811198685 [119905] =
1
12011987211990551205744120573
5minus
1
12011990551205742120575120582120573
41205735
minus1
12011990551205742120577120588120573
41205735+
1
12011987311990551205742120573
212057341205735
minus1
1201199055120574120577120588120573
212057341205735+
1
1201198731199055120574120573
1120573212057341205735
minus1
1201199055120577120588120573
1120573212057341205735+
1
120119873119905512057321120573212057341205735
+1
12011987311990551205741205732
212057341205735minus
1
12011990551205771205881205732
212057341205735
+1
60119873119905512057311205732212057341205735+
1
12011987311990551205733212057341205735
minus1
1201199055120574120575120582120573
312057341205735minus
1
1201198731199055120574120573
1120573312057341205735
+1
1201199055120577120588120573
1120573312057341205735minus
1
120119873119905512057321120573312057341205735
minus1
120119873119905512057311205732120573312057341205735minus
1
12011990551205751205821205732
312057341205735
minus1
120119873119905512057311205732312057341205735minus
1
12011990551205741205751205821205732
41205735
minus1
12011990551205741205771205881205732
41205735+
1
1201198731199055120574120573
2120573241205735
minus1
1201199055120577120588120573
2120573241205735+
1
120119873119905512057311205732120573241205735
+1
120119873119905512057322120573241205735minus
1
1201199055120575120582120573
3120573241205735
minus1
120119873119905512057311205733120573241205735minus
1
12011990551205751205821205733
41205735
minus1
12011990551205771205881205733
41205735+
1
12011987311990551205732120573341205735
+1
120119872119905512057431205732
5minus
1
1201199055120574120575120582120573
412057325
minus1
1201199055120574120577120588120573
412057325+
1
1201198721199055120574120573
2120573412057325
+1
1201198731199055120574120573
2120573412057325minus
1
1201199055120577120588120573
2120573412057325
+1
120119873119905512057311205732120573412057325+
1
120119873119905512057322120573412057325
Abstract and Applied Analysis 11
minus1
1201199055120575120582120573
3120573412057325minus
1
120119873119905512057311205733120573412057325
minus1
12011990551205751205821205732
412057325minus
1
12011990551205771205881205732
412057325
+1
120119873119905512057321205732412057325+
1
120119872119905512057421205733
5
minus1
1201199055120575120582120573
412057335minus
1
1201199055120577120588120573
412057335
+1
12011987311990551205732120573412057335+
1
12011987211990551205741205734
5
+1
12011987211990551205735
5
(33)
In general by the repeated application of the above proce-dure we have that
119867119878119899=119905119899
1198991198861119899 119867
119864119899=119905119899
1198991198862119899 119867
119868119899=119905119899
1198991198863119899
119881119878119899=119905119899
1198991198864119899 119881
119868119899=119905119899
1198991198865119899
(34)
where 1198861119899 1198862119899 1198863119899 1198864119899 1198865119899
depend on fixed set of empiricalparameters Hence the approximate solution of the modelabove is given as
119867119878(119905) =
infin
sum119899=0
119905119899
1198991198861119899
119867119864(119905) =
infin
sum119899=0
119905119899
1198991198862119899
119867119868(119905) =
infin
sum119899=0
119905119899
1198991198863119899
119881119878(119905) =
infin
sum119899=0
119905119899
1198991198864119899
119881119868(119905) =
infin
sum119899=0
119905119899
1198991198865119899
(35)
We therefore use the following theoretical parameters 1205731=
05 1205732= 05 120573
3= 05 120573
4= 05 120573
5= 05 120574 = 1 120577 = 1
119873 = 10 119872 = 10 120588 = 1 120575 = 1 and 120582 = 1 to examinethe behavior of the five different compartmental numericalsolutions in the system Figure 3 shows the behavior of thenumerical solution
7 Conclusion
Onchocerciasis model was examined and the mathematicalformulation underpinning the spread and control of thisdisease has been proposed on one hand On the otherhand we made use of some new analytical methods calledthe homotopy decomposition method (HDM) to derive thesolution of the resulting set of (1) to (5)The numerical resultswere presented to test the efficiency and the accuracy of themethod used The mathematical analysis of the resulting setof (1) to (5) above revealed that the stability analysis of the
05 10 15 20Time (per year)
2
4
6
8
10
Appr
oxim
ate s
olut
ions VS
HS
VI
HI
HE
Figure 3 Numerical solution of the model
five roots of the characteristics equations has all negativereal part In particular given that the stability analysis showsthat the DFE is asymptotically stable it would be possiblethat the parasite Onchocerca volvulus be eradicated in a finitetime On the other hand the numerical solution confirmedthat as the number of humans and vectors in the susceptiblecompartments reduces the number of infective humansand vectors in the infectious compartment increases Thenumerical solutions in both cases displayed the biologicalbehaviour of the real world situation Therefore in orderto build a healthy and a sustainable population treatmentwith IvermectinMectizan drugs and a more hygienic envi-ronment that does not allow the breeding and survival ofblackflies could lead to a total eradication of the disease
Conflict of Interests
The authors declare that they have no conflict of interestsregarding the publication of this paper
References
[1] J H F Remme The Global Burden of Onchocerciasis in 1990World Health Organization Geneva Switzerland 2004 httpwwwwhointhealthinfoglobal burden diseaseOnchocercia-sis201990pdf
[2] WorldHealthOrganization ldquoPrevention of blindness and visualimpairment priority eye diseasesrdquo 2014 httpwwwwhointblindnesscausespriorityenindex3html
[3] World Health Organization Certification of Elimination ofHuman Onchocerciasis Criteria and Procedures World HealthOrganizationGeneva Switzerland 2001 httpwhqlibdocwhointhq2001WHO CDS CPE CEE 200118bpdf
[4] K A Lindblade B Arana G Zea-Flores et al ldquoEliminationof Onchocerca volvulus transmission in the Santa Rosa focusof Guatemalardquo The American Journal of Tropical Medicine andHygiene vol 77 pp 334ndash341 2007
[5] J C Anosike and C O E Onwuliri ldquoStudies on filariasis inBauchi State Nigeria 1 Endemicity of human onchocerciasisin Ningi Local Government Areardquo Annals of Tropical Medicineand Parasitology vol 89 no 1 pp 31ndash38 1995
[6] WorldHealthOrganizationWeekly Epidemiological Records 71WHO Geneva Switzerland 1993
12 Abstract and Applied Analysis
[7] WorldHealthOrganization httpwwwwhointneglected dis-easesen
[8] Sightsaver httpwwwsightsaversorgabout uspublicationsonchocerciasis
[9] E C Uttah ldquoOnchocerciasis in the Upper Ima River BasinrdquoIranian Journal of Parasitology vol 5 no 2 pp 33ndash41 2010
[10] J C Anosike and C O E Onwuliri ldquoStudies on filariasis inBauchi State Nigeriamdash1 endemicity of human onchocerciasisin Ningi Local Government Areardquo Annals of Tropical Medicineand Parasitology vol 89 no 1 pp 31ndash38 1995
[11] L D Edungbola and S O Asaolu ldquoParasitologic survey ofonchocerciasis (river blindness) in Babana district Kwara StateNigeriardquo American Journal of Tropical Medicine and Hygienevol 33 no 6 pp 1147ndash1154 1984
[12] O C Nwaorgu A Ohaegbula I E Onweluzo et al ldquoResultsof a large scale onchocercosis survey in Enugu State NigeriardquoJournal of Helminthology vol 68 no 2 pp 155ndash159 1994
[13] F O Akinbo and C E Okaka ldquoPrevalence and socio-economiceffects of onchocerciasis in Ovia North East Local GovernmentArea of Edo State Nigeriardquo Rivista Di Parasitologie vol 22 no3 pp 215ndash221 2005
[14] M D Wogu and C E Okaka ldquoPrevalence and socio-economic effects of onchocerciasis in Okpuje Owan WestLocal Government Area Edo State Nigeriardquo InternationalJournal of Biomedical and Health Sciences vol 4 no 3 2008httpwwwasopahorg
[15] S C O Noutchie ldquoHomotopy Decomposition analysis of atuberculosis modelrdquo Life Science Journal vol 10 no 3 pp 459ndash465 2013
[16] A Atangana and A Secer ldquoThe time-fractional coupled-Korteweg-de-Vries equationsrdquo Abstract and Applied Analysisvol 2013 Article ID 947986 8 pages 2013
[17] A Atangana and N Bildik ldquoApproximate solution of tubercu-losis disease population dynamics modelrdquo Abstract and AppliedAnalysis vol 2013 Article ID 759801 8 pages 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 7
minus1
611990531205751205821205732
4minus1
611990531205771205881205732
4+1
61198731199053120573212057324
+1
61198721199053120574120573
21205735
1198671198643(119905) = minus
1
611990531205771205881205732
1+1
6119873119905312057331minus1
61199053120577120588120573
11205732
+1
31198731199053120573211205732+1
61198731199053120573112057322
minus1
61199053120577120588120573
11205733+1
61198731199053120573211205733
+1
61198731199053120573112057321205733+1
611990531205751205821205732
3
+1
61198731199053120573112057323+1
61198721199053120574120573
11205735
1198671198783 [119905] =
1
611990531205771205881205732
1minus1
6119873119905312057331+1
31199053120577120588120573
11205732
minus1
21198731199053120573211205732+1
611990531205771205881205732
2minus1
21198731199053120573112057322
minus1
6119873119905312057332minus1
611987211990531205742120573
5
minus1
61198721199053120574120573
11205735
minus1
61198721199053120574120573
21205735minus1
611987211990531205741205732
5
1198811198783 [119905] =
1
61199053120577120588120573
21205734minus1
61198731199053120573112057321205734minus1
61198731199053120573221205734
+1
61199053120575120582120573
31205734+1
61198731199053120573112057331205734+1
611990531205751205821205732
4
+1
611990531205771205881205732
4minus1
61198731199053120573212057324+1
61199053120575120582120573
41205735
+1
61199053120577120588120573
41205735minus1
61198731199053120573212057341205735minus1
611987211990531205733
5
1198811198683 [119905] =
1
611987211990531205742120573
5minus1
61199053120575120582120573
41205735minus1
61199053120577120588120573
41205735
+1
61198731199053120573212057341205735+1
611987211990531205741205732
5+1
611987211990531205733
5
1198671198784 [119905] = minus
1
2411990541205771205881205733
1+1
24119873119905412057341minus1
811990541205771205881205732
11205732
+1
61198731199054120573311205732minus1
81199054120577120588120573
112057322+1
411987311990541205732112057322
minus1
2411990541205771205881205733
2+1
61198731199054120573112057332+1
24119873119905412057342
+1
2411987211990541205743120573
5+1
2411987211990541205742120573
11205735
+1
2411987211990541205741205732
11205735+1
2411987211990541205742120573
21205735
+1
121198721199054120574120573
112057321205735+1
2411987211990541205741205732
21205735
minus1
241199054120574120575120582120573
41205735minus1
241199054120574120577120588120573
41205735
+1
241198731199054120574120573
212057341205735+1
24119872119905412057421205732
5
+1
241198721199054120574120573
112057325+1
241198721199054120574120573
212057325+1
2411987211990541205741205733
5
1198671198644(119905) =
1
2411990541205771205881205733
1minus1
24119873119905412057341+1
1211990541205771205881205732
11205732
minus1
81198731199054120573311205732+1
241199054120577120588120573
112057322minus1
811987311990541205732112057322
minus1
241198731199054120573112057332+1
2411990541205771205881205732
11205733minus1
241198731199054120573311205733
+1
241199054120577120588120573
112057321205733minus1
1211987311990541205732112057321205733minus1
2411987311990541205731120573221205733
+1
241199054120577120588120573
112057323minus1
2411987311990541205732112057323minus1
2411987311990541205731120573212057323
minus1
2411990541205751205821205733
3minus1
241198731199054120573112057333minus1
2411987211990541205742120573
11205735
minus1
2411987211990541205741205732
11205735minus1
241198721199054120574120573
112057321205735
minus1
241198721199054120574120573
112057331205735minus1
241198721199054120574120573
112057325
1198671198684(119905) =
1
2411990541205771205881205732
11205732minus1
241198731199054120573311205732+1
121199054120577120588120573
112057322
minus1
811987311990541205732112057322+1
2411990541205771205881205733
2minus1
81198731199054120573112057332
minus1
24119873119905412057342minus1
2411990541205771205881205732
11205733+1
241198731199054120573311205733
minus1
241199054120577120588120573
112057321205733+1
1211987311990541205732112057321205733
+1
2411987311990541205731120573221205733
minus1
241199054120577120588120573
112057323+1
2411987311990541205732112057323+1
2411987311990541205731120573212057323
+1
2411990541205751205821205733
3+1
241198731199054120573112057333+1
241199054120577120588120573
112057321205734
minus1
2411987311990541205732112057321205734+1
2411990541205771205881205732
21205734minus1
1211987311990541205731120573221205734
minus1
241198731199054120573321205734minus1
241199054120577120588120573
112057331205734
+1
2411987311990541205732112057331205734+1
2411987311990541205731120573212057331205734
+1
2411990541205751205821205732
31205734+1
2411987311990541205731120573231205734
+1
241199054120577120588120573
212057324minus1
2411987311990541205731120573212057324minus1
2411987311990541205732212057324
+1
241199054120575120582120573
312057324+1
2411987311990541205731120573312057324+1
2411990541205751205821205733
4
+1
2411990541205771205881205733
4minus1
241198731199054120573212057334minus1
2411987211990541205742120573
21205735
minus1
241198721199054120574120573
112057321205735minus1
2411987211990541205741205732
21205735
+1
241198721199054120574120573
112057331205735
minus1
241198721199054120574120573
212057341205735minus1
241198721199054120574120573
212057325
1198811198784 [119905] = minus
1
241199054120577120588120573
112057321205734+1
2411987311990541205732112057321205734
minus1
2411990541205771205881205732
21205734+1
1211987311990541205731120573221205734
8 Abstract and Applied Analysis
+1
241198731199054120573321205734+1
241199054120577120588120573
112057331205734
minus1
2411987311990541205732112057331205734minus1
2411987311990541205731120573212057331205734
minus1
2411990541205751205821205732
31205734
minus1
2411987311990541205731120573231205734minus1
241199054120577120588120573
212057324
+1
2411987311990541205731120573212057324+1
2411987311990541205732212057324
minus1
241199054120575120582120573
312057324minus1
2411987311990541205731120573312057324
minus1
2411990541205751205821205733
4minus1
2411990541205771205881205733
4
+1
241198731199054120573212057334
+1
241198721199054120574120573
212057341205735minus1
241199054120577120588120573
212057341205735
+1
2411987311990541205731120573212057341205735+1
2411987311990541205732212057341205735
minus1
241199054120575120582120573
312057341205735minus1
2411987311990541205731120573312057341205735
minus1
2411990541205751205821205732
41205735minus1
2411990541205771205881205732
41205735+1
2411987311990541205732120573241205735
minus1
241199054120575120582120573
412057325minus1
241199054120577120588120573
412057325
+1
2411987311990541205732120573412057325+1
2411987211990541205734
5
1198811198684 [119905] = minus
1
2411987211990541205743120573
5+1
241199054120574120575120582120573
41205735+1
241199054120574120577120588120573
41205735
minus1
241198731199054120574120573
212057341205735+1
241199054120577120588120573
212057341205735
minus1
2411987311990541205731120573212057341205735minus1
2411987311990541205732212057341205735
+1
241199054120575120582120573
312057341205735+1
2411987311990541205731120573312057341205735
+1
2411990541205751205821205732
41205735+1
2411990541205771205881205732
41205735
minus1
2411987311990541205732120573241205735minus1
24119872119905412057421205732
5
+1
241199054120575120582120573
412057325+1
241199054120577120588120573
412057325
minus1
2411987311990541205732120573412057325minus1
2411987211990541205741205733
5minus1
2411987211990541205734
5
1198671198785 [119905] =
1
12011990551205771205881205734
1minus
1
120119873119905512057351+1
3011990551205771205881205733
11205732
minus1
241198731199055120573411205732+1
2011990551205771205881205732
112057322minus1
1211987311990551205733112057322
+1
301199055120577120588120573
112057332minus1
1211987311990551205732112057332+
1
12011990551205771205881205734
2
minus1
241198731199055120573112057342minus
1
120119873119905512057352minus
1
12011987211990551205744120573
5
minus1
12011987211990551205743120573
11205735minus
1
120119872119905512057421205732
11205735
minus1
12011987211990551205741205733
11205735minus
1
12011987211990551205743120573
21205735
minus1
6011987211990551205742120573
112057321205735minus1
4011987211990551205741205732
112057321205735
minus1
120119872119905512057421205732
21205735minus1
401198721199055120574120573
1120573221205735
minus1
12011987211990551205741205733
21205735+
1
12011990551205742120575120582120573
41205735
+1
12011990551205742120577120588120573
41205735+
1
1201199055120574120575120582120573
112057341205735
+1
1201199055120574120577120588120573
112057341205735minus
1
12011987311990551205742120573
212057341205735
+1
1201199055120574120575120582120573
212057341205735+1
601199055120574120577120588120573
212057341205735
minus1
601198731199055120574120573
1120573212057341205735minus1
6011987311990551205741205732
212057341205735
+1
1201199055120574120575120582120573
312057341205735+
1
1201198731199055120574120573
1120573312057341205735
+1
12011990551205741205751205821205732
41205735+
1
12011990551205741205771205881205732
41205735
minus1
1201198731199055120574120573
2120573241205735minus
1
120119872119905512057431205732
5
minus1
12011987211990551205742120573
112057325minus
1
12011987211990551205741205732
112057325
minus1
12011987211990551205742120573
212057325minus1
601198721199055120574120573
1120573212057325
minus1
12011987211990551205741205732
212057325+
1
1201199055120574120575120582120573
412057325
+1
1201199055120574120577120588120573
412057325minus
1
1201198731199055120574120573
2120573412057325
minus1
120119872119905512057421205733
5minus
1
1201198721199055120574120573
112057335
minus1
1201198721199055120574120573
212057335minus
1
12011987211990551205741205734
5
1198671198645 [119905] = minus
1
12011990551205771205881205734
1+
1
120119873119905512057351
minus1
4011990551205771205881205733
11205732+1
301198731199055120573411205732
minus1
4011990551205771205881205732
112057322+1
2011987311990551205733112057322
minus1
1201199055120577120588120573
112057332+1
3011987311990551205732112057332
+1
1201198731199055120573112057342
minus1
12011990551205771205881205733
11205733+
1
1201198731199055120573411205733
minus1
6011990551205771205881205732
112057321205733+1
4011987311990551205733112057321205733
minus1
1201199055120577120588120573
1120573221205733+1
40119873119905512057321120573221205733
+1
12011987311990551205731120573321205733minus
1
12011990551205771205881205732
112057323
+1
12011987311990551205733112057323minus
1
1201199055120577120588120573
1120573212057323
Abstract and Applied Analysis 9
+1
60119873119905512057321120573212057323+
1
120119873119905512057311205732212057323
minus1
1201199055120577120588120573
112057333+
1
12011987311990551205732112057333
+1
12011987311990551205731120573212057333+
1
12011990551205751205821205734
3
+1
1201198731199055120573112057343+
1
12011987211990551205743120573
11205735
+1
120119872119905512057421205732
11205735+
1
12011987211990551205741205733
11205735
+1
12011987211990551205742120573
112057321205735+1
6011987211990551205741205732
112057321205735
+1
1201198721199055120574120573
1120573221205735+
1
12011987211990551205742120573
112057331205735
+1
12011987211990551205741205732
112057331205735+
1
1201198721199055120574120573
1120573212057331205735
+1
1201198721199055120574120573
1120573231205735minus
1
1201199055120574120575120582120573
112057341205735
minus1
1201199055120574120577120588120573
112057341205735+
1
1201198731199055120574120573
1120573212057341205735
+1
12011987211990551205742120573
112057325+
1
12011987211990551205741205732
112057325
+1
1201198721199055120574120573
1120573212057325+
1
1201198721199055120574120573
1120573312057325
+1
1201198721199055120574120573
112057335
1198671198685 [119905]= minus
1
12011990551205771205881205733
11205732+
1
1201198731199055120573411205732minus1
4011990551205771205881205732
112057322
+1
3011987311990551205733112057322minus1
401199055120577120588120573
112057332+1
2011987311990551205732112057332
minus1
12011990551205771205881205734
2+1
301198731199055120573112057342+
1
120119873119905512057352
+1
12011990551205771205881205733
11205733minus
1
1201198731199055120573411205733
+1
6011990551205771205881205732
112057321205733minus1
4011987311990551205733112057321205733
+1
1201199055120577120588120573
1120573221205733minus1
40119873119905512057321120573221205733
minus1
12011987311990551205731120573321205733+
1
12011990551205771205881205732
112057323
minus1
12011987311990551205733112057323+
1
1201199055120577120588120573
1120573212057323
minus1
60119873119905512057321120573212057323minus
1
120119873119905512057311205732212057323
+1
1201199055120577120588120573
112057333minus
1
12011987311990551205732112057333
minus1
12011987311990551205731120573212057333minus
1
12011990551205751205821205734
3
minus1
1201198731199055120573112057343minus
1
12011990551205771205881205732
112057321205734
+1
12011987311990551205733112057321205734minus1
601199055120577120588120573
1120573221205734
+1
40119873119905512057321120573221205734minus
1
12011990551205771205881205733
21205734
+1
4011987311990551205731120573321205734+
1
1201198731199055120573421205734
+1
12011990551205771205881205732
112057331205734minus
1
12011987311990551205733112057331205734
+1
1201199055120577120588120573
1120573212057331205734minus1
60119873119905512057321120573212057331205734
minus1
120119873119905512057311205732212057331205734+
1
1201199055120577120588120573
1120573231205734
minus1
120119873119905512057321120573231205734minus
1
120119873119905512057311205732120573231205734
minus1
12011990551205751205821205733
31205734minus
1
12011987311990551205731120573331205734
minus1
1201199055120577120588120573
1120573212057324+
1
120119873119905512057321120573212057324
minus1
12011990551205771205881205732
212057324+1
60119873119905512057311205732212057324
+1
12011987311990551205733212057324+
1
1201199055120577120588120573
1120573312057324
minus1
120119873119905512057321120573312057324minus
1
120119873119905512057311205732120573312057324
minus1
12011990551205751205821205732
312057324minus
1
120119873119905512057311205732312057324
minus1
1201199055120577120588120573
212057334+
1
12011987311990551205731120573212057334
+1
12011987311990551205732212057334minus
1
1201199055120575120582120573
312057334
minus1
12011987311990551205731120573312057334minus
1
12011990551205751205821205734
4
minus1
12011990551205771205881205734
4+
1
1201198731199055120573212057344
+1
12011987211990551205743120573
21205735+
1
12011987211990551205742120573
112057321205735
+1
12011987211990551205741205732
112057321205735+
1
120119872119905512057421205732
21205735
+1
601198721199055120574120573
1120573221205735+
1
12011987211990551205741205733
21205735
minus1
12011987211990551205742120573
112057331205735minus
1
12011987211990551205741205732
112057331205735
minus1
1201198721199055120574120573
1120573212057331205735minus
1
1201198721199055120574120573
1120573231205735
+1
12011987211990551205742120573
212057341205735minus
1
1201199055120574120575120582120573
212057341205735
minus1
1201199055120574120577120588120573
212057341205735+
1
1201198721199055120574120573
1120573212057341205735
+1
12011987211990551205741205732
212057341205735+
1
12011987311990551205741205732
212057341205735
minus1
1201198721199055120574120573
1120573312057341205735+
1
1201198721199055120574120573
2120573241205735
+1
12011987211990551205742120573
212057325+
1
1201198721199055120574120573
1120573212057325
+1
12011987211990551205741205732
212057325minus
1
1201198721199055120574120573
1120573312057325
+1
1201198721199055120574120573
2120573412057325+
1
1201198721199055120574120573
212057335
10 Abstract and Applied Analysis
1198811198785 [119905] =
1
12011990551205771205881205732
112057321205734minus
1
12011987311990551205733112057321205734
+1
601199055120577120588120573
1120573221205734minus1
40119873119905512057321120573221205734
+1
12011990551205771205881205733
21205734minus1
4011987311990551205731120573321205734
minus1
1201198731199055120573421205734minus
1
12011990551205771205881205732
112057331205734
+1
12011987311990551205733112057331205734minus
1
1201199055120577120588120573
1120573212057331205734
+1
60119873119905512057321120573212057331205734+
1
120119873119905512057311205732212057331205734
minus1
1201199055120577120588120573
1120573231205734+
1
120119873119905512057321120573231205734
+1
120119873119905512057311205732120573231205734+
1
12011990551205751205821205733
31205734
+1
12011987311990551205731120573331205734+
1
1201199055120577120588120573
1120573212057324
minus1
120119873119905512057321120573212057324+
1
12011990551205771205881205732
212057324
minus1
60119873119905512057311205732212057324minus
1
12011987311990551205733212057324
minus1
1201199055120577120588120573
1120573312057324+
1
120119873119905512057321120573312057324
+1
120119873119905512057311205732120573312057324+
1
12011990551205751205821205732
312057324
+1
120119873119905512057311205732312057324+
1
1201199055120577120588120573
212057334
minus1
12011987311990551205731120573212057334minus
1
12011987311990551205732212057334
+1
1201199055120575120582120573
312057334+
1
12011987311990551205731120573312057334
+1
12011990551205751205821205734
4+
1
12011990551205771205881205734
4minus
1
1201198731199055120573212057344
minus1
12011987211990551205742120573
212057341205735minus
1
1201198721199055120574120573
1120573212057341205735
+1
1201199055120577120588120573
1120573212057341205735minus
1
120119873119905512057321120573212057341205735
minus1
12011987211990551205741205732
212057341205735+
1
12011990551205771205881205732
212057341205735
minus1
60119873119905512057311205732212057341205735minus
1
12011987311990551205733212057341205735
+1
1201198721199055120574120573
1120573312057341205735minus
1
1201199055120577120588120573
1120573312057341205735
+1
120119873119905512057321120573312057341205735+
1
120119873119905512057311205732120573312057341205735
+1
12011990551205751205821205732
312057341205735+
1
120119873119905512057311205732312057341205735
minus1
1201198721199055120574120573
2120573241205735+
1
1201199055120577120588120573
2120573241205735
minus1
120119873119905512057311205732120573241205735minus
1
120119873119905512057322120573241205735
+1
1201199055120575120582120573
3120573241205735+
1
120119873119905512057311205733120573241205735
+1
12011990551205751205821205733
41205735+
1
12011990551205771205881205733
41205735
minus1
12011987311990551205732120573341205735minus1
601198721199055120574120573
2120573412057325
+1
1201199055120577120588120573
2120573412057325minus
1
120119873119905512057311205732120573412057325
minus1
120119873119905512057322120573412057325+
1
1201199055120575120582120573
3120573412057325
+1
120119873119905512057311205733120573412057325+
1
12011990551205751205821205732
412057325
+1
12011990551205771205881205732
412057325minus
1
120119873119905512057321205732412057325
+1
1201199055120575120582120573
412057335+
1
1201199055120577120588120573
412057335
minus1
12011987311990551205732120573412057335minus
1
12011987211990551205735
5
1198811198685 [119905] =
1
12011987211990551205744120573
5minus
1
12011990551205742120575120582120573
41205735
minus1
12011990551205742120577120588120573
41205735+
1
12011987311990551205742120573
212057341205735
minus1
1201199055120574120577120588120573
212057341205735+
1
1201198731199055120574120573
1120573212057341205735
minus1
1201199055120577120588120573
1120573212057341205735+
1
120119873119905512057321120573212057341205735
+1
12011987311990551205741205732
212057341205735minus
1
12011990551205771205881205732
212057341205735
+1
60119873119905512057311205732212057341205735+
1
12011987311990551205733212057341205735
minus1
1201199055120574120575120582120573
312057341205735minus
1
1201198731199055120574120573
1120573312057341205735
+1
1201199055120577120588120573
1120573312057341205735minus
1
120119873119905512057321120573312057341205735
minus1
120119873119905512057311205732120573312057341205735minus
1
12011990551205751205821205732
312057341205735
minus1
120119873119905512057311205732312057341205735minus
1
12011990551205741205751205821205732
41205735
minus1
12011990551205741205771205881205732
41205735+
1
1201198731199055120574120573
2120573241205735
minus1
1201199055120577120588120573
2120573241205735+
1
120119873119905512057311205732120573241205735
+1
120119873119905512057322120573241205735minus
1
1201199055120575120582120573
3120573241205735
minus1
120119873119905512057311205733120573241205735minus
1
12011990551205751205821205733
41205735
minus1
12011990551205771205881205733
41205735+
1
12011987311990551205732120573341205735
+1
120119872119905512057431205732
5minus
1
1201199055120574120575120582120573
412057325
minus1
1201199055120574120577120588120573
412057325+
1
1201198721199055120574120573
2120573412057325
+1
1201198731199055120574120573
2120573412057325minus
1
1201199055120577120588120573
2120573412057325
+1
120119873119905512057311205732120573412057325+
1
120119873119905512057322120573412057325
Abstract and Applied Analysis 11
minus1
1201199055120575120582120573
3120573412057325minus
1
120119873119905512057311205733120573412057325
minus1
12011990551205751205821205732
412057325minus
1
12011990551205771205881205732
412057325
+1
120119873119905512057321205732412057325+
1
120119872119905512057421205733
5
minus1
1201199055120575120582120573
412057335minus
1
1201199055120577120588120573
412057335
+1
12011987311990551205732120573412057335+
1
12011987211990551205741205734
5
+1
12011987211990551205735
5
(33)
In general by the repeated application of the above proce-dure we have that
119867119878119899=119905119899
1198991198861119899 119867
119864119899=119905119899
1198991198862119899 119867
119868119899=119905119899
1198991198863119899
119881119878119899=119905119899
1198991198864119899 119881
119868119899=119905119899
1198991198865119899
(34)
where 1198861119899 1198862119899 1198863119899 1198864119899 1198865119899
depend on fixed set of empiricalparameters Hence the approximate solution of the modelabove is given as
119867119878(119905) =
infin
sum119899=0
119905119899
1198991198861119899
119867119864(119905) =
infin
sum119899=0
119905119899
1198991198862119899
119867119868(119905) =
infin
sum119899=0
119905119899
1198991198863119899
119881119878(119905) =
infin
sum119899=0
119905119899
1198991198864119899
119881119868(119905) =
infin
sum119899=0
119905119899
1198991198865119899
(35)
We therefore use the following theoretical parameters 1205731=
05 1205732= 05 120573
3= 05 120573
4= 05 120573
5= 05 120574 = 1 120577 = 1
119873 = 10 119872 = 10 120588 = 1 120575 = 1 and 120582 = 1 to examinethe behavior of the five different compartmental numericalsolutions in the system Figure 3 shows the behavior of thenumerical solution
7 Conclusion
Onchocerciasis model was examined and the mathematicalformulation underpinning the spread and control of thisdisease has been proposed on one hand On the otherhand we made use of some new analytical methods calledthe homotopy decomposition method (HDM) to derive thesolution of the resulting set of (1) to (5)The numerical resultswere presented to test the efficiency and the accuracy of themethod used The mathematical analysis of the resulting setof (1) to (5) above revealed that the stability analysis of the
05 10 15 20Time (per year)
2
4
6
8
10
Appr
oxim
ate s
olut
ions VS
HS
VI
HI
HE
Figure 3 Numerical solution of the model
five roots of the characteristics equations has all negativereal part In particular given that the stability analysis showsthat the DFE is asymptotically stable it would be possiblethat the parasite Onchocerca volvulus be eradicated in a finitetime On the other hand the numerical solution confirmedthat as the number of humans and vectors in the susceptiblecompartments reduces the number of infective humansand vectors in the infectious compartment increases Thenumerical solutions in both cases displayed the biologicalbehaviour of the real world situation Therefore in orderto build a healthy and a sustainable population treatmentwith IvermectinMectizan drugs and a more hygienic envi-ronment that does not allow the breeding and survival ofblackflies could lead to a total eradication of the disease
Conflict of Interests
The authors declare that they have no conflict of interestsregarding the publication of this paper
References
[1] J H F Remme The Global Burden of Onchocerciasis in 1990World Health Organization Geneva Switzerland 2004 httpwwwwhointhealthinfoglobal burden diseaseOnchocercia-sis201990pdf
[2] WorldHealthOrganization ldquoPrevention of blindness and visualimpairment priority eye diseasesrdquo 2014 httpwwwwhointblindnesscausespriorityenindex3html
[3] World Health Organization Certification of Elimination ofHuman Onchocerciasis Criteria and Procedures World HealthOrganizationGeneva Switzerland 2001 httpwhqlibdocwhointhq2001WHO CDS CPE CEE 200118bpdf
[4] K A Lindblade B Arana G Zea-Flores et al ldquoEliminationof Onchocerca volvulus transmission in the Santa Rosa focusof Guatemalardquo The American Journal of Tropical Medicine andHygiene vol 77 pp 334ndash341 2007
[5] J C Anosike and C O E Onwuliri ldquoStudies on filariasis inBauchi State Nigeria 1 Endemicity of human onchocerciasisin Ningi Local Government Areardquo Annals of Tropical Medicineand Parasitology vol 89 no 1 pp 31ndash38 1995
[6] WorldHealthOrganizationWeekly Epidemiological Records 71WHO Geneva Switzerland 1993
12 Abstract and Applied Analysis
[7] WorldHealthOrganization httpwwwwhointneglected dis-easesen
[8] Sightsaver httpwwwsightsaversorgabout uspublicationsonchocerciasis
[9] E C Uttah ldquoOnchocerciasis in the Upper Ima River BasinrdquoIranian Journal of Parasitology vol 5 no 2 pp 33ndash41 2010
[10] J C Anosike and C O E Onwuliri ldquoStudies on filariasis inBauchi State Nigeriamdash1 endemicity of human onchocerciasisin Ningi Local Government Areardquo Annals of Tropical Medicineand Parasitology vol 89 no 1 pp 31ndash38 1995
[11] L D Edungbola and S O Asaolu ldquoParasitologic survey ofonchocerciasis (river blindness) in Babana district Kwara StateNigeriardquo American Journal of Tropical Medicine and Hygienevol 33 no 6 pp 1147ndash1154 1984
[12] O C Nwaorgu A Ohaegbula I E Onweluzo et al ldquoResultsof a large scale onchocercosis survey in Enugu State NigeriardquoJournal of Helminthology vol 68 no 2 pp 155ndash159 1994
[13] F O Akinbo and C E Okaka ldquoPrevalence and socio-economiceffects of onchocerciasis in Ovia North East Local GovernmentArea of Edo State Nigeriardquo Rivista Di Parasitologie vol 22 no3 pp 215ndash221 2005
[14] M D Wogu and C E Okaka ldquoPrevalence and socio-economic effects of onchocerciasis in Okpuje Owan WestLocal Government Area Edo State Nigeriardquo InternationalJournal of Biomedical and Health Sciences vol 4 no 3 2008httpwwwasopahorg
[15] S C O Noutchie ldquoHomotopy Decomposition analysis of atuberculosis modelrdquo Life Science Journal vol 10 no 3 pp 459ndash465 2013
[16] A Atangana and A Secer ldquoThe time-fractional coupled-Korteweg-de-Vries equationsrdquo Abstract and Applied Analysisvol 2013 Article ID 947986 8 pages 2013
[17] A Atangana and N Bildik ldquoApproximate solution of tubercu-losis disease population dynamics modelrdquo Abstract and AppliedAnalysis vol 2013 Article ID 759801 8 pages 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Abstract and Applied Analysis
+1
241198731199054120573321205734+1
241199054120577120588120573
112057331205734
minus1
2411987311990541205732112057331205734minus1
2411987311990541205731120573212057331205734
minus1
2411990541205751205821205732
31205734
minus1
2411987311990541205731120573231205734minus1
241199054120577120588120573
212057324
+1
2411987311990541205731120573212057324+1
2411987311990541205732212057324
minus1
241199054120575120582120573
312057324minus1
2411987311990541205731120573312057324
minus1
2411990541205751205821205733
4minus1
2411990541205771205881205733
4
+1
241198731199054120573212057334
+1
241198721199054120574120573
212057341205735minus1
241199054120577120588120573
212057341205735
+1
2411987311990541205731120573212057341205735+1
2411987311990541205732212057341205735
minus1
241199054120575120582120573
312057341205735minus1
2411987311990541205731120573312057341205735
minus1
2411990541205751205821205732
41205735minus1
2411990541205771205881205732
41205735+1
2411987311990541205732120573241205735
minus1
241199054120575120582120573
412057325minus1
241199054120577120588120573
412057325
+1
2411987311990541205732120573412057325+1
2411987211990541205734
5
1198811198684 [119905] = minus
1
2411987211990541205743120573
5+1
241199054120574120575120582120573
41205735+1
241199054120574120577120588120573
41205735
minus1
241198731199054120574120573
212057341205735+1
241199054120577120588120573
212057341205735
minus1
2411987311990541205731120573212057341205735minus1
2411987311990541205732212057341205735
+1
241199054120575120582120573
312057341205735+1
2411987311990541205731120573312057341205735
+1
2411990541205751205821205732
41205735+1
2411990541205771205881205732
41205735
minus1
2411987311990541205732120573241205735minus1
24119872119905412057421205732
5
+1
241199054120575120582120573
412057325+1
241199054120577120588120573
412057325
minus1
2411987311990541205732120573412057325minus1
2411987211990541205741205733
5minus1
2411987211990541205734
5
1198671198785 [119905] =
1
12011990551205771205881205734
1minus
1
120119873119905512057351+1
3011990551205771205881205733
11205732
minus1
241198731199055120573411205732+1
2011990551205771205881205732
112057322minus1
1211987311990551205733112057322
+1
301199055120577120588120573
112057332minus1
1211987311990551205732112057332+
1
12011990551205771205881205734
2
minus1
241198731199055120573112057342minus
1
120119873119905512057352minus
1
12011987211990551205744120573
5
minus1
12011987211990551205743120573
11205735minus
1
120119872119905512057421205732
11205735
minus1
12011987211990551205741205733
11205735minus
1
12011987211990551205743120573
21205735
minus1
6011987211990551205742120573
112057321205735minus1
4011987211990551205741205732
112057321205735
minus1
120119872119905512057421205732
21205735minus1
401198721199055120574120573
1120573221205735
minus1
12011987211990551205741205733
21205735+
1
12011990551205742120575120582120573
41205735
+1
12011990551205742120577120588120573
41205735+
1
1201199055120574120575120582120573
112057341205735
+1
1201199055120574120577120588120573
112057341205735minus
1
12011987311990551205742120573
212057341205735
+1
1201199055120574120575120582120573
212057341205735+1
601199055120574120577120588120573
212057341205735
minus1
601198731199055120574120573
1120573212057341205735minus1
6011987311990551205741205732
212057341205735
+1
1201199055120574120575120582120573
312057341205735+
1
1201198731199055120574120573
1120573312057341205735
+1
12011990551205741205751205821205732
41205735+
1
12011990551205741205771205881205732
41205735
minus1
1201198731199055120574120573
2120573241205735minus
1
120119872119905512057431205732
5
minus1
12011987211990551205742120573
112057325minus
1
12011987211990551205741205732
112057325
minus1
12011987211990551205742120573
212057325minus1
601198721199055120574120573
1120573212057325
minus1
12011987211990551205741205732
212057325+
1
1201199055120574120575120582120573
412057325
+1
1201199055120574120577120588120573
412057325minus
1
1201198731199055120574120573
2120573412057325
minus1
120119872119905512057421205733
5minus
1
1201198721199055120574120573
112057335
minus1
1201198721199055120574120573
212057335minus
1
12011987211990551205741205734
5
1198671198645 [119905] = minus
1
12011990551205771205881205734
1+
1
120119873119905512057351
minus1
4011990551205771205881205733
11205732+1
301198731199055120573411205732
minus1
4011990551205771205881205732
112057322+1
2011987311990551205733112057322
minus1
1201199055120577120588120573
112057332+1
3011987311990551205732112057332
+1
1201198731199055120573112057342
minus1
12011990551205771205881205733
11205733+
1
1201198731199055120573411205733
minus1
6011990551205771205881205732
112057321205733+1
4011987311990551205733112057321205733
minus1
1201199055120577120588120573
1120573221205733+1
40119873119905512057321120573221205733
+1
12011987311990551205731120573321205733minus
1
12011990551205771205881205732
112057323
+1
12011987311990551205733112057323minus
1
1201199055120577120588120573
1120573212057323
Abstract and Applied Analysis 9
+1
60119873119905512057321120573212057323+
1
120119873119905512057311205732212057323
minus1
1201199055120577120588120573
112057333+
1
12011987311990551205732112057333
+1
12011987311990551205731120573212057333+
1
12011990551205751205821205734
3
+1
1201198731199055120573112057343+
1
12011987211990551205743120573
11205735
+1
120119872119905512057421205732
11205735+
1
12011987211990551205741205733
11205735
+1
12011987211990551205742120573
112057321205735+1
6011987211990551205741205732
112057321205735
+1
1201198721199055120574120573
1120573221205735+
1
12011987211990551205742120573
112057331205735
+1
12011987211990551205741205732
112057331205735+
1
1201198721199055120574120573
1120573212057331205735
+1
1201198721199055120574120573
1120573231205735minus
1
1201199055120574120575120582120573
112057341205735
minus1
1201199055120574120577120588120573
112057341205735+
1
1201198731199055120574120573
1120573212057341205735
+1
12011987211990551205742120573
112057325+
1
12011987211990551205741205732
112057325
+1
1201198721199055120574120573
1120573212057325+
1
1201198721199055120574120573
1120573312057325
+1
1201198721199055120574120573
112057335
1198671198685 [119905]= minus
1
12011990551205771205881205733
11205732+
1
1201198731199055120573411205732minus1
4011990551205771205881205732
112057322
+1
3011987311990551205733112057322minus1
401199055120577120588120573
112057332+1
2011987311990551205732112057332
minus1
12011990551205771205881205734
2+1
301198731199055120573112057342+
1
120119873119905512057352
+1
12011990551205771205881205733
11205733minus
1
1201198731199055120573411205733
+1
6011990551205771205881205732
112057321205733minus1
4011987311990551205733112057321205733
+1
1201199055120577120588120573
1120573221205733minus1
40119873119905512057321120573221205733
minus1
12011987311990551205731120573321205733+
1
12011990551205771205881205732
112057323
minus1
12011987311990551205733112057323+
1
1201199055120577120588120573
1120573212057323
minus1
60119873119905512057321120573212057323minus
1
120119873119905512057311205732212057323
+1
1201199055120577120588120573
112057333minus
1
12011987311990551205732112057333
minus1
12011987311990551205731120573212057333minus
1
12011990551205751205821205734
3
minus1
1201198731199055120573112057343minus
1
12011990551205771205881205732
112057321205734
+1
12011987311990551205733112057321205734minus1
601199055120577120588120573
1120573221205734
+1
40119873119905512057321120573221205734minus
1
12011990551205771205881205733
21205734
+1
4011987311990551205731120573321205734+
1
1201198731199055120573421205734
+1
12011990551205771205881205732
112057331205734minus
1
12011987311990551205733112057331205734
+1
1201199055120577120588120573
1120573212057331205734minus1
60119873119905512057321120573212057331205734
minus1
120119873119905512057311205732212057331205734+
1
1201199055120577120588120573
1120573231205734
minus1
120119873119905512057321120573231205734minus
1
120119873119905512057311205732120573231205734
minus1
12011990551205751205821205733
31205734minus
1
12011987311990551205731120573331205734
minus1
1201199055120577120588120573
1120573212057324+
1
120119873119905512057321120573212057324
minus1
12011990551205771205881205732
212057324+1
60119873119905512057311205732212057324
+1
12011987311990551205733212057324+
1
1201199055120577120588120573
1120573312057324
minus1
120119873119905512057321120573312057324minus
1
120119873119905512057311205732120573312057324
minus1
12011990551205751205821205732
312057324minus
1
120119873119905512057311205732312057324
minus1
1201199055120577120588120573
212057334+
1
12011987311990551205731120573212057334
+1
12011987311990551205732212057334minus
1
1201199055120575120582120573
312057334
minus1
12011987311990551205731120573312057334minus
1
12011990551205751205821205734
4
minus1
12011990551205771205881205734
4+
1
1201198731199055120573212057344
+1
12011987211990551205743120573
21205735+
1
12011987211990551205742120573
112057321205735
+1
12011987211990551205741205732
112057321205735+
1
120119872119905512057421205732
21205735
+1
601198721199055120574120573
1120573221205735+
1
12011987211990551205741205733
21205735
minus1
12011987211990551205742120573
112057331205735minus
1
12011987211990551205741205732
112057331205735
minus1
1201198721199055120574120573
1120573212057331205735minus
1
1201198721199055120574120573
1120573231205735
+1
12011987211990551205742120573
212057341205735minus
1
1201199055120574120575120582120573
212057341205735
minus1
1201199055120574120577120588120573
212057341205735+
1
1201198721199055120574120573
1120573212057341205735
+1
12011987211990551205741205732
212057341205735+
1
12011987311990551205741205732
212057341205735
minus1
1201198721199055120574120573
1120573312057341205735+
1
1201198721199055120574120573
2120573241205735
+1
12011987211990551205742120573
212057325+
1
1201198721199055120574120573
1120573212057325
+1
12011987211990551205741205732
212057325minus
1
1201198721199055120574120573
1120573312057325
+1
1201198721199055120574120573
2120573412057325+
1
1201198721199055120574120573
212057335
10 Abstract and Applied Analysis
1198811198785 [119905] =
1
12011990551205771205881205732
112057321205734minus
1
12011987311990551205733112057321205734
+1
601199055120577120588120573
1120573221205734minus1
40119873119905512057321120573221205734
+1
12011990551205771205881205733
21205734minus1
4011987311990551205731120573321205734
minus1
1201198731199055120573421205734minus
1
12011990551205771205881205732
112057331205734
+1
12011987311990551205733112057331205734minus
1
1201199055120577120588120573
1120573212057331205734
+1
60119873119905512057321120573212057331205734+
1
120119873119905512057311205732212057331205734
minus1
1201199055120577120588120573
1120573231205734+
1
120119873119905512057321120573231205734
+1
120119873119905512057311205732120573231205734+
1
12011990551205751205821205733
31205734
+1
12011987311990551205731120573331205734+
1
1201199055120577120588120573
1120573212057324
minus1
120119873119905512057321120573212057324+
1
12011990551205771205881205732
212057324
minus1
60119873119905512057311205732212057324minus
1
12011987311990551205733212057324
minus1
1201199055120577120588120573
1120573312057324+
1
120119873119905512057321120573312057324
+1
120119873119905512057311205732120573312057324+
1
12011990551205751205821205732
312057324
+1
120119873119905512057311205732312057324+
1
1201199055120577120588120573
212057334
minus1
12011987311990551205731120573212057334minus
1
12011987311990551205732212057334
+1
1201199055120575120582120573
312057334+
1
12011987311990551205731120573312057334
+1
12011990551205751205821205734
4+
1
12011990551205771205881205734
4minus
1
1201198731199055120573212057344
minus1
12011987211990551205742120573
212057341205735minus
1
1201198721199055120574120573
1120573212057341205735
+1
1201199055120577120588120573
1120573212057341205735minus
1
120119873119905512057321120573212057341205735
minus1
12011987211990551205741205732
212057341205735+
1
12011990551205771205881205732
212057341205735
minus1
60119873119905512057311205732212057341205735minus
1
12011987311990551205733212057341205735
+1
1201198721199055120574120573
1120573312057341205735minus
1
1201199055120577120588120573
1120573312057341205735
+1
120119873119905512057321120573312057341205735+
1
120119873119905512057311205732120573312057341205735
+1
12011990551205751205821205732
312057341205735+
1
120119873119905512057311205732312057341205735
minus1
1201198721199055120574120573
2120573241205735+
1
1201199055120577120588120573
2120573241205735
minus1
120119873119905512057311205732120573241205735minus
1
120119873119905512057322120573241205735
+1
1201199055120575120582120573
3120573241205735+
1
120119873119905512057311205733120573241205735
+1
12011990551205751205821205733
41205735+
1
12011990551205771205881205733
41205735
minus1
12011987311990551205732120573341205735minus1
601198721199055120574120573
2120573412057325
+1
1201199055120577120588120573
2120573412057325minus
1
120119873119905512057311205732120573412057325
minus1
120119873119905512057322120573412057325+
1
1201199055120575120582120573
3120573412057325
+1
120119873119905512057311205733120573412057325+
1
12011990551205751205821205732
412057325
+1
12011990551205771205881205732
412057325minus
1
120119873119905512057321205732412057325
+1
1201199055120575120582120573
412057335+
1
1201199055120577120588120573
412057335
minus1
12011987311990551205732120573412057335minus
1
12011987211990551205735
5
1198811198685 [119905] =
1
12011987211990551205744120573
5minus
1
12011990551205742120575120582120573
41205735
minus1
12011990551205742120577120588120573
41205735+
1
12011987311990551205742120573
212057341205735
minus1
1201199055120574120577120588120573
212057341205735+
1
1201198731199055120574120573
1120573212057341205735
minus1
1201199055120577120588120573
1120573212057341205735+
1
120119873119905512057321120573212057341205735
+1
12011987311990551205741205732
212057341205735minus
1
12011990551205771205881205732
212057341205735
+1
60119873119905512057311205732212057341205735+
1
12011987311990551205733212057341205735
minus1
1201199055120574120575120582120573
312057341205735minus
1
1201198731199055120574120573
1120573312057341205735
+1
1201199055120577120588120573
1120573312057341205735minus
1
120119873119905512057321120573312057341205735
minus1
120119873119905512057311205732120573312057341205735minus
1
12011990551205751205821205732
312057341205735
minus1
120119873119905512057311205732312057341205735minus
1
12011990551205741205751205821205732
41205735
minus1
12011990551205741205771205881205732
41205735+
1
1201198731199055120574120573
2120573241205735
minus1
1201199055120577120588120573
2120573241205735+
1
120119873119905512057311205732120573241205735
+1
120119873119905512057322120573241205735minus
1
1201199055120575120582120573
3120573241205735
minus1
120119873119905512057311205733120573241205735minus
1
12011990551205751205821205733
41205735
minus1
12011990551205771205881205733
41205735+
1
12011987311990551205732120573341205735
+1
120119872119905512057431205732
5minus
1
1201199055120574120575120582120573
412057325
minus1
1201199055120574120577120588120573
412057325+
1
1201198721199055120574120573
2120573412057325
+1
1201198731199055120574120573
2120573412057325minus
1
1201199055120577120588120573
2120573412057325
+1
120119873119905512057311205732120573412057325+
1
120119873119905512057322120573412057325
Abstract and Applied Analysis 11
minus1
1201199055120575120582120573
3120573412057325minus
1
120119873119905512057311205733120573412057325
minus1
12011990551205751205821205732
412057325minus
1
12011990551205771205881205732
412057325
+1
120119873119905512057321205732412057325+
1
120119872119905512057421205733
5
minus1
1201199055120575120582120573
412057335minus
1
1201199055120577120588120573
412057335
+1
12011987311990551205732120573412057335+
1
12011987211990551205741205734
5
+1
12011987211990551205735
5
(33)
In general by the repeated application of the above proce-dure we have that
119867119878119899=119905119899
1198991198861119899 119867
119864119899=119905119899
1198991198862119899 119867
119868119899=119905119899
1198991198863119899
119881119878119899=119905119899
1198991198864119899 119881
119868119899=119905119899
1198991198865119899
(34)
where 1198861119899 1198862119899 1198863119899 1198864119899 1198865119899
depend on fixed set of empiricalparameters Hence the approximate solution of the modelabove is given as
119867119878(119905) =
infin
sum119899=0
119905119899
1198991198861119899
119867119864(119905) =
infin
sum119899=0
119905119899
1198991198862119899
119867119868(119905) =
infin
sum119899=0
119905119899
1198991198863119899
119881119878(119905) =
infin
sum119899=0
119905119899
1198991198864119899
119881119868(119905) =
infin
sum119899=0
119905119899
1198991198865119899
(35)
We therefore use the following theoretical parameters 1205731=
05 1205732= 05 120573
3= 05 120573
4= 05 120573
5= 05 120574 = 1 120577 = 1
119873 = 10 119872 = 10 120588 = 1 120575 = 1 and 120582 = 1 to examinethe behavior of the five different compartmental numericalsolutions in the system Figure 3 shows the behavior of thenumerical solution
7 Conclusion
Onchocerciasis model was examined and the mathematicalformulation underpinning the spread and control of thisdisease has been proposed on one hand On the otherhand we made use of some new analytical methods calledthe homotopy decomposition method (HDM) to derive thesolution of the resulting set of (1) to (5)The numerical resultswere presented to test the efficiency and the accuracy of themethod used The mathematical analysis of the resulting setof (1) to (5) above revealed that the stability analysis of the
05 10 15 20Time (per year)
2
4
6
8
10
Appr
oxim
ate s
olut
ions VS
HS
VI
HI
HE
Figure 3 Numerical solution of the model
five roots of the characteristics equations has all negativereal part In particular given that the stability analysis showsthat the DFE is asymptotically stable it would be possiblethat the parasite Onchocerca volvulus be eradicated in a finitetime On the other hand the numerical solution confirmedthat as the number of humans and vectors in the susceptiblecompartments reduces the number of infective humansand vectors in the infectious compartment increases Thenumerical solutions in both cases displayed the biologicalbehaviour of the real world situation Therefore in orderto build a healthy and a sustainable population treatmentwith IvermectinMectizan drugs and a more hygienic envi-ronment that does not allow the breeding and survival ofblackflies could lead to a total eradication of the disease
Conflict of Interests
The authors declare that they have no conflict of interestsregarding the publication of this paper
References
[1] J H F Remme The Global Burden of Onchocerciasis in 1990World Health Organization Geneva Switzerland 2004 httpwwwwhointhealthinfoglobal burden diseaseOnchocercia-sis201990pdf
[2] WorldHealthOrganization ldquoPrevention of blindness and visualimpairment priority eye diseasesrdquo 2014 httpwwwwhointblindnesscausespriorityenindex3html
[3] World Health Organization Certification of Elimination ofHuman Onchocerciasis Criteria and Procedures World HealthOrganizationGeneva Switzerland 2001 httpwhqlibdocwhointhq2001WHO CDS CPE CEE 200118bpdf
[4] K A Lindblade B Arana G Zea-Flores et al ldquoEliminationof Onchocerca volvulus transmission in the Santa Rosa focusof Guatemalardquo The American Journal of Tropical Medicine andHygiene vol 77 pp 334ndash341 2007
[5] J C Anosike and C O E Onwuliri ldquoStudies on filariasis inBauchi State Nigeria 1 Endemicity of human onchocerciasisin Ningi Local Government Areardquo Annals of Tropical Medicineand Parasitology vol 89 no 1 pp 31ndash38 1995
[6] WorldHealthOrganizationWeekly Epidemiological Records 71WHO Geneva Switzerland 1993
12 Abstract and Applied Analysis
[7] WorldHealthOrganization httpwwwwhointneglected dis-easesen
[8] Sightsaver httpwwwsightsaversorgabout uspublicationsonchocerciasis
[9] E C Uttah ldquoOnchocerciasis in the Upper Ima River BasinrdquoIranian Journal of Parasitology vol 5 no 2 pp 33ndash41 2010
[10] J C Anosike and C O E Onwuliri ldquoStudies on filariasis inBauchi State Nigeriamdash1 endemicity of human onchocerciasisin Ningi Local Government Areardquo Annals of Tropical Medicineand Parasitology vol 89 no 1 pp 31ndash38 1995
[11] L D Edungbola and S O Asaolu ldquoParasitologic survey ofonchocerciasis (river blindness) in Babana district Kwara StateNigeriardquo American Journal of Tropical Medicine and Hygienevol 33 no 6 pp 1147ndash1154 1984
[12] O C Nwaorgu A Ohaegbula I E Onweluzo et al ldquoResultsof a large scale onchocercosis survey in Enugu State NigeriardquoJournal of Helminthology vol 68 no 2 pp 155ndash159 1994
[13] F O Akinbo and C E Okaka ldquoPrevalence and socio-economiceffects of onchocerciasis in Ovia North East Local GovernmentArea of Edo State Nigeriardquo Rivista Di Parasitologie vol 22 no3 pp 215ndash221 2005
[14] M D Wogu and C E Okaka ldquoPrevalence and socio-economic effects of onchocerciasis in Okpuje Owan WestLocal Government Area Edo State Nigeriardquo InternationalJournal of Biomedical and Health Sciences vol 4 no 3 2008httpwwwasopahorg
[15] S C O Noutchie ldquoHomotopy Decomposition analysis of atuberculosis modelrdquo Life Science Journal vol 10 no 3 pp 459ndash465 2013
[16] A Atangana and A Secer ldquoThe time-fractional coupled-Korteweg-de-Vries equationsrdquo Abstract and Applied Analysisvol 2013 Article ID 947986 8 pages 2013
[17] A Atangana and N Bildik ldquoApproximate solution of tubercu-losis disease population dynamics modelrdquo Abstract and AppliedAnalysis vol 2013 Article ID 759801 8 pages 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 9
+1
60119873119905512057321120573212057323+
1
120119873119905512057311205732212057323
minus1
1201199055120577120588120573
112057333+
1
12011987311990551205732112057333
+1
12011987311990551205731120573212057333+
1
12011990551205751205821205734
3
+1
1201198731199055120573112057343+
1
12011987211990551205743120573
11205735
+1
120119872119905512057421205732
11205735+
1
12011987211990551205741205733
11205735
+1
12011987211990551205742120573
112057321205735+1
6011987211990551205741205732
112057321205735
+1
1201198721199055120574120573
1120573221205735+
1
12011987211990551205742120573
112057331205735
+1
12011987211990551205741205732
112057331205735+
1
1201198721199055120574120573
1120573212057331205735
+1
1201198721199055120574120573
1120573231205735minus
1
1201199055120574120575120582120573
112057341205735
minus1
1201199055120574120577120588120573
112057341205735+
1
1201198731199055120574120573
1120573212057341205735
+1
12011987211990551205742120573
112057325+
1
12011987211990551205741205732
112057325
+1
1201198721199055120574120573
1120573212057325+
1
1201198721199055120574120573
1120573312057325
+1
1201198721199055120574120573
112057335
1198671198685 [119905]= minus
1
12011990551205771205881205733
11205732+
1
1201198731199055120573411205732minus1
4011990551205771205881205732
112057322
+1
3011987311990551205733112057322minus1
401199055120577120588120573
112057332+1
2011987311990551205732112057332
minus1
12011990551205771205881205734
2+1
301198731199055120573112057342+
1
120119873119905512057352
+1
12011990551205771205881205733
11205733minus
1
1201198731199055120573411205733
+1
6011990551205771205881205732
112057321205733minus1
4011987311990551205733112057321205733
+1
1201199055120577120588120573
1120573221205733minus1
40119873119905512057321120573221205733
minus1
12011987311990551205731120573321205733+
1
12011990551205771205881205732
112057323
minus1
12011987311990551205733112057323+
1
1201199055120577120588120573
1120573212057323
minus1
60119873119905512057321120573212057323minus
1
120119873119905512057311205732212057323
+1
1201199055120577120588120573
112057333minus
1
12011987311990551205732112057333
minus1
12011987311990551205731120573212057333minus
1
12011990551205751205821205734
3
minus1
1201198731199055120573112057343minus
1
12011990551205771205881205732
112057321205734
+1
12011987311990551205733112057321205734minus1
601199055120577120588120573
1120573221205734
+1
40119873119905512057321120573221205734minus
1
12011990551205771205881205733
21205734
+1
4011987311990551205731120573321205734+
1
1201198731199055120573421205734
+1
12011990551205771205881205732
112057331205734minus
1
12011987311990551205733112057331205734
+1
1201199055120577120588120573
1120573212057331205734minus1
60119873119905512057321120573212057331205734
minus1
120119873119905512057311205732212057331205734+
1
1201199055120577120588120573
1120573231205734
minus1
120119873119905512057321120573231205734minus
1
120119873119905512057311205732120573231205734
minus1
12011990551205751205821205733
31205734minus
1
12011987311990551205731120573331205734
minus1
1201199055120577120588120573
1120573212057324+
1
120119873119905512057321120573212057324
minus1
12011990551205771205881205732
212057324+1
60119873119905512057311205732212057324
+1
12011987311990551205733212057324+
1
1201199055120577120588120573
1120573312057324
minus1
120119873119905512057321120573312057324minus
1
120119873119905512057311205732120573312057324
minus1
12011990551205751205821205732
312057324minus
1
120119873119905512057311205732312057324
minus1
1201199055120577120588120573
212057334+
1
12011987311990551205731120573212057334
+1
12011987311990551205732212057334minus
1
1201199055120575120582120573
312057334
minus1
12011987311990551205731120573312057334minus
1
12011990551205751205821205734
4
minus1
12011990551205771205881205734
4+
1
1201198731199055120573212057344
+1
12011987211990551205743120573
21205735+
1
12011987211990551205742120573
112057321205735
+1
12011987211990551205741205732
112057321205735+
1
120119872119905512057421205732
21205735
+1
601198721199055120574120573
1120573221205735+
1
12011987211990551205741205733
21205735
minus1
12011987211990551205742120573
112057331205735minus
1
12011987211990551205741205732
112057331205735
minus1
1201198721199055120574120573
1120573212057331205735minus
1
1201198721199055120574120573
1120573231205735
+1
12011987211990551205742120573
212057341205735minus
1
1201199055120574120575120582120573
212057341205735
minus1
1201199055120574120577120588120573
212057341205735+
1
1201198721199055120574120573
1120573212057341205735
+1
12011987211990551205741205732
212057341205735+
1
12011987311990551205741205732
212057341205735
minus1
1201198721199055120574120573
1120573312057341205735+
1
1201198721199055120574120573
2120573241205735
+1
12011987211990551205742120573
212057325+
1
1201198721199055120574120573
1120573212057325
+1
12011987211990551205741205732
212057325minus
1
1201198721199055120574120573
1120573312057325
+1
1201198721199055120574120573
2120573412057325+
1
1201198721199055120574120573
212057335
10 Abstract and Applied Analysis
1198811198785 [119905] =
1
12011990551205771205881205732
112057321205734minus
1
12011987311990551205733112057321205734
+1
601199055120577120588120573
1120573221205734minus1
40119873119905512057321120573221205734
+1
12011990551205771205881205733
21205734minus1
4011987311990551205731120573321205734
minus1
1201198731199055120573421205734minus
1
12011990551205771205881205732
112057331205734
+1
12011987311990551205733112057331205734minus
1
1201199055120577120588120573
1120573212057331205734
+1
60119873119905512057321120573212057331205734+
1
120119873119905512057311205732212057331205734
minus1
1201199055120577120588120573
1120573231205734+
1
120119873119905512057321120573231205734
+1
120119873119905512057311205732120573231205734+
1
12011990551205751205821205733
31205734
+1
12011987311990551205731120573331205734+
1
1201199055120577120588120573
1120573212057324
minus1
120119873119905512057321120573212057324+
1
12011990551205771205881205732
212057324
minus1
60119873119905512057311205732212057324minus
1
12011987311990551205733212057324
minus1
1201199055120577120588120573
1120573312057324+
1
120119873119905512057321120573312057324
+1
120119873119905512057311205732120573312057324+
1
12011990551205751205821205732
312057324
+1
120119873119905512057311205732312057324+
1
1201199055120577120588120573
212057334
minus1
12011987311990551205731120573212057334minus
1
12011987311990551205732212057334
+1
1201199055120575120582120573
312057334+
1
12011987311990551205731120573312057334
+1
12011990551205751205821205734
4+
1
12011990551205771205881205734
4minus
1
1201198731199055120573212057344
minus1
12011987211990551205742120573
212057341205735minus
1
1201198721199055120574120573
1120573212057341205735
+1
1201199055120577120588120573
1120573212057341205735minus
1
120119873119905512057321120573212057341205735
minus1
12011987211990551205741205732
212057341205735+
1
12011990551205771205881205732
212057341205735
minus1
60119873119905512057311205732212057341205735minus
1
12011987311990551205733212057341205735
+1
1201198721199055120574120573
1120573312057341205735minus
1
1201199055120577120588120573
1120573312057341205735
+1
120119873119905512057321120573312057341205735+
1
120119873119905512057311205732120573312057341205735
+1
12011990551205751205821205732
312057341205735+
1
120119873119905512057311205732312057341205735
minus1
1201198721199055120574120573
2120573241205735+
1
1201199055120577120588120573
2120573241205735
minus1
120119873119905512057311205732120573241205735minus
1
120119873119905512057322120573241205735
+1
1201199055120575120582120573
3120573241205735+
1
120119873119905512057311205733120573241205735
+1
12011990551205751205821205733
41205735+
1
12011990551205771205881205733
41205735
minus1
12011987311990551205732120573341205735minus1
601198721199055120574120573
2120573412057325
+1
1201199055120577120588120573
2120573412057325minus
1
120119873119905512057311205732120573412057325
minus1
120119873119905512057322120573412057325+
1
1201199055120575120582120573
3120573412057325
+1
120119873119905512057311205733120573412057325+
1
12011990551205751205821205732
412057325
+1
12011990551205771205881205732
412057325minus
1
120119873119905512057321205732412057325
+1
1201199055120575120582120573
412057335+
1
1201199055120577120588120573
412057335
minus1
12011987311990551205732120573412057335minus
1
12011987211990551205735
5
1198811198685 [119905] =
1
12011987211990551205744120573
5minus
1
12011990551205742120575120582120573
41205735
minus1
12011990551205742120577120588120573
41205735+
1
12011987311990551205742120573
212057341205735
minus1
1201199055120574120577120588120573
212057341205735+
1
1201198731199055120574120573
1120573212057341205735
minus1
1201199055120577120588120573
1120573212057341205735+
1
120119873119905512057321120573212057341205735
+1
12011987311990551205741205732
212057341205735minus
1
12011990551205771205881205732
212057341205735
+1
60119873119905512057311205732212057341205735+
1
12011987311990551205733212057341205735
minus1
1201199055120574120575120582120573
312057341205735minus
1
1201198731199055120574120573
1120573312057341205735
+1
1201199055120577120588120573
1120573312057341205735minus
1
120119873119905512057321120573312057341205735
minus1
120119873119905512057311205732120573312057341205735minus
1
12011990551205751205821205732
312057341205735
minus1
120119873119905512057311205732312057341205735minus
1
12011990551205741205751205821205732
41205735
minus1
12011990551205741205771205881205732
41205735+
1
1201198731199055120574120573
2120573241205735
minus1
1201199055120577120588120573
2120573241205735+
1
120119873119905512057311205732120573241205735
+1
120119873119905512057322120573241205735minus
1
1201199055120575120582120573
3120573241205735
minus1
120119873119905512057311205733120573241205735minus
1
12011990551205751205821205733
41205735
minus1
12011990551205771205881205733
41205735+
1
12011987311990551205732120573341205735
+1
120119872119905512057431205732
5minus
1
1201199055120574120575120582120573
412057325
minus1
1201199055120574120577120588120573
412057325+
1
1201198721199055120574120573
2120573412057325
+1
1201198731199055120574120573
2120573412057325minus
1
1201199055120577120588120573
2120573412057325
+1
120119873119905512057311205732120573412057325+
1
120119873119905512057322120573412057325
Abstract and Applied Analysis 11
minus1
1201199055120575120582120573
3120573412057325minus
1
120119873119905512057311205733120573412057325
minus1
12011990551205751205821205732
412057325minus
1
12011990551205771205881205732
412057325
+1
120119873119905512057321205732412057325+
1
120119872119905512057421205733
5
minus1
1201199055120575120582120573
412057335minus
1
1201199055120577120588120573
412057335
+1
12011987311990551205732120573412057335+
1
12011987211990551205741205734
5
+1
12011987211990551205735
5
(33)
In general by the repeated application of the above proce-dure we have that
119867119878119899=119905119899
1198991198861119899 119867
119864119899=119905119899
1198991198862119899 119867
119868119899=119905119899
1198991198863119899
119881119878119899=119905119899
1198991198864119899 119881
119868119899=119905119899
1198991198865119899
(34)
where 1198861119899 1198862119899 1198863119899 1198864119899 1198865119899
depend on fixed set of empiricalparameters Hence the approximate solution of the modelabove is given as
119867119878(119905) =
infin
sum119899=0
119905119899
1198991198861119899
119867119864(119905) =
infin
sum119899=0
119905119899
1198991198862119899
119867119868(119905) =
infin
sum119899=0
119905119899
1198991198863119899
119881119878(119905) =
infin
sum119899=0
119905119899
1198991198864119899
119881119868(119905) =
infin
sum119899=0
119905119899
1198991198865119899
(35)
We therefore use the following theoretical parameters 1205731=
05 1205732= 05 120573
3= 05 120573
4= 05 120573
5= 05 120574 = 1 120577 = 1
119873 = 10 119872 = 10 120588 = 1 120575 = 1 and 120582 = 1 to examinethe behavior of the five different compartmental numericalsolutions in the system Figure 3 shows the behavior of thenumerical solution
7 Conclusion
Onchocerciasis model was examined and the mathematicalformulation underpinning the spread and control of thisdisease has been proposed on one hand On the otherhand we made use of some new analytical methods calledthe homotopy decomposition method (HDM) to derive thesolution of the resulting set of (1) to (5)The numerical resultswere presented to test the efficiency and the accuracy of themethod used The mathematical analysis of the resulting setof (1) to (5) above revealed that the stability analysis of the
05 10 15 20Time (per year)
2
4
6
8
10
Appr
oxim
ate s
olut
ions VS
HS
VI
HI
HE
Figure 3 Numerical solution of the model
five roots of the characteristics equations has all negativereal part In particular given that the stability analysis showsthat the DFE is asymptotically stable it would be possiblethat the parasite Onchocerca volvulus be eradicated in a finitetime On the other hand the numerical solution confirmedthat as the number of humans and vectors in the susceptiblecompartments reduces the number of infective humansand vectors in the infectious compartment increases Thenumerical solutions in both cases displayed the biologicalbehaviour of the real world situation Therefore in orderto build a healthy and a sustainable population treatmentwith IvermectinMectizan drugs and a more hygienic envi-ronment that does not allow the breeding and survival ofblackflies could lead to a total eradication of the disease
Conflict of Interests
The authors declare that they have no conflict of interestsregarding the publication of this paper
References
[1] J H F Remme The Global Burden of Onchocerciasis in 1990World Health Organization Geneva Switzerland 2004 httpwwwwhointhealthinfoglobal burden diseaseOnchocercia-sis201990pdf
[2] WorldHealthOrganization ldquoPrevention of blindness and visualimpairment priority eye diseasesrdquo 2014 httpwwwwhointblindnesscausespriorityenindex3html
[3] World Health Organization Certification of Elimination ofHuman Onchocerciasis Criteria and Procedures World HealthOrganizationGeneva Switzerland 2001 httpwhqlibdocwhointhq2001WHO CDS CPE CEE 200118bpdf
[4] K A Lindblade B Arana G Zea-Flores et al ldquoEliminationof Onchocerca volvulus transmission in the Santa Rosa focusof Guatemalardquo The American Journal of Tropical Medicine andHygiene vol 77 pp 334ndash341 2007
[5] J C Anosike and C O E Onwuliri ldquoStudies on filariasis inBauchi State Nigeria 1 Endemicity of human onchocerciasisin Ningi Local Government Areardquo Annals of Tropical Medicineand Parasitology vol 89 no 1 pp 31ndash38 1995
[6] WorldHealthOrganizationWeekly Epidemiological Records 71WHO Geneva Switzerland 1993
12 Abstract and Applied Analysis
[7] WorldHealthOrganization httpwwwwhointneglected dis-easesen
[8] Sightsaver httpwwwsightsaversorgabout uspublicationsonchocerciasis
[9] E C Uttah ldquoOnchocerciasis in the Upper Ima River BasinrdquoIranian Journal of Parasitology vol 5 no 2 pp 33ndash41 2010
[10] J C Anosike and C O E Onwuliri ldquoStudies on filariasis inBauchi State Nigeriamdash1 endemicity of human onchocerciasisin Ningi Local Government Areardquo Annals of Tropical Medicineand Parasitology vol 89 no 1 pp 31ndash38 1995
[11] L D Edungbola and S O Asaolu ldquoParasitologic survey ofonchocerciasis (river blindness) in Babana district Kwara StateNigeriardquo American Journal of Tropical Medicine and Hygienevol 33 no 6 pp 1147ndash1154 1984
[12] O C Nwaorgu A Ohaegbula I E Onweluzo et al ldquoResultsof a large scale onchocercosis survey in Enugu State NigeriardquoJournal of Helminthology vol 68 no 2 pp 155ndash159 1994
[13] F O Akinbo and C E Okaka ldquoPrevalence and socio-economiceffects of onchocerciasis in Ovia North East Local GovernmentArea of Edo State Nigeriardquo Rivista Di Parasitologie vol 22 no3 pp 215ndash221 2005
[14] M D Wogu and C E Okaka ldquoPrevalence and socio-economic effects of onchocerciasis in Okpuje Owan WestLocal Government Area Edo State Nigeriardquo InternationalJournal of Biomedical and Health Sciences vol 4 no 3 2008httpwwwasopahorg
[15] S C O Noutchie ldquoHomotopy Decomposition analysis of atuberculosis modelrdquo Life Science Journal vol 10 no 3 pp 459ndash465 2013
[16] A Atangana and A Secer ldquoThe time-fractional coupled-Korteweg-de-Vries equationsrdquo Abstract and Applied Analysisvol 2013 Article ID 947986 8 pages 2013
[17] A Atangana and N Bildik ldquoApproximate solution of tubercu-losis disease population dynamics modelrdquo Abstract and AppliedAnalysis vol 2013 Article ID 759801 8 pages 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
10 Abstract and Applied Analysis
1198811198785 [119905] =
1
12011990551205771205881205732
112057321205734minus
1
12011987311990551205733112057321205734
+1
601199055120577120588120573
1120573221205734minus1
40119873119905512057321120573221205734
+1
12011990551205771205881205733
21205734minus1
4011987311990551205731120573321205734
minus1
1201198731199055120573421205734minus
1
12011990551205771205881205732
112057331205734
+1
12011987311990551205733112057331205734minus
1
1201199055120577120588120573
1120573212057331205734
+1
60119873119905512057321120573212057331205734+
1
120119873119905512057311205732212057331205734
minus1
1201199055120577120588120573
1120573231205734+
1
120119873119905512057321120573231205734
+1
120119873119905512057311205732120573231205734+
1
12011990551205751205821205733
31205734
+1
12011987311990551205731120573331205734+
1
1201199055120577120588120573
1120573212057324
minus1
120119873119905512057321120573212057324+
1
12011990551205771205881205732
212057324
minus1
60119873119905512057311205732212057324minus
1
12011987311990551205733212057324
minus1
1201199055120577120588120573
1120573312057324+
1
120119873119905512057321120573312057324
+1
120119873119905512057311205732120573312057324+
1
12011990551205751205821205732
312057324
+1
120119873119905512057311205732312057324+
1
1201199055120577120588120573
212057334
minus1
12011987311990551205731120573212057334minus
1
12011987311990551205732212057334
+1
1201199055120575120582120573
312057334+
1
12011987311990551205731120573312057334
+1
12011990551205751205821205734
4+
1
12011990551205771205881205734
4minus
1
1201198731199055120573212057344
minus1
12011987211990551205742120573
212057341205735minus
1
1201198721199055120574120573
1120573212057341205735
+1
1201199055120577120588120573
1120573212057341205735minus
1
120119873119905512057321120573212057341205735
minus1
12011987211990551205741205732
212057341205735+
1
12011990551205771205881205732
212057341205735
minus1
60119873119905512057311205732212057341205735minus
1
12011987311990551205733212057341205735
+1
1201198721199055120574120573
1120573312057341205735minus
1
1201199055120577120588120573
1120573312057341205735
+1
120119873119905512057321120573312057341205735+
1
120119873119905512057311205732120573312057341205735
+1
12011990551205751205821205732
312057341205735+
1
120119873119905512057311205732312057341205735
minus1
1201198721199055120574120573
2120573241205735+
1
1201199055120577120588120573
2120573241205735
minus1
120119873119905512057311205732120573241205735minus
1
120119873119905512057322120573241205735
+1
1201199055120575120582120573
3120573241205735+
1
120119873119905512057311205733120573241205735
+1
12011990551205751205821205733
41205735+
1
12011990551205771205881205733
41205735
minus1
12011987311990551205732120573341205735minus1
601198721199055120574120573
2120573412057325
+1
1201199055120577120588120573
2120573412057325minus
1
120119873119905512057311205732120573412057325
minus1
120119873119905512057322120573412057325+
1
1201199055120575120582120573
3120573412057325
+1
120119873119905512057311205733120573412057325+
1
12011990551205751205821205732
412057325
+1
12011990551205771205881205732
412057325minus
1
120119873119905512057321205732412057325
+1
1201199055120575120582120573
412057335+
1
1201199055120577120588120573
412057335
minus1
12011987311990551205732120573412057335minus
1
12011987211990551205735
5
1198811198685 [119905] =
1
12011987211990551205744120573
5minus
1
12011990551205742120575120582120573
41205735
minus1
12011990551205742120577120588120573
41205735+
1
12011987311990551205742120573
212057341205735
minus1
1201199055120574120577120588120573
212057341205735+
1
1201198731199055120574120573
1120573212057341205735
minus1
1201199055120577120588120573
1120573212057341205735+
1
120119873119905512057321120573212057341205735
+1
12011987311990551205741205732
212057341205735minus
1
12011990551205771205881205732
212057341205735
+1
60119873119905512057311205732212057341205735+
1
12011987311990551205733212057341205735
minus1
1201199055120574120575120582120573
312057341205735minus
1
1201198731199055120574120573
1120573312057341205735
+1
1201199055120577120588120573
1120573312057341205735minus
1
120119873119905512057321120573312057341205735
minus1
120119873119905512057311205732120573312057341205735minus
1
12011990551205751205821205732
312057341205735
minus1
120119873119905512057311205732312057341205735minus
1
12011990551205741205751205821205732
41205735
minus1
12011990551205741205771205881205732
41205735+
1
1201198731199055120574120573
2120573241205735
minus1
1201199055120577120588120573
2120573241205735+
1
120119873119905512057311205732120573241205735
+1
120119873119905512057322120573241205735minus
1
1201199055120575120582120573
3120573241205735
minus1
120119873119905512057311205733120573241205735minus
1
12011990551205751205821205733
41205735
minus1
12011990551205771205881205733
41205735+
1
12011987311990551205732120573341205735
+1
120119872119905512057431205732
5minus
1
1201199055120574120575120582120573
412057325
minus1
1201199055120574120577120588120573
412057325+
1
1201198721199055120574120573
2120573412057325
+1
1201198731199055120574120573
2120573412057325minus
1
1201199055120577120588120573
2120573412057325
+1
120119873119905512057311205732120573412057325+
1
120119873119905512057322120573412057325
Abstract and Applied Analysis 11
minus1
1201199055120575120582120573
3120573412057325minus
1
120119873119905512057311205733120573412057325
minus1
12011990551205751205821205732
412057325minus
1
12011990551205771205881205732
412057325
+1
120119873119905512057321205732412057325+
1
120119872119905512057421205733
5
minus1
1201199055120575120582120573
412057335minus
1
1201199055120577120588120573
412057335
+1
12011987311990551205732120573412057335+
1
12011987211990551205741205734
5
+1
12011987211990551205735
5
(33)
In general by the repeated application of the above proce-dure we have that
119867119878119899=119905119899
1198991198861119899 119867
119864119899=119905119899
1198991198862119899 119867
119868119899=119905119899
1198991198863119899
119881119878119899=119905119899
1198991198864119899 119881
119868119899=119905119899
1198991198865119899
(34)
where 1198861119899 1198862119899 1198863119899 1198864119899 1198865119899
depend on fixed set of empiricalparameters Hence the approximate solution of the modelabove is given as
119867119878(119905) =
infin
sum119899=0
119905119899
1198991198861119899
119867119864(119905) =
infin
sum119899=0
119905119899
1198991198862119899
119867119868(119905) =
infin
sum119899=0
119905119899
1198991198863119899
119881119878(119905) =
infin
sum119899=0
119905119899
1198991198864119899
119881119868(119905) =
infin
sum119899=0
119905119899
1198991198865119899
(35)
We therefore use the following theoretical parameters 1205731=
05 1205732= 05 120573
3= 05 120573
4= 05 120573
5= 05 120574 = 1 120577 = 1
119873 = 10 119872 = 10 120588 = 1 120575 = 1 and 120582 = 1 to examinethe behavior of the five different compartmental numericalsolutions in the system Figure 3 shows the behavior of thenumerical solution
7 Conclusion
Onchocerciasis model was examined and the mathematicalformulation underpinning the spread and control of thisdisease has been proposed on one hand On the otherhand we made use of some new analytical methods calledthe homotopy decomposition method (HDM) to derive thesolution of the resulting set of (1) to (5)The numerical resultswere presented to test the efficiency and the accuracy of themethod used The mathematical analysis of the resulting setof (1) to (5) above revealed that the stability analysis of the
05 10 15 20Time (per year)
2
4
6
8
10
Appr
oxim
ate s
olut
ions VS
HS
VI
HI
HE
Figure 3 Numerical solution of the model
five roots of the characteristics equations has all negativereal part In particular given that the stability analysis showsthat the DFE is asymptotically stable it would be possiblethat the parasite Onchocerca volvulus be eradicated in a finitetime On the other hand the numerical solution confirmedthat as the number of humans and vectors in the susceptiblecompartments reduces the number of infective humansand vectors in the infectious compartment increases Thenumerical solutions in both cases displayed the biologicalbehaviour of the real world situation Therefore in orderto build a healthy and a sustainable population treatmentwith IvermectinMectizan drugs and a more hygienic envi-ronment that does not allow the breeding and survival ofblackflies could lead to a total eradication of the disease
Conflict of Interests
The authors declare that they have no conflict of interestsregarding the publication of this paper
References
[1] J H F Remme The Global Burden of Onchocerciasis in 1990World Health Organization Geneva Switzerland 2004 httpwwwwhointhealthinfoglobal burden diseaseOnchocercia-sis201990pdf
[2] WorldHealthOrganization ldquoPrevention of blindness and visualimpairment priority eye diseasesrdquo 2014 httpwwwwhointblindnesscausespriorityenindex3html
[3] World Health Organization Certification of Elimination ofHuman Onchocerciasis Criteria and Procedures World HealthOrganizationGeneva Switzerland 2001 httpwhqlibdocwhointhq2001WHO CDS CPE CEE 200118bpdf
[4] K A Lindblade B Arana G Zea-Flores et al ldquoEliminationof Onchocerca volvulus transmission in the Santa Rosa focusof Guatemalardquo The American Journal of Tropical Medicine andHygiene vol 77 pp 334ndash341 2007
[5] J C Anosike and C O E Onwuliri ldquoStudies on filariasis inBauchi State Nigeria 1 Endemicity of human onchocerciasisin Ningi Local Government Areardquo Annals of Tropical Medicineand Parasitology vol 89 no 1 pp 31ndash38 1995
[6] WorldHealthOrganizationWeekly Epidemiological Records 71WHO Geneva Switzerland 1993
12 Abstract and Applied Analysis
[7] WorldHealthOrganization httpwwwwhointneglected dis-easesen
[8] Sightsaver httpwwwsightsaversorgabout uspublicationsonchocerciasis
[9] E C Uttah ldquoOnchocerciasis in the Upper Ima River BasinrdquoIranian Journal of Parasitology vol 5 no 2 pp 33ndash41 2010
[10] J C Anosike and C O E Onwuliri ldquoStudies on filariasis inBauchi State Nigeriamdash1 endemicity of human onchocerciasisin Ningi Local Government Areardquo Annals of Tropical Medicineand Parasitology vol 89 no 1 pp 31ndash38 1995
[11] L D Edungbola and S O Asaolu ldquoParasitologic survey ofonchocerciasis (river blindness) in Babana district Kwara StateNigeriardquo American Journal of Tropical Medicine and Hygienevol 33 no 6 pp 1147ndash1154 1984
[12] O C Nwaorgu A Ohaegbula I E Onweluzo et al ldquoResultsof a large scale onchocercosis survey in Enugu State NigeriardquoJournal of Helminthology vol 68 no 2 pp 155ndash159 1994
[13] F O Akinbo and C E Okaka ldquoPrevalence and socio-economiceffects of onchocerciasis in Ovia North East Local GovernmentArea of Edo State Nigeriardquo Rivista Di Parasitologie vol 22 no3 pp 215ndash221 2005
[14] M D Wogu and C E Okaka ldquoPrevalence and socio-economic effects of onchocerciasis in Okpuje Owan WestLocal Government Area Edo State Nigeriardquo InternationalJournal of Biomedical and Health Sciences vol 4 no 3 2008httpwwwasopahorg
[15] S C O Noutchie ldquoHomotopy Decomposition analysis of atuberculosis modelrdquo Life Science Journal vol 10 no 3 pp 459ndash465 2013
[16] A Atangana and A Secer ldquoThe time-fractional coupled-Korteweg-de-Vries equationsrdquo Abstract and Applied Analysisvol 2013 Article ID 947986 8 pages 2013
[17] A Atangana and N Bildik ldquoApproximate solution of tubercu-losis disease population dynamics modelrdquo Abstract and AppliedAnalysis vol 2013 Article ID 759801 8 pages 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 11
minus1
1201199055120575120582120573
3120573412057325minus
1
120119873119905512057311205733120573412057325
minus1
12011990551205751205821205732
412057325minus
1
12011990551205771205881205732
412057325
+1
120119873119905512057321205732412057325+
1
120119872119905512057421205733
5
minus1
1201199055120575120582120573
412057335minus
1
1201199055120577120588120573
412057335
+1
12011987311990551205732120573412057335+
1
12011987211990551205741205734
5
+1
12011987211990551205735
5
(33)
In general by the repeated application of the above proce-dure we have that
119867119878119899=119905119899
1198991198861119899 119867
119864119899=119905119899
1198991198862119899 119867
119868119899=119905119899
1198991198863119899
119881119878119899=119905119899
1198991198864119899 119881
119868119899=119905119899
1198991198865119899
(34)
where 1198861119899 1198862119899 1198863119899 1198864119899 1198865119899
depend on fixed set of empiricalparameters Hence the approximate solution of the modelabove is given as
119867119878(119905) =
infin
sum119899=0
119905119899
1198991198861119899
119867119864(119905) =
infin
sum119899=0
119905119899
1198991198862119899
119867119868(119905) =
infin
sum119899=0
119905119899
1198991198863119899
119881119878(119905) =
infin
sum119899=0
119905119899
1198991198864119899
119881119868(119905) =
infin
sum119899=0
119905119899
1198991198865119899
(35)
We therefore use the following theoretical parameters 1205731=
05 1205732= 05 120573
3= 05 120573
4= 05 120573
5= 05 120574 = 1 120577 = 1
119873 = 10 119872 = 10 120588 = 1 120575 = 1 and 120582 = 1 to examinethe behavior of the five different compartmental numericalsolutions in the system Figure 3 shows the behavior of thenumerical solution
7 Conclusion
Onchocerciasis model was examined and the mathematicalformulation underpinning the spread and control of thisdisease has been proposed on one hand On the otherhand we made use of some new analytical methods calledthe homotopy decomposition method (HDM) to derive thesolution of the resulting set of (1) to (5)The numerical resultswere presented to test the efficiency and the accuracy of themethod used The mathematical analysis of the resulting setof (1) to (5) above revealed that the stability analysis of the
05 10 15 20Time (per year)
2
4
6
8
10
Appr
oxim
ate s
olut
ions VS
HS
VI
HI
HE
Figure 3 Numerical solution of the model
five roots of the characteristics equations has all negativereal part In particular given that the stability analysis showsthat the DFE is asymptotically stable it would be possiblethat the parasite Onchocerca volvulus be eradicated in a finitetime On the other hand the numerical solution confirmedthat as the number of humans and vectors in the susceptiblecompartments reduces the number of infective humansand vectors in the infectious compartment increases Thenumerical solutions in both cases displayed the biologicalbehaviour of the real world situation Therefore in orderto build a healthy and a sustainable population treatmentwith IvermectinMectizan drugs and a more hygienic envi-ronment that does not allow the breeding and survival ofblackflies could lead to a total eradication of the disease
Conflict of Interests
The authors declare that they have no conflict of interestsregarding the publication of this paper
References
[1] J H F Remme The Global Burden of Onchocerciasis in 1990World Health Organization Geneva Switzerland 2004 httpwwwwhointhealthinfoglobal burden diseaseOnchocercia-sis201990pdf
[2] WorldHealthOrganization ldquoPrevention of blindness and visualimpairment priority eye diseasesrdquo 2014 httpwwwwhointblindnesscausespriorityenindex3html
[3] World Health Organization Certification of Elimination ofHuman Onchocerciasis Criteria and Procedures World HealthOrganizationGeneva Switzerland 2001 httpwhqlibdocwhointhq2001WHO CDS CPE CEE 200118bpdf
[4] K A Lindblade B Arana G Zea-Flores et al ldquoEliminationof Onchocerca volvulus transmission in the Santa Rosa focusof Guatemalardquo The American Journal of Tropical Medicine andHygiene vol 77 pp 334ndash341 2007
[5] J C Anosike and C O E Onwuliri ldquoStudies on filariasis inBauchi State Nigeria 1 Endemicity of human onchocerciasisin Ningi Local Government Areardquo Annals of Tropical Medicineand Parasitology vol 89 no 1 pp 31ndash38 1995
[6] WorldHealthOrganizationWeekly Epidemiological Records 71WHO Geneva Switzerland 1993
12 Abstract and Applied Analysis
[7] WorldHealthOrganization httpwwwwhointneglected dis-easesen
[8] Sightsaver httpwwwsightsaversorgabout uspublicationsonchocerciasis
[9] E C Uttah ldquoOnchocerciasis in the Upper Ima River BasinrdquoIranian Journal of Parasitology vol 5 no 2 pp 33ndash41 2010
[10] J C Anosike and C O E Onwuliri ldquoStudies on filariasis inBauchi State Nigeriamdash1 endemicity of human onchocerciasisin Ningi Local Government Areardquo Annals of Tropical Medicineand Parasitology vol 89 no 1 pp 31ndash38 1995
[11] L D Edungbola and S O Asaolu ldquoParasitologic survey ofonchocerciasis (river blindness) in Babana district Kwara StateNigeriardquo American Journal of Tropical Medicine and Hygienevol 33 no 6 pp 1147ndash1154 1984
[12] O C Nwaorgu A Ohaegbula I E Onweluzo et al ldquoResultsof a large scale onchocercosis survey in Enugu State NigeriardquoJournal of Helminthology vol 68 no 2 pp 155ndash159 1994
[13] F O Akinbo and C E Okaka ldquoPrevalence and socio-economiceffects of onchocerciasis in Ovia North East Local GovernmentArea of Edo State Nigeriardquo Rivista Di Parasitologie vol 22 no3 pp 215ndash221 2005
[14] M D Wogu and C E Okaka ldquoPrevalence and socio-economic effects of onchocerciasis in Okpuje Owan WestLocal Government Area Edo State Nigeriardquo InternationalJournal of Biomedical and Health Sciences vol 4 no 3 2008httpwwwasopahorg
[15] S C O Noutchie ldquoHomotopy Decomposition analysis of atuberculosis modelrdquo Life Science Journal vol 10 no 3 pp 459ndash465 2013
[16] A Atangana and A Secer ldquoThe time-fractional coupled-Korteweg-de-Vries equationsrdquo Abstract and Applied Analysisvol 2013 Article ID 947986 8 pages 2013
[17] A Atangana and N Bildik ldquoApproximate solution of tubercu-losis disease population dynamics modelrdquo Abstract and AppliedAnalysis vol 2013 Article ID 759801 8 pages 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
12 Abstract and Applied Analysis
[7] WorldHealthOrganization httpwwwwhointneglected dis-easesen
[8] Sightsaver httpwwwsightsaversorgabout uspublicationsonchocerciasis
[9] E C Uttah ldquoOnchocerciasis in the Upper Ima River BasinrdquoIranian Journal of Parasitology vol 5 no 2 pp 33ndash41 2010
[10] J C Anosike and C O E Onwuliri ldquoStudies on filariasis inBauchi State Nigeriamdash1 endemicity of human onchocerciasisin Ningi Local Government Areardquo Annals of Tropical Medicineand Parasitology vol 89 no 1 pp 31ndash38 1995
[11] L D Edungbola and S O Asaolu ldquoParasitologic survey ofonchocerciasis (river blindness) in Babana district Kwara StateNigeriardquo American Journal of Tropical Medicine and Hygienevol 33 no 6 pp 1147ndash1154 1984
[12] O C Nwaorgu A Ohaegbula I E Onweluzo et al ldquoResultsof a large scale onchocercosis survey in Enugu State NigeriardquoJournal of Helminthology vol 68 no 2 pp 155ndash159 1994
[13] F O Akinbo and C E Okaka ldquoPrevalence and socio-economiceffects of onchocerciasis in Ovia North East Local GovernmentArea of Edo State Nigeriardquo Rivista Di Parasitologie vol 22 no3 pp 215ndash221 2005
[14] M D Wogu and C E Okaka ldquoPrevalence and socio-economic effects of onchocerciasis in Okpuje Owan WestLocal Government Area Edo State Nigeriardquo InternationalJournal of Biomedical and Health Sciences vol 4 no 3 2008httpwwwasopahorg
[15] S C O Noutchie ldquoHomotopy Decomposition analysis of atuberculosis modelrdquo Life Science Journal vol 10 no 3 pp 459ndash465 2013
[16] A Atangana and A Secer ldquoThe time-fractional coupled-Korteweg-de-Vries equationsrdquo Abstract and Applied Analysisvol 2013 Article ID 947986 8 pages 2013
[17] A Atangana and N Bildik ldquoApproximate solution of tubercu-losis disease population dynamics modelrdquo Abstract and AppliedAnalysis vol 2013 Article ID 759801 8 pages 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
