Research Article HSV-2 and Substance Abuse amongst...

Research ArticleHSV-2 and Substance Abuse amongst AdolescentsInsights through Mathematical Modelling

A Mhlanga C P Bhunu and S Mushayabasa

Department of Mathematics University of Zimbabwe PO Box MP 167 Mount Pleasant Harare Zimbabwe

Correspondence should be addressed to C P Bhunu cpbhunugmailcom

Received 5 July 2014 Accepted 5 October 2014 Published 13 November 2014

Academic Editor Peter G L Leach

Copyright copy 2014 A Mhlanga et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Herpes simplex virus infection is mostly spread and occurs more commonly among substance abusing adolescents as compared tothe nonsubstance abusing In this paper amathematical model for the spread ofHSV-2within a community with substance abusingadolescents is developed and analysed The impacts of condom use and educational campaigns are examined The study suggeststhat condom use is highly effective among adolescents when we have more of them quitting than becoming substance abusersMeasures such as educational campaigns can be put in place to try and reduce adolescents from becoming substance abusersFurther we applied optimal control theory to the proposed modelThe controls represent condom use and educational campaignsThe objective is based onmaximising the susceptible nonsubstance abusing adolescents whileminimising the susceptible substanceabusing adolescents the infectious nonsubstance abusing adolescents and the infectious substance abusing adolescents We usedPontryginrsquos maximum principle to characterise the optimal levels of the two controls The resulting optimality system is solvednumerically Overall the application of the optimal control theory suggests that more effort should be devoted to condom use ascompared to educational campaigns

1 Introduction

Herpes simplex virus type 2 (HSV-2) a double-strandedDNA virus is a sexually transmitted disease that inflictssevere public health burden globally The most commonsymptoms of HSV-2 usually appear within 1-2 weeks aftersexual exposure to the virus The first signs are a tinglingsensation in the affected areas (genitalia buttocks andthighs) and group of small red bumps that develop intoblisters Herpes simplex virus-2 (HSV-2) is a highly prevalentSTD that causes severe public health burden globally with thehighest prevalence in the Sub-SaharanAfrica and someAsiancountries There are 15 million new herpes simplex virus-2 (HSV-2) infections among 15ndash19 year old females in Sub-Saharan African every year [1] In 2003 for instance up to536 million people aged 15ndash49 years were living with HSV-2 globally and 236 million new infections were recorded inthat year [2] In the United States of America (USA) annualhealth costs for STIs has reachedUS $17Billion [3] withHSV-2 chewing up $541millionmaking it the thirdmost costly STIafter HIV-1 and human papillomavirus (HPV) [4]

Adolescents are the range of people who are 10ndash19years old [5] making up approximately 20 of the worldrsquospopulation and about 85 living in developing countries [6]During the period of adolescence many individuals begin toengage in risky sexual behaviours hence making them theage group at greatest risk for nearly all STIs [7] The reasonsfor this trend are many including cognitive developmentphysiological susceptibility peer pressure logistic issues andspecific sexual behaviours [7] Adolescents aged 15ndash19 yearsaccount for approximately 3 million cases meaning one outof four sexually active teenagers reports an STI every year[8 9] Adolescents aremore vulnerable to STIs because of lackof education yet in some schools sex education in schoolsbegins late when adolescents have already been initiatedinto sexual intercourse [10] Biologically adolescent girls areat higher risk of contracting STIs because in adolescentgirls the cervix and vagina undergo dramatic histologicalchanges due to estrogen exposure [11] Adolescent girls arealso the ones at greater risk of HSV-2 infections [12 13]Some studies have found out that being young at age really

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2014 Article ID 104819 17 pageshttpdxdoiorg1011552014104819

2 Journal of Applied Mathematics

increased the likelihood of acquiring HSV-2 infection [14ndash16] implying that we have more adolescents being infectedPrevalence estimates for adolescents are available throughcommunity-based studies One study in Zimbabwe amongadolescent females aged 15ndash19 years reported that 12 ofparticipants tested positive forHSV-2 [17] A study inwesternKenya found HSV-2 prevalence to be 9 for females and4 for males aged 13-14 years and 28 females and 17for males aged 15ndash19 years [18] It was shown that sexualactivity in Namibia begins at a very young age often aslow as 10 years [10] and also some 61 percent of twelfthgraders in United States of America (USA) have had sexualintercourse at least once [19] Many adolescents experimentwith substances parents teachers and policy makers want toknow the consequences of adolescent substance abuse Sincethe abuse of substances affects their thinking capacity mostof them who are substance abusers have higher rates of HSV-2 infection [15 20] Those adolescents who are infected withHSV-2 type 2 (HSV-2) infection may be at greater risk fortransmission and acquisition of human immunodeficiencyvirus (HIV) [21] Interventions to control HSV-2 infectioncould prevent as many as 50 to 60 of new HIV infec-tions [22] Furthermore pregnant adolescents infected withgenital HSV (particularly those with a primary infection)can transmit infection to the neonate which can lead toserious complications for the neonate such as neurologicproblems and even death [23 24] Since currently there isno vaccination for HSV-2 [25] current intervention optionsfor controlling HSV-2 are limited to patient education andpromotion of condom use or treatment with oral medication[26] Some preventive efforts should be made to targetchildren before initiation of sexual activity since preventionis always better than cure

A number of mathematical models have looked intomathematical modelling of HSV-2 transmission dynamicsfocusing on a number of different issues (see [36ndash42] tomention just a few but none has looked into the effectsof substance abuse on the transmission dynamics of HSV-2among adolescents) In contrast to other STIs such as chlamy-dia gonorrhea syphilis and trichomoniasis infection withHSV-2 is lifelong and once established there is currentlyno treatment to eliminate [43 44] The other problem withHSV-2 infection in adolescents is that it is somewhat difficultto determine given that many infections are asymptomaticor go unrecognised hence it differs with most of otherSTIs in general To the best of our knowledge as authorsno mathematical model has looked into the transmissiondynamics of HSV-2 and substance abuse with a focus onadolescents It is against this background that our study findsrelevance and motivation by formulating a mathematicalmodel to investigate the impact of substance abuse amongadolescents on the transmission dynamics of HSV-2 Themodel incorporates some key epidemiological features suchas the impact of condom use and educational campaignsamong the adolescents The main objective in this study is toforecast future trends in the incidence ofHSV-2 epidemic andalso to quantify the association between substance abuse andHSV-2 epidemic amongst adolescents in the community

The paper is structured as follows The HSV-2 transmis-sion model is formulated in the next section Analytic resultsof the model system are presented in Section 3 Simulationresults and projection profiles of HSV-2 are presented inSection 4 In Section 5 optimal control theory has beenapplied to the model formulation in Section 2 Summary andconcluding remarks round up the paper

2 Model Formulation

The total sexually active population at time 119905 denoted by119873is subdivided into mutually exclusive compartments namelysusceptible adolescents who are nonsubstance abusers (119878

119899)

susceptible adolescents who are substance abusers (119878119886) ado-

lescents infected with HSV-2 who are nonsubstance abusers(119868119899) and adolescents infected with HSV-2 who are substance

abusers (119868119886) Although there are many causes of substance

abuse amongst adolescents here we dwell on peer pressure asthe main one The total population is given by

119873 = 119878119899(119905) + 119878

119886(119905) + 119868

119899(119905) + 119868

119886(119905) (1)

The susceptible is increased by a constant inflow into thepopulation at rate Λ A fraction 120587

0of these adolescents are

assumed to be nonsubstance abusers and the complementaryfraction 120587

1= 1 minus 120587

0being substance abusers This is

reasonable because the underlying population is made ofadolescents only into these two classes (influence from somepeers is probably the main drive behind the adolescentsabusing substances and hence their recruitment into thesubstance abusers class) It has been found that adolescentsstart abusing substances for four main reasons to improvetheir mood to receive social rewards to reduce negativefeelings and to avoid social rejection [45] and these makethem fall victims to peer influence easily Individuals indifferent human subgroups suffer from natural death at rate120583 Assuming homogeneous mixing of the population thesusceptible individuals acquireHSV-2 infection at rate120582 with

120582 (119905) =120573 (119868119886(119905) + (1 minus 120578) 119868

119899(119905))

119873 (119905) (2)

120578 isin (0 1) is a modification parameter accounting for reducednumber a nonsubstance abuser would infect as comparedto a substance abuser [15 20] perhaps because adolescentswho engage in substance abuse also engage in other riskybehaviours Substance abuse diminishes judgement leadingto the engagement in the high risk behaviours [46] and thesame explanation also applies for the modification parameter120572 isin (0 1) 119901 is the probability of being infected from asexual partner and 119888 is the rate at which an individual acquiressexual partners per unit time and 120573 = 119901119888 is the effectivecontact rate for HSV-2 infection (contact sufficient to resultin HSV-2 infection) The role of condom use is representedby 120579 If 120579 = 0 then condom use is not effective 120579 = 1

corresponds to completely effective condom use while 0 lt120579 lt 1 implies that condom use would be effective to somedegree Substance abusers experience related diseases likeliver cirrhosis pancreatitis and heart disease among other

Journal of Applied Mathematics 3

Λ1205870 Λ1205871

120588

120588

120601

120601 SaSn

120583 + 120583

120583 + 120583

(1 minus 120579)120582120572(1 minus 120579)120582

IaIn

Figure 1 Structure of the model

substance abuse induced diseases and die at a rate V becauseof these diseases [28] Individuals are assumed to becomesubstance abusers mostly by peer pressure maybe by choiceor any other reasons at rate 120588 and likewise individuals maycease being substance abusers due to educational campaigns(mostly) health reasons pressure at work or school povertyor any other reason at rate 120601 The model flow diagram isdepicted in Figure 1

From the descriptions and assumptions on the dynamicsof the epidemic made above the following are the modelequations

1198781015840

119899= 1205870Λ minus 120572 (1 minus 120579) 120582119878

119899+ 120601119878119886minus (120583 + 120588) 119878

119899

1198781015840

119886= 1205871Λ minus (1 minus 120579) 120582119878

119886+ 120588119878119899minus (120583 + V + 120601) 119878

119886

1198681015840

119899= 120572 (1 minus 120579) 120582119878

119899+ 120601119868119886minus (120583 + 120588) 119868

119899

1198681015840

119886= (1 minus 120579) 120582119878

119886+ 120588119868119899minus (120583 + V + 120601) 119868

119886

(3)

21 Positivity and Boundedness of Solutions Model system (3)describes human population and therefore it is necessary toprove that all the variables 119878

119899(119905) 119878119886(119905) 119868119899(119905) and 119868

119886(119905) are

nonnegative for all time Solutions of the model system (3)with positive initial data remain positive for all time 119905 ge 0

and are bounded inG

Theorem 1 Let 119878119899(119905) ge 0 119878

119886(119905) ge 0 119868

119899(119905) ge 0 and 119868

119886(119905) ge 0

The solutions 119878119899 119878119886 119868119899 and 119868

119886of model system (3) are positive

for all 119905 ge 0 For model system (3) the region G is positivelyinvariant and all solutions starting inG approach enter or stayinG

Proof Under the given conditions it is easy to prove thateach solution component of model system (3) is positiveotherwise assume by contradiction that there exists a firsttime

1199051 119878119899(1199051) = 0 1198781015840

119899(1199051) lt 0 and 119878

119899(119905) gt 0 119878

119886(119905) gt 0 119868

119899(119905) gt

0 119868119886(119905) gt 0 for 0 lt 119905 lt 119905

1or there exists a

1199052 119878119886(1199052) = 0 1198781015840

119899(1199052) lt 0 and 119878

119899(119905) gt 0 119878

119886(119905) gt 0 119868

119899(119905) gt

0 119868119886(119905) gt 0 for 0 lt 119905 lt 119905

2or there exists a

1199053 119868119899(1199053) = 0 1198681015840

119899(1199053) lt 0 and 119878

119899(119905) gt 0 119878

119886(119905) gt 0 119868

119899(119905) gt

0 119868119886(119905) gt 0 for 0 lt 119905 lt 119905

3or there exists a

1199054 119868119886(1199054) = 0 1198681015840

119886(1199054) lt 0 and 119878

119899(119905) gt 0 119878

119886(119905) gt 0 119868

119899(119905) gt

0 119868119886(119905) gt 0 for 0 lt 119905 lt 119905

4 In the first case we have

1198781015840

119899(1199051) = 1205870Λ + 120601119878

119886(1199051) gt 0 (4)

which is a contradiction and consequently 119878119899remains posi-

tive In the second case we have

1198781015840

119886(1199052) = 1205871Λ + 120588119878

119899(1199052) gt 0 (5)

which is a contradiction therefore 119878119886is positive In the third

case we have

1198681015840

119899(1199053) = 120572 (1 minus 120579) 120582 (119905

3) 119878119899(1199053) + 120601119868

119886(1199053) gt 0 (6)

which is a contradiction therefore 119868119899is positive In the final

case we have

1198681015840

119886(1199054) = (1 minus 120579) 120582 (119905

4) 119878119886(1199054) + 120588119868

119904(1199054) gt 0 (7)

which is a contradiction that is 119868119886remains positive Thus in

all cases 119878119899 119878119886 119868119899 and 119868

119886remain positive

Since119873(119905) ge 119878119886(119905) and119873(119905) ge 119868

119886(119905) then

Λ minus (120583 + 2V)119873 le 1198731015840

(119905) le Λ minus 120583119873 (8)

implies that 119873(119905) is bounded and all solutions starting in Gapproach enter or stay inG

For system (3) the first octant in the state space ispositively invariant and attracting that is solutions that startin this octant where all the variables are nonnegative staythere Thus system (3) will be analysed in a suitable regionG sub R4

+

G = (119878119899 119878119886 119868119899 119868119886) isin R4

+ 119873 le

Λ

120583 (9)

which is positively invariant and attracting It is sufficientto consider solutions in G where existence uniqueness andcontinuation results for system (3) hold

3 Model Analysis

31 Disease-Free Equilibriumand Its StabilityAnalysis Modelsystem (3) has the disease-free equilibrium

E0 = (Λ (1205870(120583 + V) + 120601)

V (120583 + 120588) + 120583 (120583 + 120588 + 120601)

Λ (1205831205871+ 120588)

V (120583 + 120588) + 120583 (120583 + 120588 + 120601) 0 0)

(10)


The linear stability ofE0 is governed by the basic reproductivenumber 119877

0which is defined as the spectral radius of an

irreducible or primitive nonnegativematrix [47] Biologicallythe basic reproductive number can be interpreted as theexpected number of secondary infections produced by asingle infectious individual during hisher infectious periodwhen introduced in a completely susceptible populationUsing the notation in [48] the nonnegative matrix 119865 and thenonsingular matrix 119881 for the new infection terms and theremaining transfer terms are respectively given (at DFE) by119865

=[[

[

120572120573 (1 minus 120579) (1 minus 120578) (1205870(120583 + V) + 120601)

120583 + 120588 + 120601 + V1205870

120572120573 (1 minus 120579) (1205870(120583 + V) + 120601)

120583 + 120588 + 120601 + V1205870

120573 (1 minus 120579) (1 minus 120578) (1205831205871+ 120588)

120583 + 120588 + 120601 + V1205870

120573 (1 minus 120579) (1205831205871+ 120588)

120583 + 120588 + 120601 + V1205870

]]

]

(11)

119881 = [120583 + 120588 minus120601

minus120588 120583 + 120601 + V] (12)

We now consider different possibilities in detail

Case a (there are no substance abusers in the community)We set V = 120601 = 120588 = 120587

1= 0 and 120587

0= 1 Then the spectral

radius is given by

1198770= 119877119899=120572120573 (1 minus 120579) (1 minus 120578)

120583 (13)

This reproductive rate sometimes referred to as the back ofthe napkin [49] is simply the ratio of the per capita rate ofinfection and the average lifetime of an individual in class119868119899 It is defined as the number of secondary HSV-2 cases

produced by a single infected individual during hisher entireinfectious period in a totally naive (susceptible) population inthe absence of substance abusers

Case b (the entire population is made up of substanceabusers) We set 120601 = 120588 = 120587

0= 0 and 120587

1= 1 Then the

spectral radius is given by

119877119886=120573 (1 minus 120579)

120583 + V (14)

biologically 119877119886measures the average number of new infec-

tions generated by a single HSV-2 infective who is alsoa substance abuser during hisher entire infectious periodwhen heshe is introduced to a susceptible population ofsubstance abusers

Case c (there is no condom use) We set 120579 = 0 Then thespectral radius is given by

119877119888=

120573 (1198961+ 1198962+ 1198963)

(120583 + 120588 + 120601 + V1205870) ((120583 + V) (120583 + 120588) + 120583120601)

(15)

where1198961= V120572 ((1 minus 120578) (2120583120587

0+ 120601) + 120587

0(120588 + (1 minus 120578) 120601))

1198962= (120583120587

1+ 120588) (120583 + 120588 + (1 minus 120578) 120601) + V2120572 (1 minus 120578) 120587

0

1198963= 120572 (120583120587

0+ 120601) (120588 + (1 minus 120588) (120583 + 120601) + 120588)

(16)

biologically 119877119888measures the average number of new sec-

ondary HSV-2 infections generated by a single HSV-2 infec-tive during hisher infectious periodwhilst in a community ofnonsubstance abusers and substance abusers in the absenceof condom use

Case d (the general case) Existence of nonsubstance abusersand substance abusers Then the spectral radius is given by

119877119899119886=

120573 (1 minus 120579) (1198961+ 1198962+ 1198963)

(120583 + 120588 + 120601 + V1205870) ((120583 + V) (120583 + 120588) + 120583120601)

(17)

where

1198961= V120572 ((1 minus 120578) (2120583120587

0+ 120601) + 120587

0(120588 + (1 minus 120578) 120601))

1198962= (120583120587

1+ 120588) (120583 + 120588 + (1 minus 120578) 120601) + V2120572 (1 minus 120578) 120587

0

1198963= 120572 (120583120587

0+ 120601) (120588 + (1 minus 120588) (120583 + 120601) + 120588)

(18)

biologically 119877119899119886

measures the average number of new HSV-2 infections generated by a single HSV-2 infective duringhisher infectious periodwithin a community in the presenceof condomuse nonsubstance abusers and substance abusersTheorem 2 follows from van den Driessche and Watmough[48]

Theorem2 Thedisease-free equilibriumofmodel system (3) islocally asymptotically stable if119877

119899119886le 1 and unstable otherwise

Using a proof based on the comparison theorem we caneven show the global stability of the disease-free equilibriumin the case that the disease-free equilibrium is less than unity

311 Global Stability of the Disease-Free Equilibrium

Theorem 3 The disease-free equilibrium E0 of system (3) isglobally asymptotically stable (GAS) if 119877

119899119886le 1 and unstable if

119877119899119886gt 1

Proof The proof is based on using a comparison theorem[50] Note that the equations of the infected components insystem (3) can be written as

[[[

[

119889119868119899

119889119905

119889119868119886

119889119905

]]]

]

= [119865 minus 119881] [119868119899

119868119886

] minus 119867[1 minus 120578 1

0 0] [119868119899

119868119886

] (19)

where 119865 and 119881 are as defined earlier in (11) and (12)respectively and also where

119867 = [120572120573 (1 minus 120579) (1205870(120583 + V) + 120601

120583 + 120601 + 120588 + V1205870

minus119878119899

119873)

+120573 (1 minus 120579) (1205831205871+ 120588

120583 + 120601 + 120588 + V1205870

minus119878119886

119873)]

(20)


Since 119867 ge 0 (full proof for 119867 ge 0 see Appendix A) (for alltime 119905 ge 0) inG it follows that

[[[

[

119889119868119899

119889119905

119889119868119886

119889119905

]]]

]

le [119865 minus 119881] [119868119899

119868119886

] (21)

Using the fact that the eigenvalues of the matrix 119865 minus 119881

all have negative real parts it follows that the linearizeddifferential inequality system (21) is stable whenever 119877

119899119886le 1

Consequently (119868119899 119868119886) rarr (0 0) as 119905 rarr infin Thus by a

comparison theorem [50] (119868119899 119868119886) rarr (0 0) as 119905 rarr infin and

evaluating system (3) at 119868119899= 119868119886= 0 gives 119878

119899rarr 1198780

119899and

119878119886rarr 1198780

119886for 119877119899119886le 1 Hence the disease-free equilibrium

(DFE) is globally asymptotically stable for 119877119899119886le 1

32 Endemic Equilibrium and Stability Analysis Model sys-tem (3) has three possible endemic equilibria the (HSV-2only) substance abuse-free endemic equilibria with a popula-tion of nonsubstance abusers only the endemic equilibriumwhen the whole population is made up of substance abusersonly and lastly the equilibrium where nonsubstance abusersand substance abusers co-exist herein referred to as theinterior equilibrium point

321 Nonsubstance Abusers (Only) Endemic Equilibrium Weset 1205871= 120588 = 120601 = 0 and 120587

0= 1 so that there are no substance

abusers in the communityThus model system (3) reduces to

1198781015840

119899= Λ minus

120572 (1 minus 120579) (1 minus 120578) 120573119878lowast

119899119868lowast

119899

119873lowast

119899

minus 120583119878lowast

119899

1198681015840

119899=120572 (1 minus 120579) (1 minus 120578) 120573119878

lowast

119899119868lowast

119899

119873lowast

119899


119899

(22)

For system (22) it can be shown that the region

G119899= (119878

119899 119868119899) isin R2

+ 119873119899leΛ

120583 (23)

is invariant and attracting Thus the dynamics of nonsub-stance abusing (only) adolescents would be considered inG

119899

Summing the two equation in system (22) above we get119873119899= Λ120583 Furthermore from the second equation of system

(22) 119868119899= 0 or

0 =120572 (1 minus 120579) (1 minus 120578) 120573119878

lowast

119899119868lowast

119899

119873lowast

119899


119899

997904rArr 120572 (1 minus 120579) (1 minus 120578) 120573119878lowast

119899= 120583119873

lowast

119899

997904rArr 119878lowast

119899=

Λ

120583119877119899

(24)

with 119877119899as defined in (13)

Since119873lowast119899= 119878lowast

119899+ 119868lowast

119899 hence 119868lowast

119899= 119873lowast

119899minus 119878lowast

119899 and thus

119868lowast

119899=Λ

120583minusΛ

120583119877119899

= (119877119899minus 1) 119878

lowast

119899

(25)

Thus system (22) has an endemic equilibrium Elowast119899= (119878lowast

119899 119868lowast

119899)

whichmakes biological sense only when 119877119899gt 1This leads to

Theorem 4 below

Theorem4 The substance abuse-free endemic equilibriumElowast119899

exists whenever 119877119899gt 1

The local stability ofElowast119899is given by the Jacobian evaluated

at the point119869 (Elowast119899)

=

[[[

[

minus(


119899

119873lowast

119899

+ 120583)


119899

119873lowast

119899


119899

119873lowast

119899


119899

119873lowast

119899

minus 120583

]]]

]

(26)From (22) we note that 120572(1 minus 120579)(1 minus 120578)120573119878lowast

119899119873lowast

119899= 120583 which

is obtained from the second equation of system (22) with theright-hand side set equaling zero Thus 119869(Elowast

119899) can be written

as

119869 (Elowast

119899) =

[[[[

[

minus(120572 (1 minus 120579) (1 minus 120578) 120573119868

lowast

119899

119873lowast

119899

+ 120583) minus120583


119899

119873lowast

119899

0

]]]]

]

(27)

Hence the characteristic polynomial of the linearised systemis given by

Υ2

+ (120572 (1 minus 120579) (1 minus 120578) 120573119868

lowast

119899

119873lowast

119899

+ 120583)Υ

+120572 (1 minus 120579) (1 minus 120578) 120583120573119868

lowast

119899

119873lowast

119899

= 0

997904rArr Υ1= minus120583

Υ2= minus 120572 (1 minus 120572) (1 minus 120578) (119877

119873minus 1)

120573119868lowast

119899

119873lowast

119899

(28)

Thus Elowast119899is locally asymptotically stable for 119877

119899gt 1 We

summarise the result in Theorem 5

Theorem 5 The endemic equilibrium (Elowast119899) is locally asymp-

totically stable whenever 119877119899gt 1

We now study the global stability of system (22) usingthe Poincare-BendixonTheorem [51]We claim the followingresult

Theorem6 Theendemic equilibriumof theHSV-2 onlymodelsystem (22) is globally asymptotically stable in G

119899whenever

119877119899gt 1

Proof From model system (22) we note that 119873119899= Λ120583

Using the fact that 119878119899= 119873119899minus119868119899 it follows that 119878

119899= (Λ120583)minus119868

119899

and substituting this result into (22) we obtain

1198681015840

119899=120572 (1 minus 120579) (1 minus 120578) 120573119868

119899

Λ120583(Λ

120583minus 119868119899) minus 120583119868

119899 (29)


Using Dulacrsquos multiplier 1119868119899 it follows that

120597

120597119868119899

[120572 (1 minus 120579) (1 minus 120578) 120573

Λ120583(Λ

120583minus 119868119899) minus 120583]

= minus120572 (1 minus 120579) (1 minus 120578) 120583120573

Λlt 0


119873lt 0

(30)

Thus by Dulacrsquos criterion there are no periodic orbits inG119899

SinceG119899is positively invariant and the endemic equilibrium

exists whenever 119877119899gt 1 then it follows from the Poincare-

Bendixson Theorem [51] that all solutions of the limitingsystem originating in G remain in G

119899 for all 119905 ge 0

Further the absence of periodic orbits inG119899implies that the

endemic equilibrium of the HSV-2 only model is globallyasymptotically stable

322 Substance Abusers (Only) Endemic Equilibrium Thisoccurs when the whole community consists of substanceabusers only Using the similar analysis as in Section 321 itcan be easily shown that the endemic equilibrium is

Elowast

119886= 119878lowast

119886=

Λ

120583119877119886

119868lowast

119886= (119877119886minus 1) 119878

lowast

119886 (31)

with 119877119886as defined in (14) and it follows that the endemic

equilibrium Elowast119886makes biological sense whenever 119877

119886gt 1

Furthermore using the analysis done in Section 321 thestability of Elowast

119886can be established We now discuss the

coexistence of nonsubstance abusers and substance abusersin the community

323 Interior Endemic Equilibrium This endemic equilib-rium refers to the case where both the nonsubstance abusersand the substance abusers coexistThis state is denoted byElowast

119899119886

with

Elowast

119899119886=

119878lowast

119899=

1205870Λ + 120601119878

lowast

119886

120572 (1 minus 120579) 120582lowast

119899119886+ 120583 + 120588

119878lowast

119886=

1205871Λ + 120588119878

lowast

119899

(1 minus 120579) 120582lowast

119899119886+ 120583 + V + 120601

119868lowast

119899=120572 (1 minus 120579) 120582

lowast

119899119886119878lowast

119899+ 120601119868lowast

119886

120583 + 120588

119868lowast

119886=(1 minus 120579) 120582

lowast

119899119886119878lowast

119886+ 120588119868lowast

119899

120583 + V + 120601

with 120582lowast119899119886=120573 (119868lowast

119886+ (1 minus 120578) 119868

lowast

119899)

119873lowast

119899119886

(32)

We now study the global stability of the interior equilibriumof system (3) using the Poincare-Bendinxon Theorem (seeAppendix B) The permanence of the disease destabilises thedisease-free equilibrium E0 since 119877

119899119886gt 1 and thus the

equilibriumElowast119899119886exists

Lemma 7 Model system (3) is uniformlypersistent onG

Proof Uniform persistence of system (3) implies that thereexists a constant 120577 gt 0 such that any solution of model (3)starts in

∘

G and the interior ofG satisfies

120577 le lim inf119905rarrinfin

119878119899 120577 le lim inf

119905rarrinfin

119878119886


119868119899 120577 le lim inf

119905rarrinfin

119868119886

(33)

Define the following Korobeinikov and Maini [52] type ofLyapunov functional

119881 (119878119899 119878119886 119868119899 119868119886) = (119878


119899ln 119878119899) + (119878


119886ln 119878119886)

+ (119868119899minus 119868lowast

119899ln 119868119899) + (119868119886minus 119868lowast

119886ln 119868119886)

(34)

which is continuous for all (119878119899 119878119886 119868119899 119868119886) gt 0 and satisfies

120597119881

120597119878119899

= (1 minus119878lowast

119899

119878119899

) 120597119881

120597119878119886


119886

119878119886

)

120597119881

120597119868119899


119899

119868119899

) 120597119881

120597119868119886


119886

119868119886

)

(35)

(see Korobeinikov and Wake [53] for more details) Conse-quently the interior equilibrium Elowast

119899119886is the only extremum

and the global minimum of the function 119881 isin R4+ Also

119881(119878119899 119878119886 119868119899 119868119886) gt 0 and 119881

1015840

(119878119899 119878119886 119868119899 119868119886) = 0 only at

Elowast119899119886 Then substituting our interior endemic equilibrium

identities into the time derivative of119881 along the solution pathof the model system (3) we have

1198811015840

(119878119899 119878119899 119868119886 119868119886) = (119878


119899)1198781015840

119899

119878119899

+ (119878119886minus 119878lowast

119886)1198781015840

119886

119878119886

+ (119868119899minus 119868lowast

119899)1198681015840

119899

119868119899

+ (119868119886minus 119868lowast

119886)1198681015840

119886

119868119886

le minus120583(119878119899minus 119878lowast

119899)2

119878119899

+ 119892 (119878119899 119878119886 119868119899 119868119886)

(36)

The function 119892 can be shown to be nonpositive using Bar-balat Lemma [54] or the approach inMcCluskey [55] Hence1198811015840

(119878119899 119878119886 119868119899 119868119886) le 0 with equality only at Elowast

119899119886 The only

invariant set in∘

G is the set consisting of the equilibriumElowast119899119886

Thus all solutions ofmodel system (3) which intersect with∘

Glimit to an invariant set the singleton Elowast

119899119886 Therefore from

the Lyapunov-Lasalle invariance principle model system (3)is uniformly persistent

From the epidemiological point of view this result meansthat the disease persists at endemic state which is not goodnews from the public health viewpoint

Theorem 8 If 119877119899119886gt 1 then the interior equilibrium Elowast

119899119886is

globally asymptotically stablein the interior of G


Table 1 Model parameters and their interpretations

Definition Symbol Baseline values (range) SourceRecruitment rate Λ 10000 yrminus1 AssumedProportion of recruited individuals 120587

0 1205871

07 03 AssumedNatural death rate 120583 002 (0015ndash002) yrminus1 [27]Contact rate 119888 3 [28]HSV-2 transmission probability 119901 001 (0001ndash003) yrminus1 [29ndash32]Substance abuse induced death rate V 0035 yrminus1 [33]Rate of change in substance abuse habit 120588 120601 Variable AssumedCondom use efficacy 120579 05 [34 35]Modification parameter 120578 06 (0-1) AssumedModification parameter 120572 04 (0-1) Assumed

Proof Denote the right-hand side of model system (3) by ΦandΨ for 119868

119899and 119868119886 respectively and choose aDulac function

as119863(119868119899 119868119886) = 1119868

119899119868119886 Then we have

120597 (119863Φ)

120597119868119899

+120597 (119863Ψ)

120597119868119886

= minus120601

1198682

119899

minus120588

1198682

119886

minus (1 minus 120579) 120573(120572119878119899

1198682

119899

+(1 minus 120578) 119878

119886

1198682

119886

) lt 0

(37)

Thus system (3) does not have a limit cycle in the interior ofG Furthermore the absence of periodic orbits in G impliesthat Elowast

119899119886is globally asymptotically stable whenever 119877

119899119886gt 1

4 Numerical Results

In order to illustrate the results of the foregoing analysis wehave simulated model (3) using the parameters in Table 1Unfortunately the scarcity of the data on HSV-2 and sub-stance abuse in correlation with a focus on adolescents limitsour ability to calibrate nevertheless we assume some ofthe parameters in the realistic range for illustrative purposeThese parsimonious assumptions reflect the lack of infor-mation currently available on HSV-2 and substance abusewith a focus on adolescents Reliable data on the risk oftransmission of HSV-2 among adolescents would enhanceour understanding and aid in the possible interventionstrategies to be implemented

Figure 2 shows the effects of varying the effective contactrate on the reproduction number Results on Figure 2 suggestthat the increase or the existence of substance abusersmay increase the reproduction number Further analysisof Figure 2 reveals that the combined use of condomsand educational campaigns as HSV-2 intervention strategiesamong substance abusing adolescents in a community havean impact on reducing the magnitude of HSV-2 cases asshown by the lowest graph for 119877

119899119886

Figure 3(a) depicts that the reproduction number can bereduced by low levels of adolescents becoming substanceabusers and high levels of adolescents quitting the substanceabusive habits Numerical results in Figure 3(b) suggest thatmore adolescents should desist from substance abuse since

02 04 06 08 10

10

20

30

40 Ra

R0

Rc

Rna

120573

R

Figure 2 Effects of varying the effective contact rate (120573) on thereproduction numbers

the trend Σ shows a decrease in the reproduction number119877119899119886

with increasing 120601 Trend line Ω shows the effects ofincreasing 120588 which results in an increase of the reproductivenumber 119877

119899119886thereby increasing the prevalence of HSV-2

in community The point 120595 depicts the balance in transferrates from being a substance abuser to a nonsubstanceabuser or from a nonsubstance abuser to a substance abuser(120588 = 120601) and this suggests that changes in the substanceabusive habit will not have an impact on the reproductionnumber Point 120595 also suggests that besides substance abuseamong adolescents other factors are influencing the spreadof HSV-2 amongst them in the community Measures such aseducational campaigns and counsellingmay be introduced soas to reduce 120588 while increasing 120601

In many epidemiological models the magnitude of thereproductive number is associated with the level of infectionThe same is true in model system (3) Sensitivity analysisassesses the amount and type of change inherent in themodelas captured by the terms that define the reproductive number


00

05

10

00

05

10

10

15

120601120588

Rna

(a)

02 04 06 08 10

10

12

14

16

18

20

22

(120601 120588)

Rna

Ω

120595

Σ

(b)

Figure 3 (a)The relationship between the reproduction number (119877119899119886) rate of becoming a substance abuser (120588) and rate of quitting substance

abuse (120601) (b) Numerical results of model system (3) showing the effect in the change of substance abusing habits Parameter values used arein Table 1 with 120601 and 120588 varying from 00 to 10

(119877119899119886) If 119877119899119886is very sensitive to a particular parameter then a

perturbation of the conditions that connect the dynamics tosuch a parameter may prove useful in identifying policies orintervention strategies that reduce epidemic prevalence

Partial rank correlation coefficients (PRCCs) were calcu-lated to estimate the degree of correlation between values of119877119899119886and the ten model parameters across 1000 random draws

from the empirical distribution of 119877119899119886

and its associatedparameters

Figure 4 illustrates the PRCCs using 119877119899119886

as the outputvariable PRCCs show that the rate of becoming substanceabusers by the adolescents has the most impact in the spreadof HSV-2 among adolescents Effective contact rate is foundto support the spread of HSV-2 as noted by its positive PRCCHowever condom use efficacy is shown to have a greaterimpact on reducing 119877

119899119886

Since the rate of becoming a substance abuser (120588)condom use efficacy (120579) and the effective contact rate (120573)have significant effects on 119877

119899119886 we examine their dependence

on 119877119899119886

in more detail We used Latin hypercube samplingand Monte Carlo simulations to run 1000 simulations whereall parameters were simultaneously drawn from across theirranges

Figure 5 illustrates the effect that varying three sampleparameters will have on 119877

119899119886 In Figure 5(a) if the rate of

becoming a substance abuser is sufficiently high then119877119899119886gt 1

and the disease will persist However if the rate of becominga substance abuser is low then 119877

119899119886gt 1 and the disease can

be controlled Figure 5(b) illustrates the effect of condom useefficacy on controlling HSV-2 The result demonstrates thatan increase in 120579 results in a decrease in the reproductionnumber Figure 5(b) demonstrates that if condomuse efficacyis more than 70 of the time then the reproduction numberis less than unity and then the disease will be controlled

0 02 04 06 08

Substance abuse induced death rate

Recruitment of substance abusers

Modification parameter

Recruitment of non-substance abusers

Natural death rate

Rate of becoming a sustance abuser


Condom use efficacy

Rate of quitting substance abusing

Effective contact rate

minus06 minus04 minus02

Figure 4 Partial rank correlation coefficients showing the effect ofparameter variations on119877

119899119886using ranges in Table 1 Parameters with

positive PRCCs will increase 119877119899119886when they are increased whereas

parameters with negative PRCCs will decrease 119877119899119886

when they areincreased

Numerical results on Figure 8 are in agreement with theearlier findings which have shown that condom use has asignificant effect on HSV-2 cases among adolescents How-ever Figure 8(b) shows that for more adolescents leavingtheir substance abuse habit than becoming substance abuserscondom use is more effective Figure 8(b) also shows thatmore adolescents should desist from substance abuse toreduce the HSV-2 cases It is also worth noting that as thecondom use efficacy is increasing the cumulative HSV-2cases also are reduced

In general simulations in Figure 7 suggest that anincrease in recruitment of substance abusing adolescents into


0 01 02 03 04 05 06 07 08 09 1

0

2

Rate of becoming a substance abuser

minus10

minus8

minus6

minus4

minus2

log(Rna)

(a)

0 01 02 03 04 05 06 07 08 09 1Condom use efficacy

0

2

minus10

minus8

minus6

minus4

minus2

log(Rna)

(b)

0 001 002 003 004 005 006 007 008 009Effective contact rate

0

2

minus10

minus8

minus6

minus4

minus2

log(Rna)

(c)

Figure 5 Monte Carlo simulations of 1000 sample values for three illustrative parameters (120588 120579 and 120573) chosen via Latin hypercube sampling

the community will increase the prevalence of HSV-2 casesIt is worth noting that the prevalence is much higher whenwe have more adolescents becoming substance abusers thanquitting the substance abusing habit that is for 120588 gt 120601

5 Optimal Condom Use andEducational Campaigns

In this section our goal is to solve the following problemgiven initial population sizes of all the four subgroups 119878

119899

119878119886 119868119899 and 119868

119886 find the best strategy in terms of combined

efforts of condom use and educational campaigns that wouldminimise the cumulative HSV-2 cases while at the sametime minimising the cost of condom use and educationalcampaigns Naturally there are various ways of expressingsuch a goal mathematically In this section for a fixed 119879 weconsider the following objective functional

119869 (1199061 1199062)

= int

119905119891

0

[119878119899minus 1198611119878119886minus 1198612119868119899minus 1198613119868119886minus (11986011199062

1+ 11986021199062

2)] 119889119905

(38)


120579 = 02

120579 = 05

120579 = 08

0 10 20 30 40 50 600

500

1000

1500

2000

2500

3000

3500

Time (years)

Cum

ulat

ive H

SV-2

case

s

(a)

120579 = 02

120579 = 05

120579 = 08

0 10 20 30 40 50 60100

200

300

400

500

600

Time (years)

Cum

ulat

ive H

SV-2

case

s

(b)

Figure 6 Simulations of model system (3) showing the effects of an increase on condom use efficacy on the HSV-2 cases (119868119886+ 119868119899) over a

period of 60 years The rest of the parameters are fixed on their baseline values with Figure 8(a) having 120588 gt 120601 and Figure 8(b) having 120601 gt 120588

0 10 20 30 40 50 600

500

1000

1500

2000

2500

3000

3500

Time (years)

Cum

ulat

ive H

SV-2

case

s

(a)

0 10 20 30 40 50 60100

200

300

400

500

600

700

800

900

Time (years)

Cum

ulat

ive H

SV-2

case

s

(b)

Figure 7 Simulations of model system (3) showing the effects of increasing the proportion of recruited substance abusers (1205871) varying from

00 to 10 with a step size of 025 for the 2 cases where (a) 120588 gt 120601 and (b) 120601 gt 120588

subject to

1198781015840

119899= 1205870Λ minus 120572 (1 minus 119906

1) 120582119878119899+ 120601119878119886minus (1 minus 119906

2) 120588119878119899minus 120583119878119899

1198781015840

119886= (1 minus 119906

2) 1205871Λ minus (1 minus 119906

1) 120582119878119886

+ (1 minus 1199062) 120588119878119899minus (120583 + V + 120601) 119878

119886

1198681015840

119899= 120572 (1 minus 119906

1) 120582119878119899+ 120601119868119886minus (1 minus 119906

2) 120588119868119899minus 120583119868119899

1198681015840

119886= (1 minus 119906

1) 120582119878119886+ (1 minus 119906

2) 120588119868119899minus (120583 + V + 120601) 119868

119886

(39)

The parameters 1198611 1198612 and 119861

3represent the weight constants

of the susceptible substance abusers the infectious nonsub-stance abusers and the infectious substance abusers while


the parameters1198601and119860

2are theweights and cost of condom

use and educational campaigns for the controls 1199061and 119906

2

respectively The terms 11990621and 1199062

2reflect the effectiveness of

condom use and educational campaigns in the control ofthe disease The values 119906

1= 1199062= 1 represent maximum

effectiveness of condom use and educational campaigns Wetherefore seek an optimal control 119906lowast

1and 119906lowast

2such that

119869 (119906lowast

1 119906lowast

2) = max 119869 (119906

1 1199062) | 1199061 1199062isin U (40)

whereU = 1199061(119905) 1199062(119905) | 119906

1(119905) 1199062(119905) is measurable 0 le 119886

11le

1199061(119905) le 119887

11le 1 0 le 119886

22le 1199062(119905) le 119887

22le 1 119905 isin [0 119905

119891]

The basic framework of this problem is to characterize theoptimal control and prove the existence of the optimal controland uniqueness of the optimality system

In our analysis we assume Λ = 120583119873 V = 0 as appliedin [56] Thus the total population 119873 is constant We canalso treat the nonconstant population case by these sametechniques but we choose to present the constant populationcase here

51 Existence of an Optimal Control The existence of anoptimal control is proved by a result from Fleming and Rishel(1975) [57] The boundedness of solutions of system (39) fora finite interval is used to prove the existence of an optimalcontrol To determine existence of an optimal control to ourproblem we use a result from Fleming and Rishel (1975)(Theorem41 pages 68-69) where the following properties aresatisfied

(1) The class of all initial conditions with an optimalcontrol set 119906

1and 119906

2in the admissible control set

along with each state equation being satisfied is notempty

(2) The control setU is convex and closed(3) The right-hand side of the state is continuous is

bounded above by a sum of the bounded control andthe state and can be written as a linear function ofeach control in the optimal control set 119906

1and 119906

2with

coefficients depending on time and the state variables(4) The integrand of the functional is concave onU and is

bounded above by 1198882minus1198881(|1199061|2

+ |1199062|2

) where 1198881 1198882gt 0

An existence result in Lukes (1982) [58] (Theorem 921)for the system of (39) for bounded coefficients is used to givecondition 1The control set is closed and convex by definitionThe right-hand side of the state system equation (39) satisfiescondition 3 since the state solutions are a priori boundedThe integrand in the objective functional 119878

119899minus 1198611119878119886minus 1198612119868119899minus

1198613119868119886minus (11986011199062

1+11986021199062

2) is Lebesgue integrable and concave on

U Furthermore 1198881 1198882gt 0 and 119861

1 1198612 1198613 1198601 1198602gt 1 hence

satisfying

119878119899minus 1198611119878119886minus 1198612119868119899minus 1198613119868119886minus (11986011199062

1+ 11986021199062

2)

le 1198882minus 1198881(10038161003816100381610038161199061

1003816100381610038161003816

2

+10038161003816100381610038161199062

1003816100381610038161003816

2

)

(41)

hence the optimal control exists since the states are bounded

52 Characterisation Since there exists an optimal controlfor maximising the functional (42) subject to (39) we usePontryaginrsquos maximum principle to derive the necessaryconditions for this optimal controlThe Lagrangian is definedas

119871 = 119878119899minus 1198611119878119886minus 1198612119868119899minus 1198613119868119886minus (11986011199062

1+ 11986021199062

2)

+ 1205821[1205870Λ minus

120572 (1 minus 1199061) 120573119868119886119878119899

119873

minus120572 (1 minus 119906

1) (1 minus 120578) 120573119868

119899119878119899

119873

+120601119878119886minus (1 minus 119906

2) 120588119878119899minus 120583119878119899]

+ 1205822[(1 minus 119906

2) 1205871Λ minus

(1 minus 1199061) 120573119868119886119878119886

119873

minus(1 minus 119906

1) (1 minus 120578) 120573119868

119899119878119886

119873

+ (1 minus 1199062) 120588119878119899minus (120583 + V + 120601) 119878

119886]

+ 1205823[120572 (1 minus 119906

1) 120573119868119886119878119899

119873

+120572 (1 minus 119906

1) (1 minus 120578) 120573119868

119899119878119899

119873

+120601119868119886minus (1 minus 119906

2) 120588119868119899minus 120583119868119899]

+ 1205824[(1 minus 119906

1) 120573119868119886119878119886

119873+(1 minus 119906

1) (1 minus 120578) 120573119868

119899119878119886

119873

+ (1 minus 1199062) 120588119868119899minus (120583 + V + 120601) 119868

119886]

+ 12059611(119905) (1198871minus 1199061(119905)) + 120596

12(119905) (1199061(119905) minus 119886

1)

+ 12059621(119905) (1198872minus 1199062(119905)) + 120596

22(119905) (1199062(119905) minus 119886

2)

(42)

where 12059611(119905) 12059612(119905) ge 0 120596

21(119905) and 120596

22(119905) ge 0 are penalty

multipliers satisfying 12059611(119905)(1198871minus 1199061(119905)) = 0 120596

12(119905)(1199061(119905) minus

1198861) = 0 120596

21(119905)(1198872minus 1199062(119905)) = 0 and 120596

22(119905)(1199062(119905) minus 119886

2) = 0 at

the optimal point 119906lowast1and 119906lowast

2

Theorem9 Given optimal controls 119906lowast1and 119906lowast2and solutions of

the corresponding state system (39) there exist adjoint variables120582119894 119894 = 1 4 satisfying1198891205821

119889119905= minus

120597119871

120597119878119899

= minus1 +120572 (1 minus 119906

1) 120573119868119886[1205821minus 1205823]

119873

+120572 (1 minus 119906

1) (1 minus 120578) 120573119868

119899[1205821minus 1205823]

119873

+ (1 minus 1199062) 120588 [1205821minus 1205822] + 120583120582

1


1198891205822

119889119905= minus

120597119871

120597119878119886

= 1198611+ 120601 [120582

2minus 1205821] +

(1 minus 1199061) 120573119868119886[1205822minus 1205824]

119873

+(1 minus 119906

1) (1 minus 120578) 120573119868

119899[1205822minus 1205824]

119873+ (120583 + V) 120582

2

1198891205822

119889119905= minus

120597119871

120597119868119899

= 1198612+120572 (1 minus 119906

1) (1 minus 120578) 120573119878

119899[1205821minus 1205823]

119873

+(1 minus 119906

1) (1 minus 120578) 120573119878

119886[1205822minus 1205824]

119873

+ (1 minus 1199062) 120588 [1205823minus 1205824] + 120583120582

3

1198891205822

119889119905= minus

120597119871

120597119868119899

= 1198613+120572 (1 minus 119906

1) 120573119878119899[1205821minus 1205823]

119873

+(1 minus 119906

1) 120573119878119886[1205822minus 1205824]

119873+ 120601 [120582

4minus 1205823] + (120583 + V) 120582

4

(43)

Proof The form of the adjoint equation and transversalityconditions are standard results from Pontryaginrsquos maximumprinciple [57 59] therefore solutions to the adjoint systemexist and are bounded Todetermine the interiormaximumofour Lagrangian we take the partial derivative 119871 with respectto 1199061(119905) and 119906

2(119905) and set to zeroThus we have the following

1199061(119905)lowast is subject of formulae

1199061(119905)lowast

=120573 (119868119886+ (1 minus 120578) 119868

119899)

21198601119873

[120572119878119899(1205821minus 1205823) + 119878119886(1205822minus 1205824)]

+12059612(119905) minus 120596

11(119905)

21198601

(44)


1199062(119905)lowast

=120588

21198602

[119878119899(1205821minus 1205822) + 119868119899(1205823minus 1205824)] minus

1205871Λ1205822

21198602

+12059622minus 12059621

21198602

(45)

To determine the explicit expression for the controlwithout 120596

1(119905) and 120596

2(119905) a standard optimality technique is

utilised The specific characterisation of the optimal controls1199061(119905)lowast and 119906

2(119905)lowast Consider

1199061(119905)lowast

= minmax 1198861120573 (119868119886+ (1 minus 120578) 119868

119899)

21198601119873

times [120572119878119899(1205821minus 1205823) + 119878119886(1205822minus 1205824)] 119887

1

1199062(119905)lowast

= minmax1198862120588

21198602

[119878119899(1205821minus 1205822) + 119868119899(1205823minus 1205824)]

minus1205871Λ1205822

21198602

1198872

(46)

The optimality system consists of the state system coupledwith the adjoint system with the initial conditions thetransversality conditions and the characterisation of theoptimal control Substituting 119906lowast

1 and 119906lowast

2for 1199061(119905) and 119906

2(119905)

in (43) gives the optimality system The state system andadjoint system have finite upper bounds These bounds areneeded in the uniqueness proof of the optimality system Dueto a priori boundedness of the state and adjoint functionsand the resulting Lipschitz structure of the ODEs we obtainthe uniqueness of the optimal control for small 119905

119891[60]

The uniqueness of the optimal control follows from theuniqueness of the optimality system

53 Numerical Simulations The optimality system is solvedusing an iterative method with Runge-Kutta fourth-orderscheme Starting with a guess for the adjoint variables thestate equations are solved forward in time Then these statevalues are used to solve the adjoint equations backwardin time and the iterations continue until convergence Thesimulations were carried out using parameter values inTable 1 and the following are the values 119860

1= 0005 119860

2=

0005 1198611= 006 119861

2= 005 and 119861

3= 005 The assumed

initial conditions for the differential equations are 1198781198990

= 041198781198860

= 03 1198681198990

= 02 and 1198681198860

= 01Figure 8(a) illustrates the population of the suscepti-

ble nonsubstance abusing adolescents in the presence andabsence of the controls over a period of 20 years For theperiod 0ndash3 years in the absence of the control the susceptiblenonsubstance abusing adolescents have a sharp increase andfor the same period when there is no control the populationof the nonsubstance abusing adolescents decreases sharplyFurther we note that for the case when there is no controlfor the period 3ndash20 years the population of the susceptiblenonsubstance abusing adolescents increases slightly

Figure 8(b) illustrates the population of the susceptiblesubstance abusers adolescents in the presence and absenceof the controls over a period of 20 years For the period0ndash3 years in the presence of the controls the susceptiblesubstance abusing adolescents have a sharp decrease and forthe same period when there are controls the population of


036

038

04

042

044

046

048

0 2 4 6 8 10 12 14 16 18 20Time (years)

S n(t)

(a)

024

026

028

03

032

034

036

0 2 4 6 8 10 12 14 16 18 20Time (years)

S a(t)

(b)

005

01

015

02

025

03

035

Without control With control

00 2 4 6 8 10 12 14 16 18 20

Time (years)

I n(t)

(c)


0

005

01

015

02

025

03

035

0 2 4 6 8 10 12 14 16 18 20Time (years)

I a(t)

(d)Figure 8 Time series plots showing the effects of optimal control on the susceptible nonsubstance abusing adolescents 119878

119899 susceptible

substance abusing adolescents 119878119886 infectious nonsubstance abusing adolescents 119868

119899 and the infectious substance abusing 119868

119886 over a period

of 20 years

the susceptible substance abusers increases sharply It is worthnoting that after 3 years the population for the substanceabuserswith control remains constantwhile the population ofthe susceptible substance abusers without control continuesincreasing gradually

Figure 8(c) highlights the impact of controls on thepopulation of infectious nonsubstance abusing adolescentsFor the period 0ndash2 years in the absence of the controlcumulativeHSV-2 cases for the nonsubstance abusing adoles-cents decrease rapidly before starting to increase moderatelyfor the remaining years under consideration The cumula-tive HSV-2 cases for the nonsubstance abusing adolescents

decrease sharply for the whole 20 years under considerationIt is worth noting that for the infectious nonsubstanceabusing adolescents the controls yield positive results aftera period of 2 years hence the controls have an impact incontrolling cumulative HSV-2 cases within a community

Figure 8(d) highlights the impact of controls on thepopulation of infectious substance abusing adolescents Forthe whole period under investigation in the absence ofthe control cumulative infectious substance abusing HSV-2 cases increase rapidly and for the similar period whenthere is a control the population of the exposed individualsdecreases sharply Results on Figure 8(d) clearly suggest that


0 2 4 6 8 10 12 14 16 18 200

02

04

06

08

1

Time (years)

u1

(a)

0 2 4 6 8 10 12 14 16 18 200

02

04

06

08

1

Time (years)

u2

(b)

Figure 9 The optimal control graphs for the two controls namely condom use (1199061) and educational campaigns (119906

2)

the presence of controls has an impact on reducing cumu-lative HSV-2 cases within a society with substance abusingadolescents

Figure 9 represents the controls 1199061and 119906

2 The condom

use control 1199061is at the upper bound 119887

1for approximately 20

years and has a steady drop until it reaches the lower bound1198861= 0 Treatment control 119906

2 is at the upper bound for

approximately half a year and has a sharp drop until it reachesthe lower bound 119886

2= 0

These results suggest that more effort should be devotedto condom use (119906

1) as compared to educational campaigns

(1199062)

6 Discussion

Adolescents who abuse substances have higher rates of HSV-2 infection since adolescents who engage in substance abusealso engage in risky behaviours hence most of them do notuse condoms In general adolescents are not fully educatedonHSV-2 Amathematical model for investigating the effectsof substance abuse amongst adolescents on the transmissiondynamics of HSV-2 in the community is formulated andanalysed We computed and compared the reproductionnumbers Results from the analysis of the reproduction num-bers suggest that condom use and educational campaignshave a great impact in the reduction of HSV-2 cases in thecommunity The PRCCs were calculated to estimate the cor-relation between values of the reproduction number and theepidemiological parameters It has been seen that an increasein condom use and a reduction on the number of adolescentsbecoming substance abusers have the greatest influence onreducing the magnitude of the reproduction number thanany other epidemiological parameters (Figure 6) This result

is further supported by numerical simulations which showthat condom use is highly effective when we are havingmore adolescents quitting substance abusing habit than thosebecoming ones whereas it is less effective when we havemore adolescents becoming substance abusers than quittingSubstance abuse amongst adolescents increases disease trans-mission and prevalence of disease increases with increasedrate of becoming a substance abuser Thus in the eventwhen we have more individuals becoming heavy substanceabusers there is urgent need for intervention strategies suchas counselling and educational campaigns in order to curtailthe HSV-2 spread Numerical simulations also show andsuggest that an increase in recruitment of substance abusersin the community will increase the prevalence of HSV-2cases

The optimal control results show how a cost effectivecombination of aforementioned HSV-2 intervention strate-gies (condom use and educational campaigns) amongstadolescents may influence cumulative cases over a period of20 years Overall optimal control theory results suggest thatmore effort should be devoted to condom use compared toeducational campaigns

Appendices

A Positivity of 119867

To show that119867 ge 0 it is sufficient to show that

1205870(120583 + V) + 120601

120583 + 120601 + 120588 + V1205870

ge119878119899

119873

1205831205871+ 120588

120583 + 120601 + 120588 + V1205870

ge119878119886

119873

(A1)


To do this we prove it by contradiction Assume the followingstatements are true

(i)1205870(120583 + V) + 120601

120583 + 120601 + 120588 + V1205870

lt119878119899

119873

(ii)1205831205871+ 120588

120583 + 120601 + 120588 + V1205870

lt119878119886

119873

(A2)

Adding the inequalities (i) and (ii) in (A2) we have that

120583 (1205870+ 1205871) + 120601 + 120588 + 120587

0

120583 + 120601 + 120588 + V1205870

lt119878119886+ 119878119899

119873997904rArr 1 lt 1 (A3)

which is not true hence a contradiction (since a numbercannot be less than itself)

Equations (A2) and (A5) should justify that 119867 ge 0 Inorder to prove this we shall use the direct proof for exampleif

(i) 1198671ge 2

(ii) 1198672ge 1

(A4)

Adding items (i) and (ii) one gets1198671+ 1198672ge 3

Using the above argument119867 ge 0 if and only if

(i)1205870(120583 + V) + 120601

120583 + 120601 + 120588 + V1205870

ge119878119899

119873

(ii)1205831205871+ 120588

120583 + 120601 + 120588 + V1205870

ge119878119886

119873

(A5)

Adding items (i) and (ii) one gets

120583 (1205870+ 1205871) + 120601 + 120588 + V120587

0

120583 + 120601 + 120588 + V1205870

ge119878119886+ 119878119899

119873

997904rArr120583 + 120601 + 120588 + V120587

0

120583 + 120601 + 120588 + V1205870

ge119878119886+ 119878119899

119873997904rArr 1 ge 1

(A6)

Thus from the above argument119867 ge 0

B Poincareacute-Bendixsonrsquos Negative Criterion

Consider the nonlinear autonomous system

1199091015840

= 119891 (119909) (B1)

where 119891 isin 1198621

(R rarr R) If 1199090isin R119899 let 119909 = 119909(119905 119909

0) be

the solution of (B1) which satisfies 119909(0 1199090) = 1199090 Bendixonrsquos

negative criterion when 119899 = 2 a sufficient condition forthe nonexistence of nonconstant periodic solutions of (B1)is that for each 119909 isin R2 div119891(119909) = 0

Theorem B1 Suppose that one of the inequalities120583(120597119891[2]

120597119909) lt 0 120583(minus120597119891[2]

120597119909) lt 0 holds for all 119909 isin R119899 Thenthe system (B1) has nonconstant periodic solutions

For a detailed discussion of the Bendixonrsquos negativecriterion we refer the reader to [51 61]

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the handling editor andanonymous reviewers for their comments and suggestionswhich improved the paper

References

[1] K J Looker G P Garnett and G P Schmid ldquoAn estimate of theglobal prevalence and incidence of herpes simplex virus type 2infectionrdquoBulletin of theWorldHealth Organization vol 86 no10 pp 805Andash812A 2008

[2] K J Looker G P Garnett and G P Schmid ldquoAn estimate of theglobal prevalence and incidence of herpes simplex virus type 2infectionrdquoBulletin of theWorldHealth Organization vol 86 no10 pp 805ndash812 2008

[3] CDC [Centers for Disease Control and Prevention] Divisionof Sexually Transmitted Disease Surveillance 1999 AtlantaDepartment of Health and Human Services Centers for DiseaseControl and Prevention (CDC) 2000

[4] K Owusu-Edusei Jr H W Chesson T L Gift et al ldquoTheestimated direct medical cost of selected sexually transmittedinfections in the United States 2008rdquo Sexually TransmittedDiseases vol 40 no 3 pp 197ndash201 2013

[5] WHO [World Health Organisation] The Second DecadeImproving Adolescent Health and Development WHO Adoles-cent and Development Programme (WHOFRHADH9819)Geneva Switzerland 1998

[6] K L Dehne and G Riedner Sexually Transmitted Infectionsamong Adolescents The Need for Adequate Health Care WHOGeneva Switzerland 2005

[7] Y L Hill and F M Biro Adolescents and Sexually TransmittedInfections CME Feature 2009

[8] CDC [Centers for Disease Control and Prevention] Divisionof STD Prevention Sexually Transmitted Disease Surveillance1999 Department of Health and Human Services Centers forDisease Control and Prevention (CDC) Atlanta Ga USA2000

[9] W Cates and M C Pheeters ldquoAdolescents and sexuallytransmitted diseases current risks and future consequencesrdquoin Proceedings of the Workshop on Adolescent Sexuality andReproductive Health in Developing Countries Trends and Inter-ventions National Research Council Washington DC USAMarch 1997

[10] K C Chinsembu ldquoSexually transmitted infections in adoles-centsrdquo The Open Infectious Diseases Journal vol 3 pp 107ndash1172009

[11] S Y Cheng and K K Lo ldquoSexually transmitted infections inadolescentsrdquo Hong Kong Journal of Paediatrics vol 7 no 2 pp76ndash84 2002

[12] S L Rosenthal L R Stanberry FM Biro et al ldquoSeroprevalenceof herpes simplex virus types 1 and 2 and cytomegalovirus inadolescentsrdquo Clinical Infectious Diseases vol 24 no 2 pp 135ndash139 1997


[13] G Sucato C Celum D Dithmer R Ashley and A WaldldquoDemographic rather than behavioral risk factors predict her-pes simplex virus type 2 infection in sexually active adolescentsrdquoPediatric Infectious Disease Journal vol 20 no 4 pp 422ndash4262001

[14] J Noell P Rohde L Ochs et al ldquoIncidence and prevalence ofChlamydia herpes and viral hepatitis in a homeless adolescentpopulationrdquo Sexually Transmitted Diseases vol 28 no 1 pp 4ndash10 2001

[15] R L Cook N K Pollock A K Rao andD B Clark ldquoIncreasedprevalence of herpes simplex virus type 2 among adolescentwomen with alcohol use disordersrdquo Journal of AdolescentHealth vol 30 no 3 pp 169ndash174 2002

[16] C D Abraham C J Conde-Glez A Cruz-Valdez L Sanchez-Zamorano C Hernandez-Marquez and E Lazcano-PonceldquoSexual and demographic risk factors for herpes simplex virustype 2 according to schooling level among mexican youthsrdquoSexually Transmitted Diseases vol 30 no 7 pp 549ndash555 2003

[17] I J Birdthistle S Floyd A MacHingura N MudziwapasiS Gregson and J R Glynn ldquoFrom affected to infectedOrphanhood and HIV risk among female adolescents in urbanZimbabwerdquo AIDS vol 22 no 6 pp 759ndash766 2008

[18] P N Amornkul H Vandenhoudt P Nasokho et al ldquoHIVprevalence and associated risk factors among individuals aged13ndash34 years in rural Western Kenyardquo PLoS ONE vol 4 no 7Article ID e6470 2009

[19] Centers for Disease Control and Prevention ldquoYouth riskbehaviour surveillancemdashUnited Statesrdquo Morbidity and Mortal-ity Weekly Report vol 53 pp 1ndash100 2004

[20] K Buchacz W McFarland M Hernandez et al ldquoPrevalenceand correlates of herpes simplex virus type 2 infection ina population-based survey of young women in low-incomeneighborhoods of Northern Californiardquo Sexually TransmittedDiseases vol 27 no 7 pp 393ndash400 2000

[21] L Corey AWald C L Celum and T C Quinn ldquoThe effects ofherpes simplex virus-2 on HIV-1 acquisition and transmissiona review of two overlapping epidemicsrdquo Journal of AcquiredImmune Deficiency Syndromes vol 35 no 5 pp 435ndash445 2004

[22] E E Freeman H A Weiss J R Glynn P L Cross J AWhitworth and R J Hayes ldquoHerpes simplex virus 2 infectionincreases HIV acquisition in men and women systematicreview andmeta-analysis of longitudinal studiesrdquoAIDS vol 20no 1 pp 73ndash83 2006

[23] S Stagno and R JWhitley ldquoHerpes virus infections in neonatesand children cytomegalovirus and herpes simplex virusrdquo inSexually Transmitted Diseases K K Holmes P F Sparling PM Mardh et al Eds pp 1191ndash1212 McGraw-Hill New YorkNY USA 3rd edition 1999

[24] A Nahmias W E Josey Z M Naib M G Freeman R JFernandez and J H Wheeler ldquoPerinatal risk associated withmaternal genital herpes simplex virus infectionrdquoThe AmericanJournal of Obstetrics and Gynecology vol 110 no 6 pp 825ndash8371971

[25] C Johnston D M Koelle and A Wald ldquoCurrent status andprospects for development of an HSV vaccinerdquo Vaccine vol 32no 14 pp 1553ndash1560 2014

[26] S D Moretlwe and C Celum ldquoHSV-2 treatment interventionsto control HIV hope for the futurerdquo HIV Prevention Coun-selling and Testing pp 649ndash658 2004

[27] C P Bhunu and S Mushayabasa ldquoAssessing the effects ofintravenous drug use on hepatitis C transmission dynamicsrdquoJournal of Biological Systems vol 19 no 3 pp 447ndash460 2011

[28] C P Bhunu and S Mushayabasa ldquoProstitution and drug (alco-hol) misuse the menacing combinationrdquo Journal of BiologicalSystems vol 20 no 2 pp 177ndash193 2012

[29] Y J Bryson M Dillon D I Bernstein J Radolf P Zakowskiand E Garratty ldquoRisk of acquisition of genital herpes simplexvirus type 2 in sex partners of persons with genital herpes aprospective couple studyrdquo Journal of Infectious Diseases vol 167no 4 pp 942ndash946 1993

[30] L Corey A Wald R Patel et al ldquoOnce-daily valacyclovir toreduce the risk of transmission of genital herpesrdquo The NewEngland Journal of Medicine vol 350 no 1 pp 11ndash20 2004

[31] G J Mertz R W Coombs R Ashley et al ldquoTransmissionof genital herpes in couples with one symptomatic and oneasymptomatic partner a prospective studyrdquo The Journal ofInfectious Diseases vol 157 no 6 pp 1169ndash1177 1988

[32] AWald A G Langenberg K Link et al ldquoEffect of condoms onreducing the transmission of herpes simplex virus type 2 frommen to womenrdquo Journal of the American Medical Associationvol 285 no 24 pp 3100ndash3106 2001

[33] A H Mokdad J S Marks D F Stroup and J L GerberdingldquoActual causes of death in the United States 2000rdquoThe Journalof the American Medical Association vol 291 no 10 pp 1238ndash1245 2004


[35] E T Martin E Krantz S L Gottlieb et al ldquoA pooled analysisof the effect of condoms in preventing HSV-2 acquisitionrdquoArchives of Internal Medicine vol 169 no 13 pp 1233ndash12402009

[36] A C Ghani and S O Aral ldquoPatterns of sex worker-clientcontacts and their implications for the persistence of sexuallytransmitted infectionsrdquo Journal of Infectious Diseases vol 191no 1 pp S34ndashS41 2005

[37] G P Garnett G DubinM Slaoui and T Darcis ldquoThe potentialepidemiological impact of a genital herpes vaccine for womenrdquoSexually Transmitted Infections vol 80 no 1 pp 24ndash29 2004

[38] E J Schwartz and S Blower ldquoPredicting the potentialindividual- and population-level effects of imperfect herpessimplex virus type 2 vaccinesrdquoThe Journal of Infectious Diseasesvol 191 no 10 pp 1734ndash1746 2005

[39] C N Podder andA B Gumel ldquoQualitative dynamics of a vacci-nation model for HSV-2rdquo IMA Journal of Applied Mathematicsvol 75 no 1 pp 75ndash107 2010

[40] R M Alsallaq J T Schiffer I M Longini A Wald L Coreyand L J Abu-Raddad ldquoPopulation level impact of an imperfectprophylactic vaccinerdquo Sexually Transmitted Diseases vol 37 no5 pp 290ndash297 2010

[41] Y Lou R Qesmi Q Wang M Steben J Wu and J MHeffernan ldquoEpidemiological impact of a genital herpes type 2vaccine for young femalesrdquo PLoS ONE vol 7 no 10 Article IDe46027 2012

[42] A M Foss P T Vickerman Z Chalabi P Mayaud M Alaryand C H Watts ldquoDynamic modeling of herpes simplex virustype-2 (HSV-2) transmission issues in structural uncertaintyrdquoBulletin of Mathematical Biology vol 71 no 3 pp 720ndash7492009

[43] A M Geretti ldquoGenital herpesrdquo Sexually Transmitted Infectionsvol 82 no 4 pp iv31ndashiv34 2006

[44] R Gupta TWarren and AWald ldquoGenital herpesrdquoThe Lancetvol 370 no 9605 pp 2127ndash2137 2007


[45] E Kuntsche R Knibbe G Gmel and R Engels ldquoWhy do youngpeople drinkA reviewof drinkingmotivesrdquoClinical PsychologyReview vol 25 no 7 pp 841ndash861 2005

[46] B A Auslander F M Biro and S L Rosenthal ldquoGenital herpesin adolescentsrdquo Seminars in Pediatric Infectious Diseases vol 16no 1 pp 24ndash30 2005

[47] HWHethcote ldquoThemathematics of infectious diseasesrdquo SIAMReview vol 42 no 4 pp 599ndash653 2000

[48] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180no 1-2 pp 29ndash48 2002

[49] J Arino F Brauer P van den Driessche J Watmough andJ Wu ldquoA model for influenza with vaccination and antiviraltreatmentrdquo Journal ofTheoretical Biology vol 253 no 1 pp 118ndash130 2008

[50] V Lakshmikantham S Leela and A A Martynyuk StabilityAnalysis of Nonlinear Systems vol 125 of Pure and AppliedMathematics A Series of Monographs and Textbooks MarcelDekker New York NY USA 1989

[51] L PerkoDifferential Equations andDynamical Systems vol 7 ofTexts in Applied Mathematics Springer Berlin Germany 2000

[52] A Korobeinikov and P K Maini ldquoA Lyapunov function andglobal properties for SIR and SEIR epidemiologicalmodels withnonlinear incidencerdquo Mathematical Biosciences and Engineer-ing MBE vol 1 no 1 pp 57ndash60 2004

[53] A Korobeinikov and G C Wake ldquoLyapunov functions andglobal stability for SIR SIRS and SIS epidemiological modelsrdquoApplied Mathematics Letters vol 15 no 8 pp 955ndash960 2002

[54] I Barbalat ldquoSysteme drsquoequation differentielle drsquooscillationnonlineairesrdquo Revue Roumaine de Mathematique Pures etAppliquees vol 4 pp 267ndash270 1959

[55] C C McCluskey ldquoLyapunov functions for tuberculosis modelswith fast and slow progressionrdquo Mathematical Biosciences andEngineering MBE vol 3 no 4 pp 603ndash614 2006

[56] E Jung S Lenhart and Z Feng ldquoOptimal control of treatmentsin a two-strain tuberculosis modelrdquo Discrete and ContinuousDynamical Systems Series B A Journal Bridging Mathematicsand Sciences vol 2 no 4 pp 473ndash482 2002

[57] W H Fleming and R W Rishel Deterministic and StochasticOptimal Control Springer New York NY USA 1975

[58] D L Lukes Differential Equations Classical to ControlledMathematics in Science and Engineering vol 162 AcademicPress New York NY USA 1982

[59] L S Pontryagin V T Boltyanskii R V Gamkrelidze and E FMishchevkoTheMathematicalTheory of Optimal Processes vol4 Gordon and Breach Science Publishers 1985

[60] G Magombedze W Garira E Mwenje and C P BhunuldquoOptimal control for HIV-1 multi-drug therapyrdquo InternationalJournal of Computer Mathematics vol 88 no 2 pp 314ndash3402011

[61] M Fiedler ldquoAdditive compound matrices and an inequalityfor eigenvalues of symmetric stochastic matricesrdquo CzechoslovakMathematical Journal vol 24(99) pp 392ndash402 1974

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


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OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


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Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


increased the likelihood of acquiring HSV-2 infection [14ndash16] implying that we have more adolescents being infectedPrevalence estimates for adolescents are available throughcommunity-based studies One study in Zimbabwe amongadolescent females aged 15ndash19 years reported that 12 ofparticipants tested positive forHSV-2 [17] A study inwesternKenya found HSV-2 prevalence to be 9 for females and4 for males aged 13-14 years and 28 females and 17for males aged 15ndash19 years [18] It was shown that sexualactivity in Namibia begins at a very young age often aslow as 10 years [10] and also some 61 percent of twelfthgraders in United States of America (USA) have had sexualintercourse at least once [19] Many adolescents experimentwith substances parents teachers and policy makers want toknow the consequences of adolescent substance abuse Sincethe abuse of substances affects their thinking capacity mostof them who are substance abusers have higher rates of HSV-2 infection [15 20] Those adolescents who are infected withHSV-2 type 2 (HSV-2) infection may be at greater risk fortransmission and acquisition of human immunodeficiencyvirus (HIV) [21] Interventions to control HSV-2 infectioncould prevent as many as 50 to 60 of new HIV infec-tions [22] Furthermore pregnant adolescents infected withgenital HSV (particularly those with a primary infection)can transmit infection to the neonate which can lead toserious complications for the neonate such as neurologicproblems and even death [23 24] Since currently there isno vaccination for HSV-2 [25] current intervention optionsfor controlling HSV-2 are limited to patient education andpromotion of condom use or treatment with oral medication[26] Some preventive efforts should be made to targetchildren before initiation of sexual activity since preventionis always better than cure

A number of mathematical models have looked intomathematical modelling of HSV-2 transmission dynamicsfocusing on a number of different issues (see [36ndash42] tomention just a few but none has looked into the effectsof substance abuse on the transmission dynamics of HSV-2among adolescents) In contrast to other STIs such as chlamy-dia gonorrhea syphilis and trichomoniasis infection withHSV-2 is lifelong and once established there is currentlyno treatment to eliminate [43 44] The other problem withHSV-2 infection in adolescents is that it is somewhat difficultto determine given that many infections are asymptomaticor go unrecognised hence it differs with most of otherSTIs in general To the best of our knowledge as authorsno mathematical model has looked into the transmissiondynamics of HSV-2 and substance abuse with a focus onadolescents It is against this background that our study findsrelevance and motivation by formulating a mathematicalmodel to investigate the impact of substance abuse amongadolescents on the transmission dynamics of HSV-2 Themodel incorporates some key epidemiological features suchas the impact of condom use and educational campaignsamong the adolescents The main objective in this study is toforecast future trends in the incidence ofHSV-2 epidemic andalso to quantify the association between substance abuse andHSV-2 epidemic amongst adolescents in the community

The paper is structured as follows The HSV-2 transmis-sion model is formulated in the next section Analytic resultsof the model system are presented in Section 3 Simulationresults and projection profiles of HSV-2 are presented inSection 4 In Section 5 optimal control theory has beenapplied to the model formulation in Section 2 Summary andconcluding remarks round up the paper

2 Model Formulation

The total sexually active population at time 119905 denoted by119873is subdivided into mutually exclusive compartments namelysusceptible adolescents who are nonsubstance abusers (119878

119899)

susceptible adolescents who are substance abusers (119878119886) ado-

lescents infected with HSV-2 who are nonsubstance abusers(119868119899) and adolescents infected with HSV-2 who are substance

abusers (119868119886) Although there are many causes of substance

abuse amongst adolescents here we dwell on peer pressure asthe main one The total population is given by

119873 = 119878119899(119905) + 119878

119886(119905) + 119868

119899(119905) + 119868

119886(119905) (1)

The susceptible is increased by a constant inflow into thepopulation at rate Λ A fraction 120587

0of these adolescents are

assumed to be nonsubstance abusers and the complementaryfraction 120587

1= 1 minus 120587

0being substance abusers This is

reasonable because the underlying population is made ofadolescents only into these two classes (influence from somepeers is probably the main drive behind the adolescentsabusing substances and hence their recruitment into thesubstance abusers class) It has been found that adolescentsstart abusing substances for four main reasons to improvetheir mood to receive social rewards to reduce negativefeelings and to avoid social rejection [45] and these makethem fall victims to peer influence easily Individuals indifferent human subgroups suffer from natural death at rate120583 Assuming homogeneous mixing of the population thesusceptible individuals acquireHSV-2 infection at rate120582 with

120582 (119905) =120573 (119868119886(119905) + (1 minus 120578) 119868

119899(119905))

119873 (119905) (2)

120578 isin (0 1) is a modification parameter accounting for reducednumber a nonsubstance abuser would infect as comparedto a substance abuser [15 20] perhaps because adolescentswho engage in substance abuse also engage in other riskybehaviours Substance abuse diminishes judgement leadingto the engagement in the high risk behaviours [46] and thesame explanation also applies for the modification parameter120572 isin (0 1) 119901 is the probability of being infected from asexual partner and 119888 is the rate at which an individual acquiressexual partners per unit time and 120573 = 119901119888 is the effectivecontact rate for HSV-2 infection (contact sufficient to resultin HSV-2 infection) The role of condom use is representedby 120579 If 120579 = 0 then condom use is not effective 120579 = 1

corresponds to completely effective condom use while 0 lt120579 lt 1 implies that condom use would be effective to somedegree Substance abusers experience related diseases likeliver cirrhosis pancreatitis and heart disease among other


Λ1205870 Λ1205871

120588

120588

120601

120601 SaSn

120583 + 120583

120583 + 120583

(1 minus 120579)120582120572(1 minus 120579)120582

IaIn




1198781015840

119899= 1205870Λ minus 120572 (1 minus 120579) 120582119878

119899+ 120601119878119886minus (120583 + 120588) 119878

119899

1198781015840

119886= 1205871Λ minus (1 minus 120579) 120582119878

119886+ 120588119878119899minus (120583 + V + 120601) 119878

119886

1198681015840

119899= 120572 (1 minus 120579) 120582119878

119899+ 120601119868119886minus (120583 + 120588) 119868

119899

1198681015840

119886= (1 minus 120579) 120582119878

119886+ 120588119868119899minus (120583 + V + 120601) 119868

119886

(3)


119899(119905) 119878119886(119905) 119868119899(119905) and 119868

119886(119905) are


and are bounded inG

Theorem 1 Let 119878119899(119905) ge 0 119878

119886(119905) ge 0 119868

119899(119905) ge 0 and 119868

119886(119905) ge 0





1199051 119878119899(1199051) = 0 1198781015840

119899(1199051) lt 0 and 119878

119899(119905) gt 0 119878

119886(119905) gt 0 119868

119899(119905) gt

0 119868119886(119905) gt 0 for 0 lt 119905 lt 119905

1or there exists a

1199052 119878119886(1199052) = 0 1198781015840

119899(1199052) lt 0 and 119878

119899(119905) gt 0 119878

119886(119905) gt 0 119868

119899(119905) gt

0 119868119886(119905) gt 0 for 0 lt 119905 lt 119905

2or there exists a

1199053 119868119899(1199053) = 0 1198681015840

119899(1199053) lt 0 and 119878

119899(119905) gt 0 119878

119886(119905) gt 0 119868

119899(119905) gt

0 119868119886(119905) gt 0 for 0 lt 119905 lt 119905

3or there exists a

1199054 119868119886(1199054) = 0 1198681015840

119886(1199054) lt 0 and 119878

119899(119905) gt 0 119878

119886(119905) gt 0 119868

119899(119905) gt

0 119868119886(119905) gt 0 for 0 lt 119905 lt 119905


1198781015840

119899(1199051) = 1205870Λ + 120601119878

119886(1199051) gt 0 (4)



1198781015840

119886(1199052) = 1205871Λ + 120588119878

119899(1199052) gt 0 (5)


case we have

1198681015840

119899(1199053) = 120572 (1 minus 120579) 120582 (119905

3) 119878119899(1199053) + 120601119868

119886(1199053) gt 0 (6)


case we have

1198681015840

119886(1199054) = (1 minus 120579) 120582 (119905

4) 119878119886(1199054) + 120588119868

119904(1199054) gt 0 (7)


all cases 119878119899 119878119886 119868119899 and 119868


Since119873(119905) ge 119878119886(119905) and119873(119905) ge 119868

119886(119905) then

Λ minus (120583 + 2V)119873 le 1198731015840

(119905) le Λ minus 120583119873 (8)



+

G = (119878119899 119878119886 119868119899 119868119886) isin R4

+ 119873 le

Λ

120583 (9)


3 Model Analysis


E0 = (Λ (1205870(120583 + V) + 120601)

V (120583 + 120588) + 120583 (120583 + 120588 + 120601)

Λ (1205831205871+ 120588)

V (120583 + 120588) + 120583 (120583 + 120588 + 120601) 0 0)

(10)





=[[

[

120572120573 (1 minus 120579) (1 minus 120578) (1205870(120583 + V) + 120601)

120583 + 120588 + 120601 + V1205870

120572120573 (1 minus 120579) (1205870(120583 + V) + 120601)

120583 + 120588 + 120601 + V1205870

120573 (1 minus 120579) (1 minus 120578) (1205831205871+ 120588)

120583 + 120588 + 120601 + V1205870

120573 (1 minus 120579) (1205831205871+ 120588)

120583 + 120588 + 120601 + V1205870

]]

]

(11)

119881 = [120583 + 120588 minus120601

minus120588 120583 + 120601 + V] (12)



1= 0 and 120587


radius is given by

1198770= 119877119899=120572120573 (1 minus 120579) (1 minus 120578)

120583 (13)




0= 0 and 120587

1= 1 Then the


119877119886=120573 (1 minus 120579)

120583 + V (14)




119877119888=

120573 (1198961+ 1198962+ 1198963)

(120583 + 120588 + 120601 + V1205870) ((120583 + V) (120583 + 120588) + 120583120601)

(15)

where1198961= V120572 ((1 minus 120578) (2120583120587

0+ 120601) + 120587

0(120588 + (1 minus 120578) 120601))

1198962= (120583120587

1+ 120588) (120583 + 120588 + (1 minus 120578) 120601) + V2120572 (1 minus 120578) 120587

0

1198963= 120572 (120583120587

0+ 120601) (120588 + (1 minus 120588) (120583 + 120601) + 120588)

(16)




119877119899119886=

120573 (1 minus 120579) (1198961+ 1198962+ 1198963)

(120583 + 120588 + 120601 + V1205870) ((120583 + V) (120583 + 120588) + 120583120601)

(17)

where

1198961= V120572 ((1 minus 120578) (2120583120587

0+ 120601) + 120587

0(120588 + (1 minus 120578) 120601))

1198962= (120583120587

1+ 120588) (120583 + 120588 + (1 minus 120578) 120601) + V2120572 (1 minus 120578) 120587

0

1198963= 120572 (120583120587

0+ 120601) (120588 + (1 minus 120588) (120583 + 120601) + 120588)

(18)

biologically 119877119899119886








119877119899119886gt 1


[[[

[

119889119868119899

119889119905

119889119868119886

119889119905

]]]

]

= [119865 minus 119881] [119868119899

119868119886

] minus 119867[1 minus 120578 1

0 0] [119868119899

119868119886

] (19)


119867 = [120572120573 (1 minus 120579) (1205870(120583 + V) + 120601

120583 + 120601 + 120588 + V1205870

minus119878119899

119873)

+120573 (1 minus 120579) (1205831205871+ 120588

120583 + 120601 + 120588 + V1205870

minus119878119886

119873)]

(20)



[[[

[

119889119868119899

119889119905

119889119868119886

119889119905

]]]

]

le [119865 minus 119881] [119868119899

119868119886

] (21)



119899119886le 1




119899rarr 1198780

119899and

119878119886rarr 1198780







1198781015840

119899= Λ minus


119899119868lowast

119899

119873lowast

119899


119899

1198681015840

119899=120572 (1 minus 120579) (1 minus 120578) 120573119878

lowast

119899119868lowast

119899

119873lowast

119899


119899

(22)


G119899= (119878

119899 119868119899) isin R2

+ 119873119899leΛ

120583 (23)


119899


(22) 119868119899= 0 or

0 =120572 (1 minus 120579) (1 minus 120578) 120573119878

lowast

119899119868lowast

119899

119873lowast

119899


119899


119899= 120583119873

lowast

119899


119899=

Λ

120583119877119899

(24)



119899+ 119868lowast


119899= 119873lowast


119899 and thus

119868lowast

119899=Λ

120583minusΛ

120583119877119899

= (119877119899minus 1) 119878

lowast

119899

(25)


119899 119868lowast

119899)


Theorem 4 below





=

[[[

[

minus(


119899

119873lowast

119899

+ 120583)


119899

119873lowast

119899


119899

119873lowast

119899


119899

119873lowast

119899

minus 120583

]]]

]


119899119873lowast

119899= 120583 which



as

119869 (Elowast

119899) =

[[[[

[


lowast

119899

119873lowast

119899

+ 120583) minus120583


119899

119873lowast

119899

0

]]]]

]

(27)


Υ2

+ (120572 (1 minus 120579) (1 minus 120578) 120573119868

lowast

119899

119873lowast

119899

+ 120583)Υ

+120572 (1 minus 120579) (1 minus 120578) 120583120573119868

lowast

119899

119873lowast

119899

= 0

997904rArr Υ1= minus120583


119873minus 1)

120573119868lowast

119899

119873lowast

119899

(28)


119899gt 1 We






119899whenever

119877119899gt 1



119899= (Λ120583)minus119868

119899


1198681015840

119899=120572 (1 minus 120579) (1 minus 120578) 120573119868

119899

Λ120583(Λ

120583minus 119868119899) minus 120583119868

119899 (29)



120597

120597119868119899

[120572 (1 minus 120579) (1 minus 120578) 120573

Λ120583(Λ

120583minus 119868119899) minus 120583]


Λlt 0


119873lt 0

(30)





119899 for all 119905 ge 0




Elowast

119886= 119878lowast

119886=

Λ

120583119877119886

119868lowast

119886= (119877119886minus 1) 119878

lowast

119886 (31)



119886gt 1





119899119886

with

Elowast

119899119886=

119878lowast

119899=

1205870Λ + 120601119878

lowast

119886

120572 (1 minus 120579) 120582lowast

119899119886+ 120583 + 120588

119878lowast

119886=

1205871Λ + 120588119878

lowast

119899

(1 minus 120579) 120582lowast

119899119886+ 120583 + V + 120601

119868lowast

119899=120572 (1 minus 120579) 120582

lowast

119899119886119878lowast

119899+ 120601119868lowast

119886

120583 + 120588

119868lowast

119886=(1 minus 120579) 120582

lowast

119899119886119878lowast

119886+ 120588119868lowast

119899

120583 + V + 120601


119886+ (1 minus 120578) 119868

lowast

119899)

119873lowast

119899119886

(32)






∘



119878119899 120577 le lim inf

119905rarrinfin

119878119886


119868119899 120577 le lim inf

119905rarrinfin

119868119886

(33)


119881 (119878119899 119878119886 119868119899 119868119886) = (119878


119899ln 119878119899) + (119878


119886ln 119878119886)

+ (119868119899minus 119868lowast

119899ln 119868119899) + (119868119886minus 119868lowast

119886ln 119868119886)

(34)


120597119881

120597119878119899


119899

119878119899

) 120597119881

120597119878119886


119886

119878119886

)

120597119881

120597119868119899


119899

119868119899

) 120597119881

120597119868119886


119886

119868119886

)

(35)




119881(119878119899 119878119886 119868119899 119868119886) gt 0 and 119881

1015840

(119878119899 119878119886 119868119899 119868119886) = 0 only at



1198811015840

(119878119899 119878119899 119868119886 119868119886) = (119878


119899)1198781015840

119899

119878119899

+ (119878119886minus 119878lowast

119886)1198781015840

119886

119878119886

+ (119868119899minus 119868lowast

119899)1198681015840

119899

119868119899

+ (119868119886minus 119868lowast

119886)1198681015840

119886

119868119886


119899)2

119878119899

+ 119892 (119878119899 119878119886 119868119899 119868119886)

(36)



119899119886 The only

invariant set in∘








119899119886is





0 1205871




as119863(119868119899 119868119886) = 1119868

119899119868119886 Then we have

120597 (119863Φ)

120597119868119899

+120597 (119863Ψ)

120597119868119886

= minus120601

1198682

119899

minus120588

1198682

119886

minus (1 minus 120579) 120573(120572119878119899

1198682

119899

+(1 minus 120578) 119878

119886

1198682

119886

) lt 0

(37)



119899119886gt 1

4 Numerical Results



119899119886


02 04 06 08 10

10

20

30

40 Ra

R0

Rc

Rna

120573

R








00

05

10

00

05

10

10

15

120601120588

Rna

(a)

02 04 06 08 10

10

12

14

16

18

20

22

(120601 120588)

Rna

Ω

120595

Σ

(b)










119899119886



on 119877119899119886








0 02 04 06 08





Natural death rate



Condom use efficacy












0 01 02 03 04 05 06 07 08 09 1

0

2


minus10

minus8

minus6

minus4

minus2

log(Rna)

(a)


0

2

minus10

minus8

minus6

minus4

minus2

log(Rna)

(b)


0

2

minus10

minus8

minus6

minus4

minus2

log(Rna)

(c)





119899

119878119886 119868119899 and 119868



119869 (1199061 1199062)

= int

119905119891

0


1+ 11986021199062

2)] 119889119905

(38)


120579 = 02

120579 = 05

120579 = 08

0 10 20 30 40 50 600

500

1000

1500

2000

2500

3000

3500

Time (years)

Cum

ulat

ive H

SV-2

case

s

(a)

120579 = 02

120579 = 05

120579 = 08

0 10 20 30 40 50 60100

200

300

400

500

600

Time (years)

Cum

ulat

ive H

SV-2

case

s

(b)



0 10 20 30 40 50 600

500

1000

1500

2000

2500

3000

3500

Time (years)

Cum

ulat

ive H

SV-2

case

s

(a)

0 10 20 30 40 50 60100

200

300

400

500

600

700

800

900

Time (years)

Cum

ulat

ive H

SV-2

case

s

(b)



subject to

1198781015840

119899= 1205870Λ minus 120572 (1 minus 119906

1) 120582119878119899+ 120601119878119886minus (1 minus 119906

2) 120588119878119899minus 120583119878119899

1198781015840

119886= (1 minus 119906

2) 1205871Λ minus (1 minus 119906

1) 120582119878119886

+ (1 minus 1199062) 120588119878119899minus (120583 + V + 120601) 119878

119886

1198681015840

119899= 120572 (1 minus 119906

1) 120582119878119899+ 120601119868119886minus (1 minus 119906

2) 120588119868119899minus 120583119868119899

1198681015840

119886= (1 minus 119906

1) 120582119878119886+ (1 minus 119906

2) 120588119868119899minus (120583 + V + 120601) 119868

119886

(39)








2






1and 119906lowast

2such that

119869 (119906lowast

1 119906lowast

2) = max 119869 (119906

1 1199062) | 1199061 1199062isin U (40)

whereU = 1199061(119905) 1199062(119905) | 119906

1(119905) 1199062(119905) is measurable 0 le 119886

11le

1199061(119905) le 119887

11le 1 0 le 119886

22le 1199062(119905) le 119887

22le 1 119905 isin [0 119905

119891]





1and 119906





1and 119906

2with



+ |1199062|2

) where 1198881 1198882gt 0



1198613119868119886minus (11986011199062

1+11986021199062



1 1198612 1198613 1198601 1198602gt 1 hence

satisfying


1+ 11986021199062

2)

le 1198882minus 1198881(10038161003816100381610038161199061

1003816100381610038161003816

2

+10038161003816100381610038161199062

1003816100381610038161003816

2

)

(41)




1+ 11986021199062

2)

+ 1205821[1205870Λ minus

120572 (1 minus 1199061) 120573119868119886119878119899

119873

minus120572 (1 minus 119906

1) (1 minus 120578) 120573119868

119899119878119899

119873

+120601119878119886minus (1 minus 119906

2) 120588119878119899minus 120583119878119899]

+ 1205822[(1 minus 119906

2) 1205871Λ minus

(1 minus 1199061) 120573119868119886119878119886

119873


1) (1 minus 120578) 120573119868

119899119878119886

119873

+ (1 minus 1199062) 120588119878119899minus (120583 + V + 120601) 119878

119886]

+ 1205823[120572 (1 minus 119906

1) 120573119868119886119878119899

119873

+120572 (1 minus 119906

1) (1 minus 120578) 120573119868

119899119878119899

119873

+120601119868119886minus (1 minus 119906

2) 120588119868119899minus 120583119868119899]

+ 1205824[(1 minus 119906

1) 120573119868119886119878119886

119873+(1 minus 119906

1) (1 minus 120578) 120573119868

119899119878119886

119873

+ (1 minus 1199062) 120588119868119899minus (120583 + V + 120601) 119868

119886]

+ 12059611(119905) (1198871minus 1199061(119905)) + 120596

12(119905) (1199061(119905) minus 119886

1)

+ 12059621(119905) (1198872minus 1199062(119905)) + 120596

22(119905) (1199062(119905) minus 119886

2)

(42)

where 12059611(119905) 12059612(119905) ge 0 120596

21(119905) and 120596



12(119905)(1199061(119905) minus

1198861) = 0 120596

21(119905)(1198872minus 1199062(119905)) = 0 and 120596

22(119905)(1199062(119905) minus 119886

2) = 0 at


2



119889119905= minus

120597119871

120597119878119899

= minus1 +120572 (1 minus 119906

1) 120573119868119886[1205821minus 1205823]

119873

+120572 (1 minus 119906

1) (1 minus 120578) 120573119868

119899[1205821minus 1205823]

119873

+ (1 minus 1199062) 120588 [1205821minus 1205822] + 120583120582

1


1198891205822

119889119905= minus

120597119871

120597119878119886

= 1198611+ 120601 [120582

2minus 1205821] +

(1 minus 1199061) 120573119868119886[1205822minus 1205824]

119873

+(1 minus 119906

1) (1 minus 120578) 120573119868

119899[1205822minus 1205824]

119873+ (120583 + V) 120582

2

1198891205822

119889119905= minus

120597119871

120597119868119899

= 1198612+120572 (1 minus 119906

1) (1 minus 120578) 120573119878

119899[1205821minus 1205823]

119873

+(1 minus 119906

1) (1 minus 120578) 120573119878

119886[1205822minus 1205824]

119873

+ (1 minus 1199062) 120588 [1205823minus 1205824] + 120583120582

3

1198891205822

119889119905= minus

120597119871

120597119868119899

= 1198613+120572 (1 minus 119906

1) 120573119878119899[1205821minus 1205823]

119873

+(1 minus 119906

1) 120573119878119886[1205822minus 1205824]

119873+ 120601 [120582

4minus 1205823] + (120583 + V) 120582

4

(43)




1199061(119905)lowast

=120573 (119868119886+ (1 minus 120578) 119868

119899)

21198601119873

[120572119878119899(1205821minus 1205823) + 119878119886(1205822minus 1205824)]

+12059612(119905) minus 120596

11(119905)

21198601

(44)


1199062(119905)lowast

=120588

21198602


1205871Λ1205822

21198602

+12059622minus 12059621

21198602

(45)


1(119905) and 120596




1199061(119905)lowast

= minmax 1198861120573 (119868119886+ (1 minus 120578) 119868

119899)

21198601119873


1

1199062(119905)lowast

= minmax1198862120588

21198602

[119878119899(1205821minus 1205822) + 119868119899(1205823minus 1205824)]

minus1205871Λ1205822

21198602

1198872

(46)


1 and 119906lowast

2for 1199061(119905) and 119906

2(119905)


119891[60]



1= 0005 119860

2=

0005 1198611= 006 119861

2= 005 and 119861

3= 005 The assumed


= 041198781198860

= 03 1198681198990

= 02 and 1198681198860





036

038

04

042

044

046

048

0 2 4 6 8 10 12 14 16 18 20Time (years)

S n(t)

(a)

024

026

028

03

032

034

036

0 2 4 6 8 10 12 14 16 18 20Time (years)

S a(t)

(b)

005

01

015

02

025

03

035


00 2 4 6 8 10 12 14 16 18 20

Time (years)

I n(t)

(c)


0

005

01

015

02

025

03

035

0 2 4 6 8 10 12 14 16 18 20Time (years)

I a(t)


119899 susceptible




of 20 years






0 2 4 6 8 10 12 14 16 18 200

02

04

06

08

1

Time (years)

u1

(a)

0 2 4 6 8 10 12 14 16 18 200

02

04

06

08

1

Time (years)

u2

(b)


2)



2 The condom






2= 0



(1199062)

6 Discussion




Appendices



1205870(120583 + V) + 120601

120583 + 120601 + 120588 + V1205870

ge119878119899

119873

1205831205871+ 120588

120583 + 120601 + 120588 + V1205870

ge119878119886

119873

(A1)



(i)1205870(120583 + V) + 120601

120583 + 120601 + 120588 + V1205870

lt119878119899

119873

(ii)1205831205871+ 120588

120583 + 120601 + 120588 + V1205870

lt119878119886

119873

(A2)


120583 (1205870+ 1205871) + 120601 + 120588 + 120587

0

120583 + 120601 + 120588 + V1205870

lt119878119886+ 119878119899

119873997904rArr 1 lt 1 (A3)



(i) 1198671ge 2

(ii) 1198672ge 1

(A4)



(i)1205870(120583 + V) + 120601

120583 + 120601 + 120588 + V1205870

ge119878119899

119873

(ii)1205831205871+ 120588

120583 + 120601 + 120588 + V1205870

ge119878119886

119873

(A5)


120583 (1205870+ 1205871) + 120601 + 120588 + V120587

0

120583 + 120601 + 120588 + V1205870

ge119878119886+ 119878119899

119873

997904rArr120583 + 120601 + 120588 + V120587

0

120583 + 120601 + 120588 + V1205870

ge119878119886+ 119878119899

119873997904rArr 1 ge 1

(A6)




1199091015840

= 119891 (119909) (B1)

where 119891 isin 1198621


0) be




120597119909) lt 0 120583(minus120597119891[2]





Acknowledgments


References







































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Λ1205870 Λ1205871

120588

120588

120601

120601 SaSn

120583 + 120583

120583 + 120583

(1 minus 120579)120582120572(1 minus 120579)120582

IaIn




1198781015840

119899= 1205870Λ minus 120572 (1 minus 120579) 120582119878

119899+ 120601119878119886minus (120583 + 120588) 119878

119899

1198781015840

119886= 1205871Λ minus (1 minus 120579) 120582119878

119886+ 120588119878119899minus (120583 + V + 120601) 119878

119886

1198681015840

119899= 120572 (1 minus 120579) 120582119878

119899+ 120601119868119886minus (120583 + 120588) 119868

119899

1198681015840

119886= (1 minus 120579) 120582119878

119886+ 120588119868119899minus (120583 + V + 120601) 119868

119886

(3)


119899(119905) 119878119886(119905) 119868119899(119905) and 119868

119886(119905) are


and are bounded inG

Theorem 1 Let 119878119899(119905) ge 0 119878

119886(119905) ge 0 119868

119899(119905) ge 0 and 119868

119886(119905) ge 0





1199051 119878119899(1199051) = 0 1198781015840

119899(1199051) lt 0 and 119878

119899(119905) gt 0 119878

119886(119905) gt 0 119868

119899(119905) gt

0 119868119886(119905) gt 0 for 0 lt 119905 lt 119905

1or there exists a

1199052 119878119886(1199052) = 0 1198781015840

119899(1199052) lt 0 and 119878

119899(119905) gt 0 119878

119886(119905) gt 0 119868

119899(119905) gt

0 119868119886(119905) gt 0 for 0 lt 119905 lt 119905

2or there exists a

1199053 119868119899(1199053) = 0 1198681015840

119899(1199053) lt 0 and 119878

119899(119905) gt 0 119878

119886(119905) gt 0 119868

119899(119905) gt

0 119868119886(119905) gt 0 for 0 lt 119905 lt 119905

3or there exists a

1199054 119868119886(1199054) = 0 1198681015840

119886(1199054) lt 0 and 119878

119899(119905) gt 0 119878

119886(119905) gt 0 119868

119899(119905) gt

0 119868119886(119905) gt 0 for 0 lt 119905 lt 119905


1198781015840

119899(1199051) = 1205870Λ + 120601119878

119886(1199051) gt 0 (4)



1198781015840

119886(1199052) = 1205871Λ + 120588119878

119899(1199052) gt 0 (5)


case we have

1198681015840

119899(1199053) = 120572 (1 minus 120579) 120582 (119905

3) 119878119899(1199053) + 120601119868

119886(1199053) gt 0 (6)


case we have

1198681015840

119886(1199054) = (1 minus 120579) 120582 (119905

4) 119878119886(1199054) + 120588119868

119904(1199054) gt 0 (7)


all cases 119878119899 119878119886 119868119899 and 119868


Since119873(119905) ge 119878119886(119905) and119873(119905) ge 119868

119886(119905) then

Λ minus (120583 + 2V)119873 le 1198731015840

(119905) le Λ minus 120583119873 (8)



+

G = (119878119899 119878119886 119868119899 119868119886) isin R4

+ 119873 le

Λ

120583 (9)


3 Model Analysis


E0 = (Λ (1205870(120583 + V) + 120601)

V (120583 + 120588) + 120583 (120583 + 120588 + 120601)

Λ (1205831205871+ 120588)

V (120583 + 120588) + 120583 (120583 + 120588 + 120601) 0 0)

(10)





=[[

[

120572120573 (1 minus 120579) (1 minus 120578) (1205870(120583 + V) + 120601)

120583 + 120588 + 120601 + V1205870

120572120573 (1 minus 120579) (1205870(120583 + V) + 120601)

120583 + 120588 + 120601 + V1205870

120573 (1 minus 120579) (1 minus 120578) (1205831205871+ 120588)

120583 + 120588 + 120601 + V1205870

120573 (1 minus 120579) (1205831205871+ 120588)

120583 + 120588 + 120601 + V1205870

]]

]

(11)

119881 = [120583 + 120588 minus120601

minus120588 120583 + 120601 + V] (12)



1= 0 and 120587


radius is given by

1198770= 119877119899=120572120573 (1 minus 120579) (1 minus 120578)

120583 (13)




0= 0 and 120587

1= 1 Then the


119877119886=120573 (1 minus 120579)

120583 + V (14)




119877119888=

120573 (1198961+ 1198962+ 1198963)

(120583 + 120588 + 120601 + V1205870) ((120583 + V) (120583 + 120588) + 120583120601)

(15)

where1198961= V120572 ((1 minus 120578) (2120583120587

0+ 120601) + 120587

0(120588 + (1 minus 120578) 120601))

1198962= (120583120587

1+ 120588) (120583 + 120588 + (1 minus 120578) 120601) + V2120572 (1 minus 120578) 120587

0

1198963= 120572 (120583120587

0+ 120601) (120588 + (1 minus 120588) (120583 + 120601) + 120588)

(16)




119877119899119886=

120573 (1 minus 120579) (1198961+ 1198962+ 1198963)

(120583 + 120588 + 120601 + V1205870) ((120583 + V) (120583 + 120588) + 120583120601)

(17)

where

1198961= V120572 ((1 minus 120578) (2120583120587

0+ 120601) + 120587

0(120588 + (1 minus 120578) 120601))

1198962= (120583120587

1+ 120588) (120583 + 120588 + (1 minus 120578) 120601) + V2120572 (1 minus 120578) 120587

0

1198963= 120572 (120583120587

0+ 120601) (120588 + (1 minus 120588) (120583 + 120601) + 120588)

(18)

biologically 119877119899119886








119877119899119886gt 1


[[[

[

119889119868119899

119889119905

119889119868119886

119889119905

]]]

]

= [119865 minus 119881] [119868119899

119868119886

] minus 119867[1 minus 120578 1

0 0] [119868119899

119868119886

] (19)


119867 = [120572120573 (1 minus 120579) (1205870(120583 + V) + 120601

120583 + 120601 + 120588 + V1205870

minus119878119899

119873)

+120573 (1 minus 120579) (1205831205871+ 120588

120583 + 120601 + 120588 + V1205870

minus119878119886

119873)]

(20)



[[[

[

119889119868119899

119889119905

119889119868119886

119889119905

]]]

]

le [119865 minus 119881] [119868119899

119868119886

] (21)



119899119886le 1




119899rarr 1198780

119899and

119878119886rarr 1198780







1198781015840

119899= Λ minus


119899119868lowast

119899

119873lowast

119899


119899

1198681015840

119899=120572 (1 minus 120579) (1 minus 120578) 120573119878

lowast

119899119868lowast

119899

119873lowast

119899


119899

(22)


G119899= (119878

119899 119868119899) isin R2

+ 119873119899leΛ

120583 (23)


119899


(22) 119868119899= 0 or

0 =120572 (1 minus 120579) (1 minus 120578) 120573119878

lowast

119899119868lowast

119899

119873lowast

119899


119899


119899= 120583119873

lowast

119899


119899=

Λ

120583119877119899

(24)



119899+ 119868lowast


119899= 119873lowast


119899 and thus

119868lowast

119899=Λ

120583minusΛ

120583119877119899

= (119877119899minus 1) 119878

lowast

119899

(25)


119899 119868lowast

119899)


Theorem 4 below





=

[[[

[

minus(


119899

119873lowast

119899

+ 120583)


119899

119873lowast

119899


119899

119873lowast

119899


119899

119873lowast

119899

minus 120583

]]]

]


119899119873lowast

119899= 120583 which



as

119869 (Elowast

119899) =

[[[[

[


lowast

119899

119873lowast

119899

+ 120583) minus120583


119899

119873lowast

119899

0

]]]]

]

(27)


Υ2

+ (120572 (1 minus 120579) (1 minus 120578) 120573119868

lowast

119899

119873lowast

119899

+ 120583)Υ

+120572 (1 minus 120579) (1 minus 120578) 120583120573119868

lowast

119899

119873lowast

119899

= 0

997904rArr Υ1= minus120583


119873minus 1)

120573119868lowast

119899

119873lowast

119899

(28)


119899gt 1 We






119899whenever

119877119899gt 1



119899= (Λ120583)minus119868

119899


1198681015840

119899=120572 (1 minus 120579) (1 minus 120578) 120573119868

119899

Λ120583(Λ

120583minus 119868119899) minus 120583119868

119899 (29)



120597

120597119868119899

[120572 (1 minus 120579) (1 minus 120578) 120573

Λ120583(Λ

120583minus 119868119899) minus 120583]


Λlt 0


119873lt 0

(30)





119899 for all 119905 ge 0




Elowast

119886= 119878lowast

119886=

Λ

120583119877119886

119868lowast

119886= (119877119886minus 1) 119878

lowast

119886 (31)



119886gt 1





119899119886

with

Elowast

119899119886=

119878lowast

119899=

1205870Λ + 120601119878

lowast

119886

120572 (1 minus 120579) 120582lowast

119899119886+ 120583 + 120588

119878lowast

119886=

1205871Λ + 120588119878

lowast

119899

(1 minus 120579) 120582lowast

119899119886+ 120583 + V + 120601

119868lowast

119899=120572 (1 minus 120579) 120582

lowast

119899119886119878lowast

119899+ 120601119868lowast

119886

120583 + 120588

119868lowast

119886=(1 minus 120579) 120582

lowast

119899119886119878lowast

119886+ 120588119868lowast

119899

120583 + V + 120601


119886+ (1 minus 120578) 119868

lowast

119899)

119873lowast

119899119886

(32)






∘



119878119899 120577 le lim inf

119905rarrinfin

119878119886


119868119899 120577 le lim inf

119905rarrinfin

119868119886

(33)


119881 (119878119899 119878119886 119868119899 119868119886) = (119878


119899ln 119878119899) + (119878


119886ln 119878119886)

+ (119868119899minus 119868lowast

119899ln 119868119899) + (119868119886minus 119868lowast

119886ln 119868119886)

(34)


120597119881

120597119878119899


119899

119878119899

) 120597119881

120597119878119886


119886

119878119886

)

120597119881

120597119868119899


119899

119868119899

) 120597119881

120597119868119886


119886

119868119886

)

(35)




119881(119878119899 119878119886 119868119899 119868119886) gt 0 and 119881

1015840

(119878119899 119878119886 119868119899 119868119886) = 0 only at



1198811015840

(119878119899 119878119899 119868119886 119868119886) = (119878


119899)1198781015840

119899

119878119899

+ (119878119886minus 119878lowast

119886)1198781015840

119886

119878119886

+ (119868119899minus 119868lowast

119899)1198681015840

119899

119868119899

+ (119868119886minus 119868lowast

119886)1198681015840

119886

119868119886


119899)2

119878119899

+ 119892 (119878119899 119878119886 119868119899 119868119886)

(36)



119899119886 The only

invariant set in∘








119899119886is





0 1205871




as119863(119868119899 119868119886) = 1119868

119899119868119886 Then we have

120597 (119863Φ)

120597119868119899

+120597 (119863Ψ)

120597119868119886

= minus120601

1198682

119899

minus120588

1198682

119886

minus (1 minus 120579) 120573(120572119878119899

1198682

119899

+(1 minus 120578) 119878

119886

1198682

119886

) lt 0

(37)



119899119886gt 1

4 Numerical Results



119899119886


02 04 06 08 10

10

20

30

40 Ra

R0

Rc

Rna

120573

R








00

05

10

00

05

10

10

15

120601120588

Rna

(a)

02 04 06 08 10

10

12

14

16

18

20

22

(120601 120588)

Rna

Ω

120595

Σ

(b)










119899119886



on 119877119899119886








0 02 04 06 08





Natural death rate



Condom use efficacy












0 01 02 03 04 05 06 07 08 09 1

0

2


minus10

minus8

minus6

minus4

minus2

log(Rna)

(a)


0

2

minus10

minus8

minus6

minus4

minus2

log(Rna)

(b)


0

2

minus10

minus8

minus6

minus4

minus2

log(Rna)

(c)





119899

119878119886 119868119899 and 119868



119869 (1199061 1199062)

= int

119905119891

0


1+ 11986021199062

2)] 119889119905

(38)


120579 = 02

120579 = 05

120579 = 08

0 10 20 30 40 50 600

500

1000

1500

2000

2500

3000

3500

Time (years)

Cum

ulat

ive H

SV-2

case

s

(a)

120579 = 02

120579 = 05

120579 = 08

0 10 20 30 40 50 60100

200

300

400

500

600

Time (years)

Cum

ulat

ive H

SV-2

case

s

(b)



0 10 20 30 40 50 600

500

1000

1500

2000

2500

3000

3500

Time (years)

Cum

ulat

ive H

SV-2

case

s

(a)

0 10 20 30 40 50 60100

200

300

400

500

600

700

800

900

Time (years)

Cum

ulat

ive H

SV-2

case

s

(b)



subject to

1198781015840

119899= 1205870Λ minus 120572 (1 minus 119906

1) 120582119878119899+ 120601119878119886minus (1 minus 119906

2) 120588119878119899minus 120583119878119899

1198781015840

119886= (1 minus 119906

2) 1205871Λ minus (1 minus 119906

1) 120582119878119886

+ (1 minus 1199062) 120588119878119899minus (120583 + V + 120601) 119878

119886

1198681015840

119899= 120572 (1 minus 119906

1) 120582119878119899+ 120601119868119886minus (1 minus 119906

2) 120588119868119899minus 120583119868119899

1198681015840

119886= (1 minus 119906

1) 120582119878119886+ (1 minus 119906

2) 120588119868119899minus (120583 + V + 120601) 119868

119886

(39)








2






1and 119906lowast

2such that

119869 (119906lowast

1 119906lowast

2) = max 119869 (119906

1 1199062) | 1199061 1199062isin U (40)

whereU = 1199061(119905) 1199062(119905) | 119906

1(119905) 1199062(119905) is measurable 0 le 119886

11le

1199061(119905) le 119887

11le 1 0 le 119886

22le 1199062(119905) le 119887

22le 1 119905 isin [0 119905

119891]





1and 119906





1and 119906

2with



+ |1199062|2

) where 1198881 1198882gt 0



1198613119868119886minus (11986011199062

1+11986021199062



1 1198612 1198613 1198601 1198602gt 1 hence

satisfying


1+ 11986021199062

2)

le 1198882minus 1198881(10038161003816100381610038161199061

1003816100381610038161003816

2

+10038161003816100381610038161199062

1003816100381610038161003816

2

)

(41)




1+ 11986021199062

2)

+ 1205821[1205870Λ minus

120572 (1 minus 1199061) 120573119868119886119878119899

119873

minus120572 (1 minus 119906

1) (1 minus 120578) 120573119868

119899119878119899

119873

+120601119878119886minus (1 minus 119906

2) 120588119878119899minus 120583119878119899]

+ 1205822[(1 minus 119906

2) 1205871Λ minus

(1 minus 1199061) 120573119868119886119878119886

119873


1) (1 minus 120578) 120573119868

119899119878119886

119873

+ (1 minus 1199062) 120588119878119899minus (120583 + V + 120601) 119878

119886]

+ 1205823[120572 (1 minus 119906

1) 120573119868119886119878119899

119873

+120572 (1 minus 119906

1) (1 minus 120578) 120573119868

119899119878119899

119873

+120601119868119886minus (1 minus 119906

2) 120588119868119899minus 120583119868119899]

+ 1205824[(1 minus 119906

1) 120573119868119886119878119886

119873+(1 minus 119906

1) (1 minus 120578) 120573119868

119899119878119886

119873

+ (1 minus 1199062) 120588119868119899minus (120583 + V + 120601) 119868

119886]

+ 12059611(119905) (1198871minus 1199061(119905)) + 120596

12(119905) (1199061(119905) minus 119886

1)

+ 12059621(119905) (1198872minus 1199062(119905)) + 120596

22(119905) (1199062(119905) minus 119886

2)

(42)

where 12059611(119905) 12059612(119905) ge 0 120596

21(119905) and 120596



12(119905)(1199061(119905) minus

1198861) = 0 120596

21(119905)(1198872minus 1199062(119905)) = 0 and 120596

22(119905)(1199062(119905) minus 119886

2) = 0 at


2



119889119905= minus

120597119871

120597119878119899

= minus1 +120572 (1 minus 119906

1) 120573119868119886[1205821minus 1205823]

119873

+120572 (1 minus 119906

1) (1 minus 120578) 120573119868

119899[1205821minus 1205823]

119873

+ (1 minus 1199062) 120588 [1205821minus 1205822] + 120583120582

1


1198891205822

119889119905= minus

120597119871

120597119878119886

= 1198611+ 120601 [120582

2minus 1205821] +

(1 minus 1199061) 120573119868119886[1205822minus 1205824]

119873

+(1 minus 119906

1) (1 minus 120578) 120573119868

119899[1205822minus 1205824]

119873+ (120583 + V) 120582

2

1198891205822

119889119905= minus

120597119871

120597119868119899

= 1198612+120572 (1 minus 119906

1) (1 minus 120578) 120573119878

119899[1205821minus 1205823]

119873

+(1 minus 119906

1) (1 minus 120578) 120573119878

119886[1205822minus 1205824]

119873

+ (1 minus 1199062) 120588 [1205823minus 1205824] + 120583120582

3

1198891205822

119889119905= minus

120597119871

120597119868119899

= 1198613+120572 (1 minus 119906

1) 120573119878119899[1205821minus 1205823]

119873

+(1 minus 119906

1) 120573119878119886[1205822minus 1205824]

119873+ 120601 [120582

4minus 1205823] + (120583 + V) 120582

4

(43)




1199061(119905)lowast

=120573 (119868119886+ (1 minus 120578) 119868

119899)

21198601119873

[120572119878119899(1205821minus 1205823) + 119878119886(1205822minus 1205824)]

+12059612(119905) minus 120596

11(119905)

21198601

(44)


1199062(119905)lowast

=120588

21198602


1205871Λ1205822

21198602

+12059622minus 12059621

21198602

(45)


1(119905) and 120596




1199061(119905)lowast

= minmax 1198861120573 (119868119886+ (1 minus 120578) 119868

119899)

21198601119873


1

1199062(119905)lowast

= minmax1198862120588

21198602

[119878119899(1205821minus 1205822) + 119868119899(1205823minus 1205824)]

minus1205871Λ1205822

21198602

1198872

(46)


1 and 119906lowast

2for 1199061(119905) and 119906

2(119905)


119891[60]



1= 0005 119860

2=

0005 1198611= 006 119861

2= 005 and 119861

3= 005 The assumed


= 041198781198860

= 03 1198681198990

= 02 and 1198681198860





036

038

04

042

044

046

048

0 2 4 6 8 10 12 14 16 18 20Time (years)

S n(t)

(a)

024

026

028

03

032

034

036

0 2 4 6 8 10 12 14 16 18 20Time (years)

S a(t)

(b)

005

01

015

02

025

03

035


00 2 4 6 8 10 12 14 16 18 20

Time (years)

I n(t)

(c)


0

005

01

015

02

025

03

035

0 2 4 6 8 10 12 14 16 18 20Time (years)

I a(t)


119899 susceptible




of 20 years






0 2 4 6 8 10 12 14 16 18 200

02

04

06

08

1

Time (years)

u1

(a)

0 2 4 6 8 10 12 14 16 18 200

02

04

06

08

1

Time (years)

u2

(b)


2)



2 The condom






2= 0



(1199062)

6 Discussion




Appendices



1205870(120583 + V) + 120601

120583 + 120601 + 120588 + V1205870

ge119878119899

119873

1205831205871+ 120588

120583 + 120601 + 120588 + V1205870

ge119878119886

119873

(A1)



(i)1205870(120583 + V) + 120601

120583 + 120601 + 120588 + V1205870

lt119878119899

119873

(ii)1205831205871+ 120588

120583 + 120601 + 120588 + V1205870

lt119878119886

119873

(A2)


120583 (1205870+ 1205871) + 120601 + 120588 + 120587

0

120583 + 120601 + 120588 + V1205870

lt119878119886+ 119878119899

119873997904rArr 1 lt 1 (A3)



(i) 1198671ge 2

(ii) 1198672ge 1

(A4)



(i)1205870(120583 + V) + 120601

120583 + 120601 + 120588 + V1205870

ge119878119899

119873

(ii)1205831205871+ 120588

120583 + 120601 + 120588 + V1205870

ge119878119886

119873

(A5)


120583 (1205870+ 1205871) + 120601 + 120588 + V120587

0

120583 + 120601 + 120588 + V1205870

ge119878119886+ 119878119899

119873

997904rArr120583 + 120601 + 120588 + V120587

0

120583 + 120601 + 120588 + V1205870

ge119878119886+ 119878119899

119873997904rArr 1 ge 1

(A6)




1199091015840

= 119891 (119909) (B1)

where 119891 isin 1198621


0) be




120597119909) lt 0 120583(minus120597119891[2]





Acknowledgments


References







































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













=[[

[

120572120573 (1 minus 120579) (1 minus 120578) (1205870(120583 + V) + 120601)

120583 + 120588 + 120601 + V1205870

120572120573 (1 minus 120579) (1205870(120583 + V) + 120601)

120583 + 120588 + 120601 + V1205870

120573 (1 minus 120579) (1 minus 120578) (1205831205871+ 120588)

120583 + 120588 + 120601 + V1205870

120573 (1 minus 120579) (1205831205871+ 120588)

120583 + 120588 + 120601 + V1205870

]]

]

(11)

119881 = [120583 + 120588 minus120601

minus120588 120583 + 120601 + V] (12)



1= 0 and 120587


radius is given by

1198770= 119877119899=120572120573 (1 minus 120579) (1 minus 120578)

120583 (13)




0= 0 and 120587

1= 1 Then the


119877119886=120573 (1 minus 120579)

120583 + V (14)




119877119888=

120573 (1198961+ 1198962+ 1198963)

(120583 + 120588 + 120601 + V1205870) ((120583 + V) (120583 + 120588) + 120583120601)

(15)

where1198961= V120572 ((1 minus 120578) (2120583120587

0+ 120601) + 120587

0(120588 + (1 minus 120578) 120601))

1198962= (120583120587

1+ 120588) (120583 + 120588 + (1 minus 120578) 120601) + V2120572 (1 minus 120578) 120587

0

1198963= 120572 (120583120587

0+ 120601) (120588 + (1 minus 120588) (120583 + 120601) + 120588)

(16)




119877119899119886=

120573 (1 minus 120579) (1198961+ 1198962+ 1198963)

(120583 + 120588 + 120601 + V1205870) ((120583 + V) (120583 + 120588) + 120583120601)

(17)

where

1198961= V120572 ((1 minus 120578) (2120583120587

0+ 120601) + 120587

0(120588 + (1 minus 120578) 120601))

1198962= (120583120587

1+ 120588) (120583 + 120588 + (1 minus 120578) 120601) + V2120572 (1 minus 120578) 120587

0

1198963= 120572 (120583120587

0+ 120601) (120588 + (1 minus 120588) (120583 + 120601) + 120588)

(18)

biologically 119877119899119886








119877119899119886gt 1


[[[

[

119889119868119899

119889119905

119889119868119886

119889119905

]]]

]

= [119865 minus 119881] [119868119899

119868119886

] minus 119867[1 minus 120578 1

0 0] [119868119899

119868119886

] (19)


119867 = [120572120573 (1 minus 120579) (1205870(120583 + V) + 120601

120583 + 120601 + 120588 + V1205870

minus119878119899

119873)

+120573 (1 minus 120579) (1205831205871+ 120588

120583 + 120601 + 120588 + V1205870

minus119878119886

119873)]

(20)



[[[

[

119889119868119899

119889119905

119889119868119886

119889119905

]]]

]

le [119865 minus 119881] [119868119899

119868119886

] (21)



119899119886le 1




119899rarr 1198780

119899and

119878119886rarr 1198780







1198781015840

119899= Λ minus


119899119868lowast

119899

119873lowast

119899


119899

1198681015840

119899=120572 (1 minus 120579) (1 minus 120578) 120573119878

lowast

119899119868lowast

119899

119873lowast

119899


119899

(22)


G119899= (119878

119899 119868119899) isin R2

+ 119873119899leΛ

120583 (23)


119899


(22) 119868119899= 0 or

0 =120572 (1 minus 120579) (1 minus 120578) 120573119878

lowast

119899119868lowast

119899

119873lowast

119899


119899


119899= 120583119873

lowast

119899


119899=

Λ

120583119877119899

(24)



119899+ 119868lowast


119899= 119873lowast


119899 and thus

119868lowast

119899=Λ

120583minusΛ

120583119877119899

= (119877119899minus 1) 119878

lowast

119899

(25)


119899 119868lowast

119899)


Theorem 4 below





=

[[[

[

minus(


119899

119873lowast

119899

+ 120583)


119899

119873lowast

119899


119899

119873lowast

119899


119899

119873lowast

119899

minus 120583

]]]

]


119899119873lowast

119899= 120583 which



as

119869 (Elowast

119899) =

[[[[

[


lowast

119899

119873lowast

119899

+ 120583) minus120583


119899

119873lowast

119899

0

]]]]

]

(27)


Υ2

+ (120572 (1 minus 120579) (1 minus 120578) 120573119868

lowast

119899

119873lowast

119899

+ 120583)Υ

+120572 (1 minus 120579) (1 minus 120578) 120583120573119868

lowast

119899

119873lowast

119899

= 0

997904rArr Υ1= minus120583


119873minus 1)

120573119868lowast

119899

119873lowast

119899

(28)


119899gt 1 We






119899whenever

119877119899gt 1



119899= (Λ120583)minus119868

119899


1198681015840

119899=120572 (1 minus 120579) (1 minus 120578) 120573119868

119899

Λ120583(Λ

120583minus 119868119899) minus 120583119868

119899 (29)



120597

120597119868119899

[120572 (1 minus 120579) (1 minus 120578) 120573

Λ120583(Λ

120583minus 119868119899) minus 120583]


Λlt 0


119873lt 0

(30)





119899 for all 119905 ge 0




Elowast

119886= 119878lowast

119886=

Λ

120583119877119886

119868lowast

119886= (119877119886minus 1) 119878

lowast

119886 (31)



119886gt 1





119899119886

with

Elowast

119899119886=

119878lowast

119899=

1205870Λ + 120601119878

lowast

119886

120572 (1 minus 120579) 120582lowast

119899119886+ 120583 + 120588

119878lowast

119886=

1205871Λ + 120588119878

lowast

119899

(1 minus 120579) 120582lowast

119899119886+ 120583 + V + 120601

119868lowast

119899=120572 (1 minus 120579) 120582

lowast

119899119886119878lowast

119899+ 120601119868lowast

119886

120583 + 120588

119868lowast

119886=(1 minus 120579) 120582

lowast

119899119886119878lowast

119886+ 120588119868lowast

119899

120583 + V + 120601


119886+ (1 minus 120578) 119868

lowast

119899)

119873lowast

119899119886

(32)






∘



119878119899 120577 le lim inf

119905rarrinfin

119878119886


119868119899 120577 le lim inf

119905rarrinfin

119868119886

(33)


119881 (119878119899 119878119886 119868119899 119868119886) = (119878


119899ln 119878119899) + (119878


119886ln 119878119886)

+ (119868119899minus 119868lowast

119899ln 119868119899) + (119868119886minus 119868lowast

119886ln 119868119886)

(34)


120597119881

120597119878119899


119899

119878119899

) 120597119881

120597119878119886


119886

119878119886

)

120597119881

120597119868119899


119899

119868119899

) 120597119881

120597119868119886


119886

119868119886

)

(35)




119881(119878119899 119878119886 119868119899 119868119886) gt 0 and 119881

1015840

(119878119899 119878119886 119868119899 119868119886) = 0 only at



1198811015840

(119878119899 119878119899 119868119886 119868119886) = (119878


119899)1198781015840

119899

119878119899

+ (119878119886minus 119878lowast

119886)1198781015840

119886

119878119886

+ (119868119899minus 119868lowast

119899)1198681015840

119899

119868119899

+ (119868119886minus 119868lowast

119886)1198681015840

119886

119868119886


119899)2

119878119899

+ 119892 (119878119899 119878119886 119868119899 119868119886)

(36)



119899119886 The only

invariant set in∘








119899119886is





0 1205871




as119863(119868119899 119868119886) = 1119868

119899119868119886 Then we have

120597 (119863Φ)

120597119868119899

+120597 (119863Ψ)

120597119868119886

= minus120601

1198682

119899

minus120588

1198682

119886

minus (1 minus 120579) 120573(120572119878119899

1198682

119899

+(1 minus 120578) 119878

119886

1198682

119886

) lt 0

(37)



119899119886gt 1

4 Numerical Results



119899119886


02 04 06 08 10

10

20

30

40 Ra

R0

Rc

Rna

120573

R








00

05

10

00

05

10

10

15

120601120588

Rna

(a)

02 04 06 08 10

10

12

14

16

18

20

22

(120601 120588)

Rna

Ω

120595

Σ

(b)










119899119886



on 119877119899119886








0 02 04 06 08





Natural death rate



Condom use efficacy












0 01 02 03 04 05 06 07 08 09 1

0

2


minus10

minus8

minus6

minus4

minus2

log(Rna)

(a)


0

2

minus10

minus8

minus6

minus4

minus2

log(Rna)

(b)


0

2

minus10

minus8

minus6

minus4

minus2

log(Rna)

(c)





119899

119878119886 119868119899 and 119868



119869 (1199061 1199062)

= int

119905119891

0


1+ 11986021199062

2)] 119889119905

(38)


120579 = 02

120579 = 05

120579 = 08

0 10 20 30 40 50 600

500

1000

1500

2000

2500

3000

3500

Time (years)

Cum

ulat

ive H

SV-2

case

s

(a)

120579 = 02

120579 = 05

120579 = 08

0 10 20 30 40 50 60100

200

300

400

500

600

Time (years)

Cum

ulat

ive H

SV-2

case

s

(b)



0 10 20 30 40 50 600

500

1000

1500

2000

2500

3000

3500

Time (years)

Cum

ulat

ive H

SV-2

case

s

(a)

0 10 20 30 40 50 60100

200

300

400

500

600

700

800

900

Time (years)

Cum

ulat

ive H

SV-2

case

s

(b)



subject to

1198781015840

119899= 1205870Λ minus 120572 (1 minus 119906

1) 120582119878119899+ 120601119878119886minus (1 minus 119906

2) 120588119878119899minus 120583119878119899

1198781015840

119886= (1 minus 119906

2) 1205871Λ minus (1 minus 119906

1) 120582119878119886

+ (1 minus 1199062) 120588119878119899minus (120583 + V + 120601) 119878

119886

1198681015840

119899= 120572 (1 minus 119906

1) 120582119878119899+ 120601119868119886minus (1 minus 119906

2) 120588119868119899minus 120583119868119899

1198681015840

119886= (1 minus 119906

1) 120582119878119886+ (1 minus 119906

2) 120588119868119899minus (120583 + V + 120601) 119868

119886

(39)








2






1and 119906lowast

2such that

119869 (119906lowast

1 119906lowast

2) = max 119869 (119906

1 1199062) | 1199061 1199062isin U (40)

whereU = 1199061(119905) 1199062(119905) | 119906

1(119905) 1199062(119905) is measurable 0 le 119886

11le

1199061(119905) le 119887

11le 1 0 le 119886

22le 1199062(119905) le 119887

22le 1 119905 isin [0 119905

119891]





1and 119906





1and 119906

2with



+ |1199062|2

) where 1198881 1198882gt 0



1198613119868119886minus (11986011199062

1+11986021199062



1 1198612 1198613 1198601 1198602gt 1 hence

satisfying


1+ 11986021199062

2)

le 1198882minus 1198881(10038161003816100381610038161199061

1003816100381610038161003816

2

+10038161003816100381610038161199062

1003816100381610038161003816

2

)

(41)




1+ 11986021199062

2)

+ 1205821[1205870Λ minus

120572 (1 minus 1199061) 120573119868119886119878119899

119873

minus120572 (1 minus 119906

1) (1 minus 120578) 120573119868

119899119878119899

119873

+120601119878119886minus (1 minus 119906

2) 120588119878119899minus 120583119878119899]

+ 1205822[(1 minus 119906

2) 1205871Λ minus

(1 minus 1199061) 120573119868119886119878119886

119873


1) (1 minus 120578) 120573119868

119899119878119886

119873

+ (1 minus 1199062) 120588119878119899minus (120583 + V + 120601) 119878

119886]

+ 1205823[120572 (1 minus 119906

1) 120573119868119886119878119899

119873

+120572 (1 minus 119906

1) (1 minus 120578) 120573119868

119899119878119899

119873

+120601119868119886minus (1 minus 119906

2) 120588119868119899minus 120583119868119899]

+ 1205824[(1 minus 119906

1) 120573119868119886119878119886

119873+(1 minus 119906

1) (1 minus 120578) 120573119868

119899119878119886

119873

+ (1 minus 1199062) 120588119868119899minus (120583 + V + 120601) 119868

119886]

+ 12059611(119905) (1198871minus 1199061(119905)) + 120596

12(119905) (1199061(119905) minus 119886

1)

+ 12059621(119905) (1198872minus 1199062(119905)) + 120596

22(119905) (1199062(119905) minus 119886

2)

(42)

where 12059611(119905) 12059612(119905) ge 0 120596

21(119905) and 120596



12(119905)(1199061(119905) minus

1198861) = 0 120596

21(119905)(1198872minus 1199062(119905)) = 0 and 120596

22(119905)(1199062(119905) minus 119886

2) = 0 at


2



119889119905= minus

120597119871

120597119878119899

= minus1 +120572 (1 minus 119906

1) 120573119868119886[1205821minus 1205823]

119873

+120572 (1 minus 119906

1) (1 minus 120578) 120573119868

119899[1205821minus 1205823]

119873

+ (1 minus 1199062) 120588 [1205821minus 1205822] + 120583120582

1


1198891205822

119889119905= minus

120597119871

120597119878119886

= 1198611+ 120601 [120582

2minus 1205821] +

(1 minus 1199061) 120573119868119886[1205822minus 1205824]

119873

+(1 minus 119906

1) (1 minus 120578) 120573119868

119899[1205822minus 1205824]

119873+ (120583 + V) 120582

2

1198891205822

119889119905= minus

120597119871

120597119868119899

= 1198612+120572 (1 minus 119906

1) (1 minus 120578) 120573119878

119899[1205821minus 1205823]

119873

+(1 minus 119906

1) (1 minus 120578) 120573119878

119886[1205822minus 1205824]

119873

+ (1 minus 1199062) 120588 [1205823minus 1205824] + 120583120582

3

1198891205822

119889119905= minus

120597119871

120597119868119899

= 1198613+120572 (1 minus 119906

1) 120573119878119899[1205821minus 1205823]

119873

+(1 minus 119906

1) 120573119878119886[1205822minus 1205824]

119873+ 120601 [120582

4minus 1205823] + (120583 + V) 120582

4

(43)




1199061(119905)lowast

=120573 (119868119886+ (1 minus 120578) 119868

119899)

21198601119873

[120572119878119899(1205821minus 1205823) + 119878119886(1205822minus 1205824)]

+12059612(119905) minus 120596

11(119905)

21198601

(44)


1199062(119905)lowast

=120588

21198602


1205871Λ1205822

21198602

+12059622minus 12059621

21198602

(45)


1(119905) and 120596




1199061(119905)lowast

= minmax 1198861120573 (119868119886+ (1 minus 120578) 119868

119899)

21198601119873


1

1199062(119905)lowast

= minmax1198862120588

21198602

[119878119899(1205821minus 1205822) + 119868119899(1205823minus 1205824)]

minus1205871Λ1205822

21198602

1198872

(46)


1 and 119906lowast

2for 1199061(119905) and 119906

2(119905)


119891[60]



1= 0005 119860

2=

0005 1198611= 006 119861

2= 005 and 119861

3= 005 The assumed


= 041198781198860

= 03 1198681198990

= 02 and 1198681198860





036

038

04

042

044

046

048

0 2 4 6 8 10 12 14 16 18 20Time (years)

S n(t)

(a)

024

026

028

03

032

034

036

0 2 4 6 8 10 12 14 16 18 20Time (years)

S a(t)

(b)

005

01

015

02

025

03

035


00 2 4 6 8 10 12 14 16 18 20

Time (years)

I n(t)

(c)


0

005

01

015

02

025

03

035

0 2 4 6 8 10 12 14 16 18 20Time (years)

I a(t)


119899 susceptible




of 20 years






0 2 4 6 8 10 12 14 16 18 200

02

04

06

08

1

Time (years)

u1

(a)

0 2 4 6 8 10 12 14 16 18 200

02

04

06

08

1

Time (years)

u2

(b)


2)



2 The condom






2= 0



(1199062)

6 Discussion




Appendices



1205870(120583 + V) + 120601

120583 + 120601 + 120588 + V1205870

ge119878119899

119873

1205831205871+ 120588

120583 + 120601 + 120588 + V1205870

ge119878119886

119873

(A1)



(i)1205870(120583 + V) + 120601

120583 + 120601 + 120588 + V1205870

lt119878119899

119873

(ii)1205831205871+ 120588

120583 + 120601 + 120588 + V1205870

lt119878119886

119873

(A2)


120583 (1205870+ 1205871) + 120601 + 120588 + 120587

0

120583 + 120601 + 120588 + V1205870

lt119878119886+ 119878119899

119873997904rArr 1 lt 1 (A3)



(i) 1198671ge 2

(ii) 1198672ge 1

(A4)



(i)1205870(120583 + V) + 120601

120583 + 120601 + 120588 + V1205870

ge119878119899

119873

(ii)1205831205871+ 120588

120583 + 120601 + 120588 + V1205870

ge119878119886

119873

(A5)


120583 (1205870+ 1205871) + 120601 + 120588 + V120587

0

120583 + 120601 + 120588 + V1205870

ge119878119886+ 119878119899

119873

997904rArr120583 + 120601 + 120588 + V120587

0

120583 + 120601 + 120588 + V1205870

ge119878119886+ 119878119899

119873997904rArr 1 ge 1

(A6)




1199091015840

= 119891 (119909) (B1)

where 119891 isin 1198621


0) be




120597119909) lt 0 120583(minus120597119891[2]





Acknowledgments


References







































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











[[[

[

119889119868119899

119889119905

119889119868119886

119889119905

]]]

]

le [119865 minus 119881] [119868119899

119868119886

] (21)



119899119886le 1




119899rarr 1198780

119899and

119878119886rarr 1198780







1198781015840

119899= Λ minus


119899119868lowast

119899

119873lowast

119899


119899

1198681015840

119899=120572 (1 minus 120579) (1 minus 120578) 120573119878

lowast

119899119868lowast

119899

119873lowast

119899


119899

(22)


G119899= (119878

119899 119868119899) isin R2

+ 119873119899leΛ

120583 (23)


119899


(22) 119868119899= 0 or

0 =120572 (1 minus 120579) (1 minus 120578) 120573119878

lowast

119899119868lowast

119899

119873lowast

119899


119899


119899= 120583119873

lowast

119899


119899=

Λ

120583119877119899

(24)



119899+ 119868lowast


119899= 119873lowast


119899 and thus

119868lowast

119899=Λ

120583minusΛ

120583119877119899

= (119877119899minus 1) 119878

lowast

119899

(25)


119899 119868lowast

119899)


Theorem 4 below





=

[[[

[

minus(


119899

119873lowast

119899

+ 120583)


119899

119873lowast

119899


119899

119873lowast

119899


119899

119873lowast

119899

minus 120583

]]]

]


119899119873lowast

119899= 120583 which



as

119869 (Elowast

119899) =

[[[[

[


lowast

119899

119873lowast

119899

+ 120583) minus120583


119899

119873lowast

119899

0

]]]]

]

(27)


Υ2

+ (120572 (1 minus 120579) (1 minus 120578) 120573119868

lowast

119899

119873lowast

119899

+ 120583)Υ

+120572 (1 minus 120579) (1 minus 120578) 120583120573119868

lowast

119899

119873lowast

119899

= 0

997904rArr Υ1= minus120583


119873minus 1)

120573119868lowast

119899

119873lowast

119899

(28)


119899gt 1 We






119899whenever

119877119899gt 1



119899= (Λ120583)minus119868

119899


1198681015840

119899=120572 (1 minus 120579) (1 minus 120578) 120573119868

119899

Λ120583(Λ

120583minus 119868119899) minus 120583119868

119899 (29)



120597

120597119868119899

[120572 (1 minus 120579) (1 minus 120578) 120573

Λ120583(Λ

120583minus 119868119899) minus 120583]


Λlt 0


119873lt 0

(30)





119899 for all 119905 ge 0




Elowast

119886= 119878lowast

119886=

Λ

120583119877119886

119868lowast

119886= (119877119886minus 1) 119878

lowast

119886 (31)



119886gt 1





119899119886

with

Elowast

119899119886=

119878lowast

119899=

1205870Λ + 120601119878

lowast

119886

120572 (1 minus 120579) 120582lowast

119899119886+ 120583 + 120588

119878lowast

119886=

1205871Λ + 120588119878

lowast

119899

(1 minus 120579) 120582lowast

119899119886+ 120583 + V + 120601

119868lowast

119899=120572 (1 minus 120579) 120582

lowast

119899119886119878lowast

119899+ 120601119868lowast

119886

120583 + 120588

119868lowast

119886=(1 minus 120579) 120582

lowast

119899119886119878lowast

119886+ 120588119868lowast

119899

120583 + V + 120601


119886+ (1 minus 120578) 119868

lowast

119899)

119873lowast

119899119886

(32)






∘



119878119899 120577 le lim inf

119905rarrinfin

119878119886


119868119899 120577 le lim inf

119905rarrinfin

119868119886

(33)


119881 (119878119899 119878119886 119868119899 119868119886) = (119878


119899ln 119878119899) + (119878


119886ln 119878119886)

+ (119868119899minus 119868lowast

119899ln 119868119899) + (119868119886minus 119868lowast

119886ln 119868119886)

(34)


120597119881

120597119878119899


119899

119878119899

) 120597119881

120597119878119886


119886

119878119886

)

120597119881

120597119868119899


119899

119868119899

) 120597119881

120597119868119886


119886

119868119886

)

(35)




119881(119878119899 119878119886 119868119899 119868119886) gt 0 and 119881

1015840

(119878119899 119878119886 119868119899 119868119886) = 0 only at



1198811015840

(119878119899 119878119899 119868119886 119868119886) = (119878


119899)1198781015840

119899

119878119899

+ (119878119886minus 119878lowast

119886)1198781015840

119886

119878119886

+ (119868119899minus 119868lowast

119899)1198681015840

119899

119868119899

+ (119868119886minus 119868lowast

119886)1198681015840

119886

119868119886


119899)2

119878119899

+ 119892 (119878119899 119878119886 119868119899 119868119886)

(36)



119899119886 The only

invariant set in∘








119899119886is





0 1205871




as119863(119868119899 119868119886) = 1119868

119899119868119886 Then we have

120597 (119863Φ)

120597119868119899

+120597 (119863Ψ)

120597119868119886

= minus120601

1198682

119899

minus120588

1198682

119886

minus (1 minus 120579) 120573(120572119878119899

1198682

119899

+(1 minus 120578) 119878

119886

1198682

119886

) lt 0

(37)



119899119886gt 1

4 Numerical Results



119899119886


02 04 06 08 10

10

20

30

40 Ra

R0

Rc

Rna

120573

R








00

05

10

00

05

10

10

15

120601120588

Rna

(a)

02 04 06 08 10

10

12

14

16

18

20

22

(120601 120588)

Rna

Ω

120595

Σ

(b)










119899119886



on 119877119899119886








0 02 04 06 08





Natural death rate



Condom use efficacy












0 01 02 03 04 05 06 07 08 09 1

0

2


minus10

minus8

minus6

minus4

minus2

log(Rna)

(a)


0

2

minus10

minus8

minus6

minus4

minus2

log(Rna)

(b)


0

2

minus10

minus8

minus6

minus4

minus2

log(Rna)

(c)





119899

119878119886 119868119899 and 119868



119869 (1199061 1199062)

= int

119905119891

0


1+ 11986021199062

2)] 119889119905

(38)


120579 = 02

120579 = 05

120579 = 08

0 10 20 30 40 50 600

500

1000

1500

2000

2500

3000

3500

Time (years)

Cum

ulat

ive H

SV-2

case

s

(a)

120579 = 02

120579 = 05

120579 = 08

0 10 20 30 40 50 60100

200

300

400

500

600

Time (years)

Cum

ulat

ive H

SV-2

case

s

(b)



0 10 20 30 40 50 600

500

1000

1500

2000

2500

3000

3500

Time (years)

Cum

ulat

ive H

SV-2

case

s

(a)

0 10 20 30 40 50 60100

200

300

400

500

600

700

800

900

Time (years)

Cum

ulat

ive H

SV-2

case

s

(b)



subject to

1198781015840

119899= 1205870Λ minus 120572 (1 minus 119906

1) 120582119878119899+ 120601119878119886minus (1 minus 119906

2) 120588119878119899minus 120583119878119899

1198781015840

119886= (1 minus 119906

2) 1205871Λ minus (1 minus 119906

1) 120582119878119886

+ (1 minus 1199062) 120588119878119899minus (120583 + V + 120601) 119878

119886

1198681015840

119899= 120572 (1 minus 119906

1) 120582119878119899+ 120601119868119886minus (1 minus 119906

2) 120588119868119899minus 120583119868119899

1198681015840

119886= (1 minus 119906

1) 120582119878119886+ (1 minus 119906

2) 120588119868119899minus (120583 + V + 120601) 119868

119886

(39)








2






1and 119906lowast

2such that

119869 (119906lowast

1 119906lowast

2) = max 119869 (119906

1 1199062) | 1199061 1199062isin U (40)

whereU = 1199061(119905) 1199062(119905) | 119906

1(119905) 1199062(119905) is measurable 0 le 119886

11le

1199061(119905) le 119887

11le 1 0 le 119886

22le 1199062(119905) le 119887

22le 1 119905 isin [0 119905

119891]





1and 119906





1and 119906

2with



+ |1199062|2

) where 1198881 1198882gt 0



1198613119868119886minus (11986011199062

1+11986021199062



1 1198612 1198613 1198601 1198602gt 1 hence

satisfying


1+ 11986021199062

2)

le 1198882minus 1198881(10038161003816100381610038161199061

1003816100381610038161003816

2

+10038161003816100381610038161199062

1003816100381610038161003816

2

)

(41)




1+ 11986021199062

2)

+ 1205821[1205870Λ minus

120572 (1 minus 1199061) 120573119868119886119878119899

119873

minus120572 (1 minus 119906

1) (1 minus 120578) 120573119868

119899119878119899

119873

+120601119878119886minus (1 minus 119906

2) 120588119878119899minus 120583119878119899]

+ 1205822[(1 minus 119906

2) 1205871Λ minus

(1 minus 1199061) 120573119868119886119878119886

119873


1) (1 minus 120578) 120573119868

119899119878119886

119873

+ (1 minus 1199062) 120588119878119899minus (120583 + V + 120601) 119878

119886]

+ 1205823[120572 (1 minus 119906

1) 120573119868119886119878119899

119873

+120572 (1 minus 119906

1) (1 minus 120578) 120573119868

119899119878119899

119873

+120601119868119886minus (1 minus 119906

2) 120588119868119899minus 120583119868119899]

+ 1205824[(1 minus 119906

1) 120573119868119886119878119886

119873+(1 minus 119906

1) (1 minus 120578) 120573119868

119899119878119886

119873

+ (1 minus 1199062) 120588119868119899minus (120583 + V + 120601) 119868

119886]

+ 12059611(119905) (1198871minus 1199061(119905)) + 120596

12(119905) (1199061(119905) minus 119886

1)

+ 12059621(119905) (1198872minus 1199062(119905)) + 120596

22(119905) (1199062(119905) minus 119886

2)

(42)

where 12059611(119905) 12059612(119905) ge 0 120596

21(119905) and 120596



12(119905)(1199061(119905) minus

1198861) = 0 120596

21(119905)(1198872minus 1199062(119905)) = 0 and 120596

22(119905)(1199062(119905) minus 119886

2) = 0 at


2



119889119905= minus

120597119871

120597119878119899

= minus1 +120572 (1 minus 119906

1) 120573119868119886[1205821minus 1205823]

119873

+120572 (1 minus 119906

1) (1 minus 120578) 120573119868

119899[1205821minus 1205823]

119873

+ (1 minus 1199062) 120588 [1205821minus 1205822] + 120583120582

1


1198891205822

119889119905= minus

120597119871

120597119878119886

= 1198611+ 120601 [120582

2minus 1205821] +

(1 minus 1199061) 120573119868119886[1205822minus 1205824]

119873

+(1 minus 119906

1) (1 minus 120578) 120573119868

119899[1205822minus 1205824]

119873+ (120583 + V) 120582

2

1198891205822

119889119905= minus

120597119871

120597119868119899

= 1198612+120572 (1 minus 119906

1) (1 minus 120578) 120573119878

119899[1205821minus 1205823]

119873

+(1 minus 119906

1) (1 minus 120578) 120573119878

119886[1205822minus 1205824]

119873

+ (1 minus 1199062) 120588 [1205823minus 1205824] + 120583120582

3

1198891205822

119889119905= minus

120597119871

120597119868119899

= 1198613+120572 (1 minus 119906

1) 120573119878119899[1205821minus 1205823]

119873

+(1 minus 119906

1) 120573119878119886[1205822minus 1205824]

119873+ 120601 [120582

4minus 1205823] + (120583 + V) 120582

4

(43)




1199061(119905)lowast

=120573 (119868119886+ (1 minus 120578) 119868

119899)

21198601119873

[120572119878119899(1205821minus 1205823) + 119878119886(1205822minus 1205824)]

+12059612(119905) minus 120596

11(119905)

21198601

(44)


1199062(119905)lowast

=120588

21198602


1205871Λ1205822

21198602

+12059622minus 12059621

21198602

(45)


1(119905) and 120596




1199061(119905)lowast

= minmax 1198861120573 (119868119886+ (1 minus 120578) 119868

119899)

21198601119873


1

1199062(119905)lowast

= minmax1198862120588

21198602

[119878119899(1205821minus 1205822) + 119868119899(1205823minus 1205824)]

minus1205871Λ1205822

21198602

1198872

(46)


1 and 119906lowast

2for 1199061(119905) and 119906

2(119905)


119891[60]



1= 0005 119860

2=

0005 1198611= 006 119861

2= 005 and 119861

3= 005 The assumed


= 041198781198860

= 03 1198681198990

= 02 and 1198681198860





036

038

04

042

044

046

048

0 2 4 6 8 10 12 14 16 18 20Time (years)

S n(t)

(a)

024

026

028

03

032

034

036

0 2 4 6 8 10 12 14 16 18 20Time (years)

S a(t)

(b)

005

01

015

02

025

03

035


00 2 4 6 8 10 12 14 16 18 20

Time (years)

I n(t)

(c)


0

005

01

015

02

025

03

035

0 2 4 6 8 10 12 14 16 18 20Time (years)

I a(t)


119899 susceptible




of 20 years






0 2 4 6 8 10 12 14 16 18 200

02

04

06

08

1

Time (years)

u1

(a)

0 2 4 6 8 10 12 14 16 18 200

02

04

06

08

1

Time (years)

u2

(b)


2)



2 The condom






2= 0



(1199062)

6 Discussion




Appendices



1205870(120583 + V) + 120601

120583 + 120601 + 120588 + V1205870

ge119878119899

119873

1205831205871+ 120588

120583 + 120601 + 120588 + V1205870

ge119878119886

119873

(A1)



(i)1205870(120583 + V) + 120601

120583 + 120601 + 120588 + V1205870

lt119878119899

119873

(ii)1205831205871+ 120588

120583 + 120601 + 120588 + V1205870

lt119878119886

119873

(A2)


120583 (1205870+ 1205871) + 120601 + 120588 + 120587

0

120583 + 120601 + 120588 + V1205870

lt119878119886+ 119878119899

119873997904rArr 1 lt 1 (A3)



(i) 1198671ge 2

(ii) 1198672ge 1

(A4)



(i)1205870(120583 + V) + 120601

120583 + 120601 + 120588 + V1205870

ge119878119899

119873

(ii)1205831205871+ 120588

120583 + 120601 + 120588 + V1205870

ge119878119886

119873

(A5)


120583 (1205870+ 1205871) + 120601 + 120588 + V120587

0

120583 + 120601 + 120588 + V1205870

ge119878119886+ 119878119899

119873

997904rArr120583 + 120601 + 120588 + V120587

0

120583 + 120601 + 120588 + V1205870

ge119878119886+ 119878119899

119873997904rArr 1 ge 1

(A6)




1199091015840

= 119891 (119909) (B1)

where 119891 isin 1198621


0) be




120597119909) lt 0 120583(minus120597119891[2]





Acknowledgments


References







































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











120597

120597119868119899

[120572 (1 minus 120579) (1 minus 120578) 120573

Λ120583(Λ

120583minus 119868119899) minus 120583]


Λlt 0


119873lt 0

(30)





119899 for all 119905 ge 0




Elowast

119886= 119878lowast

119886=

Λ

120583119877119886

119868lowast

119886= (119877119886minus 1) 119878

lowast

119886 (31)



119886gt 1





119899119886

with

Elowast

119899119886=

119878lowast

119899=

1205870Λ + 120601119878

lowast

119886

120572 (1 minus 120579) 120582lowast

119899119886+ 120583 + 120588

119878lowast

119886=

1205871Λ + 120588119878

lowast

119899

(1 minus 120579) 120582lowast

119899119886+ 120583 + V + 120601

119868lowast

119899=120572 (1 minus 120579) 120582

lowast

119899119886119878lowast

119899+ 120601119868lowast

119886

120583 + 120588

119868lowast

119886=(1 minus 120579) 120582

lowast

119899119886119878lowast

119886+ 120588119868lowast

119899

120583 + V + 120601


119886+ (1 minus 120578) 119868

lowast

119899)

119873lowast

119899119886

(32)






∘



119878119899 120577 le lim inf

119905rarrinfin

119878119886


119868119899 120577 le lim inf

119905rarrinfin

119868119886

(33)


119881 (119878119899 119878119886 119868119899 119868119886) = (119878


119899ln 119878119899) + (119878


119886ln 119878119886)

+ (119868119899minus 119868lowast

119899ln 119868119899) + (119868119886minus 119868lowast

119886ln 119868119886)

(34)


120597119881

120597119878119899


119899

119878119899

) 120597119881

120597119878119886


119886

119878119886

)

120597119881

120597119868119899


119899

119868119899

) 120597119881

120597119868119886


119886

119868119886

)

(35)




119881(119878119899 119878119886 119868119899 119868119886) gt 0 and 119881

1015840

(119878119899 119878119886 119868119899 119868119886) = 0 only at



1198811015840

(119878119899 119878119899 119868119886 119868119886) = (119878


119899)1198781015840

119899

119878119899

+ (119878119886minus 119878lowast

119886)1198781015840

119886

119878119886

+ (119868119899minus 119868lowast

119899)1198681015840

119899

119868119899

+ (119868119886minus 119868lowast

119886)1198681015840

119886

119868119886


119899)2

119878119899

+ 119892 (119878119899 119878119886 119868119899 119868119886)

(36)



119899119886 The only

invariant set in∘








119899119886is





0 1205871




as119863(119868119899 119868119886) = 1119868

119899119868119886 Then we have

120597 (119863Φ)

120597119868119899

+120597 (119863Ψ)

120597119868119886

= minus120601

1198682

119899

minus120588

1198682

119886

minus (1 minus 120579) 120573(120572119878119899

1198682

119899

+(1 minus 120578) 119878

119886

1198682

119886

) lt 0

(37)



119899119886gt 1

4 Numerical Results



119899119886


02 04 06 08 10

10

20

30

40 Ra

R0

Rc

Rna

120573

R








00

05

10

00

05

10

10

15

120601120588

Rna

(a)

02 04 06 08 10

10

12

14

16

18

20

22

(120601 120588)

Rna

Ω

120595

Σ

(b)










119899119886



on 119877119899119886








0 02 04 06 08





Natural death rate



Condom use efficacy












0 01 02 03 04 05 06 07 08 09 1

0

2


minus10

minus8

minus6

minus4

minus2

log(Rna)

(a)


0

2

minus10

minus8

minus6

minus4

minus2

log(Rna)

(b)


0

2

minus10

minus8

minus6

minus4

minus2

log(Rna)

(c)





119899

119878119886 119868119899 and 119868



119869 (1199061 1199062)

= int

119905119891

0


1+ 11986021199062

2)] 119889119905

(38)


120579 = 02

120579 = 05

120579 = 08

0 10 20 30 40 50 600

500

1000

1500

2000

2500

3000

3500

Time (years)

Cum

ulat

ive H

SV-2

case

s

(a)

120579 = 02

120579 = 05

120579 = 08

0 10 20 30 40 50 60100

200

300

400

500

600

Time (years)

Cum

ulat

ive H

SV-2

case

s

(b)



0 10 20 30 40 50 600

500

1000

1500

2000

2500

3000

3500

Time (years)

Cum

ulat

ive H

SV-2

case

s

(a)

0 10 20 30 40 50 60100

200

300

400

500

600

700

800

900

Time (years)

Cum

ulat

ive H

SV-2

case

s

(b)



subject to

1198781015840

119899= 1205870Λ minus 120572 (1 minus 119906

1) 120582119878119899+ 120601119878119886minus (1 minus 119906

2) 120588119878119899minus 120583119878119899

1198781015840

119886= (1 minus 119906

2) 1205871Λ minus (1 minus 119906

1) 120582119878119886

+ (1 minus 1199062) 120588119878119899minus (120583 + V + 120601) 119878

119886

1198681015840

119899= 120572 (1 minus 119906

1) 120582119878119899+ 120601119868119886minus (1 minus 119906

2) 120588119868119899minus 120583119868119899

1198681015840

119886= (1 minus 119906

1) 120582119878119886+ (1 minus 119906

2) 120588119868119899minus (120583 + V + 120601) 119868

119886

(39)








2






1and 119906lowast

2such that

119869 (119906lowast

1 119906lowast

2) = max 119869 (119906

1 1199062) | 1199061 1199062isin U (40)

whereU = 1199061(119905) 1199062(119905) | 119906

1(119905) 1199062(119905) is measurable 0 le 119886

11le

1199061(119905) le 119887

11le 1 0 le 119886

22le 1199062(119905) le 119887

22le 1 119905 isin [0 119905

119891]





1and 119906





1and 119906

2with



+ |1199062|2

) where 1198881 1198882gt 0



1198613119868119886minus (11986011199062

1+11986021199062



1 1198612 1198613 1198601 1198602gt 1 hence

satisfying


1+ 11986021199062

2)

le 1198882minus 1198881(10038161003816100381610038161199061

1003816100381610038161003816

2

+10038161003816100381610038161199062

1003816100381610038161003816

2

)

(41)




1+ 11986021199062

2)

+ 1205821[1205870Λ minus

120572 (1 minus 1199061) 120573119868119886119878119899

119873

minus120572 (1 minus 119906

1) (1 minus 120578) 120573119868

119899119878119899

119873

+120601119878119886minus (1 minus 119906

2) 120588119878119899minus 120583119878119899]

+ 1205822[(1 minus 119906

2) 1205871Λ minus

(1 minus 1199061) 120573119868119886119878119886

119873


1) (1 minus 120578) 120573119868

119899119878119886

119873

+ (1 minus 1199062) 120588119878119899minus (120583 + V + 120601) 119878

119886]

+ 1205823[120572 (1 minus 119906

1) 120573119868119886119878119899

119873

+120572 (1 minus 119906

1) (1 minus 120578) 120573119868

119899119878119899

119873

+120601119868119886minus (1 minus 119906

2) 120588119868119899minus 120583119868119899]

+ 1205824[(1 minus 119906

1) 120573119868119886119878119886

119873+(1 minus 119906

1) (1 minus 120578) 120573119868

119899119878119886

119873

+ (1 minus 1199062) 120588119868119899minus (120583 + V + 120601) 119868

119886]

+ 12059611(119905) (1198871minus 1199061(119905)) + 120596

12(119905) (1199061(119905) minus 119886

1)

+ 12059621(119905) (1198872minus 1199062(119905)) + 120596

22(119905) (1199062(119905) minus 119886

2)

(42)

where 12059611(119905) 12059612(119905) ge 0 120596

21(119905) and 120596



12(119905)(1199061(119905) minus

1198861) = 0 120596

21(119905)(1198872minus 1199062(119905)) = 0 and 120596

22(119905)(1199062(119905) minus 119886

2) = 0 at


2



119889119905= minus

120597119871

120597119878119899

= minus1 +120572 (1 minus 119906

1) 120573119868119886[1205821minus 1205823]

119873

+120572 (1 minus 119906

1) (1 minus 120578) 120573119868

119899[1205821minus 1205823]

119873

+ (1 minus 1199062) 120588 [1205821minus 1205822] + 120583120582

1


1198891205822

119889119905= minus

120597119871

120597119878119886

= 1198611+ 120601 [120582

2minus 1205821] +

(1 minus 1199061) 120573119868119886[1205822minus 1205824]

119873

+(1 minus 119906

1) (1 minus 120578) 120573119868

119899[1205822minus 1205824]

119873+ (120583 + V) 120582

2

1198891205822

119889119905= minus

120597119871

120597119868119899

= 1198612+120572 (1 minus 119906

1) (1 minus 120578) 120573119878

119899[1205821minus 1205823]

119873

+(1 minus 119906

1) (1 minus 120578) 120573119878

119886[1205822minus 1205824]

119873

+ (1 minus 1199062) 120588 [1205823minus 1205824] + 120583120582

3

1198891205822

119889119905= minus

120597119871

120597119868119899

= 1198613+120572 (1 minus 119906

1) 120573119878119899[1205821minus 1205823]

119873

+(1 minus 119906

1) 120573119878119886[1205822minus 1205824]

119873+ 120601 [120582

4minus 1205823] + (120583 + V) 120582

4

(43)




1199061(119905)lowast

=120573 (119868119886+ (1 minus 120578) 119868

119899)

21198601119873

[120572119878119899(1205821minus 1205823) + 119878119886(1205822minus 1205824)]

+12059612(119905) minus 120596

11(119905)

21198601

(44)


1199062(119905)lowast

=120588

21198602


1205871Λ1205822

21198602

+12059622minus 12059621

21198602

(45)


1(119905) and 120596




1199061(119905)lowast

= minmax 1198861120573 (119868119886+ (1 minus 120578) 119868

119899)

21198601119873


1

1199062(119905)lowast

= minmax1198862120588

21198602

[119878119899(1205821minus 1205822) + 119868119899(1205823minus 1205824)]

minus1205871Λ1205822

21198602

1198872

(46)


1 and 119906lowast

2for 1199061(119905) and 119906

2(119905)


119891[60]



1= 0005 119860

2=

0005 1198611= 006 119861

2= 005 and 119861

3= 005 The assumed


= 041198781198860

= 03 1198681198990

= 02 and 1198681198860





036

038

04

042

044

046

048

0 2 4 6 8 10 12 14 16 18 20Time (years)

S n(t)

(a)

024

026

028

03

032

034

036

0 2 4 6 8 10 12 14 16 18 20Time (years)

S a(t)

(b)

005

01

015

02

025

03

035


00 2 4 6 8 10 12 14 16 18 20

Time (years)

I n(t)

(c)


0

005

01

015

02

025

03

035

0 2 4 6 8 10 12 14 16 18 20Time (years)

I a(t)


119899 susceptible




of 20 years






0 2 4 6 8 10 12 14 16 18 200

02

04

06

08

1

Time (years)

u1

(a)

0 2 4 6 8 10 12 14 16 18 200

02

04

06

08

1

Time (years)

u2

(b)


2)



2 The condom






2= 0



(1199062)

6 Discussion




Appendices



1205870(120583 + V) + 120601

120583 + 120601 + 120588 + V1205870

ge119878119899

119873

1205831205871+ 120588

120583 + 120601 + 120588 + V1205870

ge119878119886

119873

(A1)



(i)1205870(120583 + V) + 120601

120583 + 120601 + 120588 + V1205870

lt119878119899

119873

(ii)1205831205871+ 120588

120583 + 120601 + 120588 + V1205870

lt119878119886

119873

(A2)


120583 (1205870+ 1205871) + 120601 + 120588 + 120587

0

120583 + 120601 + 120588 + V1205870

lt119878119886+ 119878119899

119873997904rArr 1 lt 1 (A3)



(i) 1198671ge 2

(ii) 1198672ge 1

(A4)



(i)1205870(120583 + V) + 120601

120583 + 120601 + 120588 + V1205870

ge119878119899

119873

(ii)1205831205871+ 120588

120583 + 120601 + 120588 + V1205870

ge119878119886

119873

(A5)


120583 (1205870+ 1205871) + 120601 + 120588 + V120587

0

120583 + 120601 + 120588 + V1205870

ge119878119886+ 119878119899

119873

997904rArr120583 + 120601 + 120588 + V120587

0

120583 + 120601 + 120588 + V1205870

ge119878119886+ 119878119899

119873997904rArr 1 ge 1

(A6)




1199091015840

= 119891 (119909) (B1)

where 119891 isin 1198621


0) be




120597119909) lt 0 120583(minus120597119891[2]





Acknowledgments


References







































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












0 1205871




as119863(119868119899 119868119886) = 1119868

119899119868119886 Then we have

120597 (119863Φ)

120597119868119899

+120597 (119863Ψ)

120597119868119886

= minus120601

1198682

119899

minus120588

1198682

119886

minus (1 minus 120579) 120573(120572119878119899

1198682

119899

+(1 minus 120578) 119878

119886

1198682

119886

) lt 0

(37)



119899119886gt 1

4 Numerical Results



119899119886


02 04 06 08 10

10

20

30

40 Ra

R0

Rc

Rna

120573

R








00

05

10

00

05

10

10

15

120601120588

Rna

(a)

02 04 06 08 10

10

12

14

16

18

20

22

(120601 120588)

Rna

Ω

120595

Σ

(b)










119899119886



on 119877119899119886








0 02 04 06 08





Natural death rate



Condom use efficacy












0 01 02 03 04 05 06 07 08 09 1

0

2


minus10

minus8

minus6

minus4

minus2

log(Rna)

(a)


0

2

minus10

minus8

minus6

minus4

minus2

log(Rna)

(b)


0

2

minus10

minus8

minus6

minus4

minus2

log(Rna)

(c)





119899

119878119886 119868119899 and 119868



119869 (1199061 1199062)

= int

119905119891

0


1+ 11986021199062

2)] 119889119905

(38)


120579 = 02

120579 = 05

120579 = 08

0 10 20 30 40 50 600

500

1000

1500

2000

2500

3000

3500

Time (years)

Cum

ulat

ive H

SV-2

case

s

(a)

120579 = 02

120579 = 05

120579 = 08

0 10 20 30 40 50 60100

200

300

400

500

600

Time (years)

Cum

ulat

ive H

SV-2

case

s

(b)



0 10 20 30 40 50 600

500

1000

1500

2000

2500

3000

3500

Time (years)

Cum

ulat

ive H

SV-2

case

s

(a)

0 10 20 30 40 50 60100

200

300

400

500

600

700

800

900

Time (years)

Cum

ulat

ive H

SV-2

case

s

(b)



subject to

1198781015840

119899= 1205870Λ minus 120572 (1 minus 119906

1) 120582119878119899+ 120601119878119886minus (1 minus 119906

2) 120588119878119899minus 120583119878119899

1198781015840

119886= (1 minus 119906

2) 1205871Λ minus (1 minus 119906

1) 120582119878119886

+ (1 minus 1199062) 120588119878119899minus (120583 + V + 120601) 119878

119886

1198681015840

119899= 120572 (1 minus 119906

1) 120582119878119899+ 120601119868119886minus (1 minus 119906

2) 120588119868119899minus 120583119868119899

1198681015840

119886= (1 minus 119906

1) 120582119878119886+ (1 minus 119906

2) 120588119868119899minus (120583 + V + 120601) 119868

119886

(39)








2






1and 119906lowast

2such that

119869 (119906lowast

1 119906lowast

2) = max 119869 (119906

1 1199062) | 1199061 1199062isin U (40)

whereU = 1199061(119905) 1199062(119905) | 119906

1(119905) 1199062(119905) is measurable 0 le 119886

11le

1199061(119905) le 119887

11le 1 0 le 119886

22le 1199062(119905) le 119887

22le 1 119905 isin [0 119905

119891]





1and 119906





1and 119906

2with



+ |1199062|2

) where 1198881 1198882gt 0



1198613119868119886minus (11986011199062

1+11986021199062



1 1198612 1198613 1198601 1198602gt 1 hence

satisfying


1+ 11986021199062

2)

le 1198882minus 1198881(10038161003816100381610038161199061

1003816100381610038161003816

2

+10038161003816100381610038161199062

1003816100381610038161003816

2

)

(41)




1+ 11986021199062

2)

+ 1205821[1205870Λ minus

120572 (1 minus 1199061) 120573119868119886119878119899

119873

minus120572 (1 minus 119906

1) (1 minus 120578) 120573119868

119899119878119899

119873

+120601119878119886minus (1 minus 119906

2) 120588119878119899minus 120583119878119899]

+ 1205822[(1 minus 119906

2) 1205871Λ minus

(1 minus 1199061) 120573119868119886119878119886

119873


1) (1 minus 120578) 120573119868

119899119878119886

119873

+ (1 minus 1199062) 120588119878119899minus (120583 + V + 120601) 119878

119886]

+ 1205823[120572 (1 minus 119906

1) 120573119868119886119878119899

119873

+120572 (1 minus 119906

1) (1 minus 120578) 120573119868

119899119878119899

119873

+120601119868119886minus (1 minus 119906

2) 120588119868119899minus 120583119868119899]

+ 1205824[(1 minus 119906

1) 120573119868119886119878119886

119873+(1 minus 119906

1) (1 minus 120578) 120573119868

119899119878119886

119873

+ (1 minus 1199062) 120588119868119899minus (120583 + V + 120601) 119868

119886]

+ 12059611(119905) (1198871minus 1199061(119905)) + 120596

12(119905) (1199061(119905) minus 119886

1)

+ 12059621(119905) (1198872minus 1199062(119905)) + 120596

22(119905) (1199062(119905) minus 119886

2)

(42)

where 12059611(119905) 12059612(119905) ge 0 120596

21(119905) and 120596



12(119905)(1199061(119905) minus

1198861) = 0 120596

21(119905)(1198872minus 1199062(119905)) = 0 and 120596

22(119905)(1199062(119905) minus 119886

2) = 0 at


2



119889119905= minus

120597119871

120597119878119899

= minus1 +120572 (1 minus 119906

1) 120573119868119886[1205821minus 1205823]

119873

+120572 (1 minus 119906

1) (1 minus 120578) 120573119868

119899[1205821minus 1205823]

119873

+ (1 minus 1199062) 120588 [1205821minus 1205822] + 120583120582

1


1198891205822

119889119905= minus

120597119871

120597119878119886

= 1198611+ 120601 [120582

2minus 1205821] +

(1 minus 1199061) 120573119868119886[1205822minus 1205824]

119873

+(1 minus 119906

1) (1 minus 120578) 120573119868

119899[1205822minus 1205824]

119873+ (120583 + V) 120582

2

1198891205822

119889119905= minus

120597119871

120597119868119899

= 1198612+120572 (1 minus 119906

1) (1 minus 120578) 120573119878

119899[1205821minus 1205823]

119873

+(1 minus 119906

1) (1 minus 120578) 120573119878

119886[1205822minus 1205824]

119873

+ (1 minus 1199062) 120588 [1205823minus 1205824] + 120583120582

3

1198891205822

119889119905= minus

120597119871

120597119868119899

= 1198613+120572 (1 minus 119906

1) 120573119878119899[1205821minus 1205823]

119873

+(1 minus 119906

1) 120573119878119886[1205822minus 1205824]

119873+ 120601 [120582

4minus 1205823] + (120583 + V) 120582

4

(43)




1199061(119905)lowast

=120573 (119868119886+ (1 minus 120578) 119868

119899)

21198601119873

[120572119878119899(1205821minus 1205823) + 119878119886(1205822minus 1205824)]

+12059612(119905) minus 120596

11(119905)

21198601

(44)


1199062(119905)lowast

=120588

21198602


1205871Λ1205822

21198602

+12059622minus 12059621

21198602

(45)


1(119905) and 120596




1199061(119905)lowast

= minmax 1198861120573 (119868119886+ (1 minus 120578) 119868

119899)

21198601119873


1

1199062(119905)lowast

= minmax1198862120588

21198602

[119878119899(1205821minus 1205822) + 119868119899(1205823minus 1205824)]

minus1205871Λ1205822

21198602

1198872

(46)


1 and 119906lowast

2for 1199061(119905) and 119906

2(119905)


119891[60]



1= 0005 119860

2=

0005 1198611= 006 119861

2= 005 and 119861

3= 005 The assumed


= 041198781198860

= 03 1198681198990

= 02 and 1198681198860





036

038

04

042

044

046

048

0 2 4 6 8 10 12 14 16 18 20Time (years)

S n(t)

(a)

024

026

028

03

032

034

036

0 2 4 6 8 10 12 14 16 18 20Time (years)

S a(t)

(b)

005

01

015

02

025

03

035


00 2 4 6 8 10 12 14 16 18 20

Time (years)

I n(t)

(c)


0

005

01

015

02

025

03

035

0 2 4 6 8 10 12 14 16 18 20Time (years)

I a(t)


119899 susceptible




of 20 years






0 2 4 6 8 10 12 14 16 18 200

02

04

06

08

1

Time (years)

u1

(a)

0 2 4 6 8 10 12 14 16 18 200

02

04

06

08

1

Time (years)

u2

(b)


2)



2 The condom






2= 0



(1199062)

6 Discussion




Appendices



1205870(120583 + V) + 120601

120583 + 120601 + 120588 + V1205870

ge119878119899

119873

1205831205871+ 120588

120583 + 120601 + 120588 + V1205870

ge119878119886

119873

(A1)



(i)1205870(120583 + V) + 120601

120583 + 120601 + 120588 + V1205870

lt119878119899

119873

(ii)1205831205871+ 120588

120583 + 120601 + 120588 + V1205870

lt119878119886

119873

(A2)


120583 (1205870+ 1205871) + 120601 + 120588 + 120587

0

120583 + 120601 + 120588 + V1205870

lt119878119886+ 119878119899

119873997904rArr 1 lt 1 (A3)



(i) 1198671ge 2

(ii) 1198672ge 1

(A4)



(i)1205870(120583 + V) + 120601

120583 + 120601 + 120588 + V1205870

ge119878119899

119873

(ii)1205831205871+ 120588

120583 + 120601 + 120588 + V1205870

ge119878119886

119873

(A5)


120583 (1205870+ 1205871) + 120601 + 120588 + V120587

0

120583 + 120601 + 120588 + V1205870

ge119878119886+ 119878119899

119873

997904rArr120583 + 120601 + 120588 + V120587

0

120583 + 120601 + 120588 + V1205870

ge119878119886+ 119878119899

119873997904rArr 1 ge 1

(A6)




1199091015840

= 119891 (119909) (B1)

where 119891 isin 1198621


0) be




120597119909) lt 0 120583(minus120597119891[2]





Acknowledgments


References







































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










00

05

10

00

05

10

10

15

120601120588

Rna

(a)

02 04 06 08 10

10

12

14

16

18

20

22

(120601 120588)

Rna

Ω

120595

Σ

(b)










119899119886



on 119877119899119886








0 02 04 06 08





Natural death rate



Condom use efficacy












0 01 02 03 04 05 06 07 08 09 1

0

2


minus10

minus8

minus6

minus4

minus2

log(Rna)

(a)


0

2

minus10

minus8

minus6

minus4

minus2

log(Rna)

(b)


0

2

minus10

minus8

minus6

minus4

minus2

log(Rna)

(c)





119899

119878119886 119868119899 and 119868



119869 (1199061 1199062)

= int

119905119891

0


1+ 11986021199062

2)] 119889119905

(38)


120579 = 02

120579 = 05

120579 = 08

0 10 20 30 40 50 600

500

1000

1500

2000

2500

3000

3500

Time (years)

Cum

ulat

ive H

SV-2

case

s

(a)

120579 = 02

120579 = 05

120579 = 08

0 10 20 30 40 50 60100

200

300

400

500

600

Time (years)

Cum

ulat

ive H

SV-2

case

s

(b)



0 10 20 30 40 50 600

500

1000

1500

2000

2500

3000

3500

Time (years)

Cum

ulat

ive H

SV-2

case

s

(a)

0 10 20 30 40 50 60100

200

300

400

500

600

700

800

900

Time (years)

Cum

ulat

ive H

SV-2

case

s

(b)



subject to

1198781015840

119899= 1205870Λ minus 120572 (1 minus 119906

1) 120582119878119899+ 120601119878119886minus (1 minus 119906

2) 120588119878119899minus 120583119878119899

1198781015840

119886= (1 minus 119906

2) 1205871Λ minus (1 minus 119906

1) 120582119878119886

+ (1 minus 1199062) 120588119878119899minus (120583 + V + 120601) 119878

119886

1198681015840

119899= 120572 (1 minus 119906

1) 120582119878119899+ 120601119868119886minus (1 minus 119906

2) 120588119868119899minus 120583119868119899

1198681015840

119886= (1 minus 119906

1) 120582119878119886+ (1 minus 119906

2) 120588119868119899minus (120583 + V + 120601) 119868

119886

(39)








2






1and 119906lowast

2such that

119869 (119906lowast

1 119906lowast

2) = max 119869 (119906

1 1199062) | 1199061 1199062isin U (40)

whereU = 1199061(119905) 1199062(119905) | 119906

1(119905) 1199062(119905) is measurable 0 le 119886

11le

1199061(119905) le 119887

11le 1 0 le 119886

22le 1199062(119905) le 119887

22le 1 119905 isin [0 119905

119891]





1and 119906





1and 119906

2with



+ |1199062|2

) where 1198881 1198882gt 0



1198613119868119886minus (11986011199062

1+11986021199062



1 1198612 1198613 1198601 1198602gt 1 hence

satisfying


1+ 11986021199062

2)

le 1198882minus 1198881(10038161003816100381610038161199061

1003816100381610038161003816

2

+10038161003816100381610038161199062

1003816100381610038161003816

2

)

(41)




1+ 11986021199062

2)

+ 1205821[1205870Λ minus

120572 (1 minus 1199061) 120573119868119886119878119899

119873

minus120572 (1 minus 119906

1) (1 minus 120578) 120573119868

119899119878119899

119873

+120601119878119886minus (1 minus 119906

2) 120588119878119899minus 120583119878119899]

+ 1205822[(1 minus 119906

2) 1205871Λ minus

(1 minus 1199061) 120573119868119886119878119886

119873


1) (1 minus 120578) 120573119868

119899119878119886

119873

+ (1 minus 1199062) 120588119878119899minus (120583 + V + 120601) 119878

119886]

+ 1205823[120572 (1 minus 119906

1) 120573119868119886119878119899

119873

+120572 (1 minus 119906

1) (1 minus 120578) 120573119868

119899119878119899

119873

+120601119868119886minus (1 minus 119906

2) 120588119868119899minus 120583119868119899]

+ 1205824[(1 minus 119906

1) 120573119868119886119878119886

119873+(1 minus 119906

1) (1 minus 120578) 120573119868

119899119878119886

119873

+ (1 minus 1199062) 120588119868119899minus (120583 + V + 120601) 119868

119886]

+ 12059611(119905) (1198871minus 1199061(119905)) + 120596

12(119905) (1199061(119905) minus 119886

1)

+ 12059621(119905) (1198872minus 1199062(119905)) + 120596

22(119905) (1199062(119905) minus 119886

2)

(42)

where 12059611(119905) 12059612(119905) ge 0 120596

21(119905) and 120596



12(119905)(1199061(119905) minus

1198861) = 0 120596

21(119905)(1198872minus 1199062(119905)) = 0 and 120596

22(119905)(1199062(119905) minus 119886

2) = 0 at


2



119889119905= minus

120597119871

120597119878119899

= minus1 +120572 (1 minus 119906

1) 120573119868119886[1205821minus 1205823]

119873

+120572 (1 minus 119906

1) (1 minus 120578) 120573119868

119899[1205821minus 1205823]

119873

+ (1 minus 1199062) 120588 [1205821minus 1205822] + 120583120582

1


1198891205822

119889119905= minus

120597119871

120597119878119886

= 1198611+ 120601 [120582

2minus 1205821] +

(1 minus 1199061) 120573119868119886[1205822minus 1205824]

119873

+(1 minus 119906

1) (1 minus 120578) 120573119868

119899[1205822minus 1205824]

119873+ (120583 + V) 120582

2

1198891205822

119889119905= minus

120597119871

120597119868119899

= 1198612+120572 (1 minus 119906

1) (1 minus 120578) 120573119878

119899[1205821minus 1205823]

119873

+(1 minus 119906

1) (1 minus 120578) 120573119878

119886[1205822minus 1205824]

119873

+ (1 minus 1199062) 120588 [1205823minus 1205824] + 120583120582

3

1198891205822

119889119905= minus

120597119871

120597119868119899

= 1198613+120572 (1 minus 119906

1) 120573119878119899[1205821minus 1205823]

119873

+(1 minus 119906

1) 120573119878119886[1205822minus 1205824]

119873+ 120601 [120582

4minus 1205823] + (120583 + V) 120582

4

(43)




1199061(119905)lowast

=120573 (119868119886+ (1 minus 120578) 119868

119899)

21198601119873

[120572119878119899(1205821minus 1205823) + 119878119886(1205822minus 1205824)]

+12059612(119905) minus 120596

11(119905)

21198601

(44)


1199062(119905)lowast

=120588

21198602


1205871Λ1205822

21198602

+12059622minus 12059621

21198602

(45)


1(119905) and 120596




1199061(119905)lowast

= minmax 1198861120573 (119868119886+ (1 minus 120578) 119868

119899)

21198601119873


1

1199062(119905)lowast

= minmax1198862120588

21198602

[119878119899(1205821minus 1205822) + 119868119899(1205823minus 1205824)]

minus1205871Λ1205822

21198602

1198872

(46)


1 and 119906lowast

2for 1199061(119905) and 119906

2(119905)


119891[60]



1= 0005 119860

2=

0005 1198611= 006 119861

2= 005 and 119861

3= 005 The assumed


= 041198781198860

= 03 1198681198990

= 02 and 1198681198860





036

038

04

042

044

046

048

0 2 4 6 8 10 12 14 16 18 20Time (years)

S n(t)

(a)

024

026

028

03

032

034

036

0 2 4 6 8 10 12 14 16 18 20Time (years)

S a(t)

(b)

005

01

015

02

025

03

035


00 2 4 6 8 10 12 14 16 18 20

Time (years)

I n(t)

(c)


0

005

01

015

02

025

03

035

0 2 4 6 8 10 12 14 16 18 20Time (years)

I a(t)


119899 susceptible




of 20 years






0 2 4 6 8 10 12 14 16 18 200

02

04

06

08

1

Time (years)

u1

(a)

0 2 4 6 8 10 12 14 16 18 200

02

04

06

08

1

Time (years)

u2

(b)


2)



2 The condom






2= 0



(1199062)

6 Discussion




Appendices



1205870(120583 + V) + 120601

120583 + 120601 + 120588 + V1205870

ge119878119899

119873

1205831205871+ 120588

120583 + 120601 + 120588 + V1205870

ge119878119886

119873

(A1)



(i)1205870(120583 + V) + 120601

120583 + 120601 + 120588 + V1205870

lt119878119899

119873

(ii)1205831205871+ 120588

120583 + 120601 + 120588 + V1205870

lt119878119886

119873

(A2)


120583 (1205870+ 1205871) + 120601 + 120588 + 120587

0

120583 + 120601 + 120588 + V1205870

lt119878119886+ 119878119899

119873997904rArr 1 lt 1 (A3)



(i) 1198671ge 2

(ii) 1198672ge 1

(A4)



(i)1205870(120583 + V) + 120601

120583 + 120601 + 120588 + V1205870

ge119878119899

119873

(ii)1205831205871+ 120588

120583 + 120601 + 120588 + V1205870

ge119878119886

119873

(A5)


120583 (1205870+ 1205871) + 120601 + 120588 + V120587

0

120583 + 120601 + 120588 + V1205870

ge119878119886+ 119878119899

119873

997904rArr120583 + 120601 + 120588 + V120587

0

120583 + 120601 + 120588 + V1205870

ge119878119886+ 119878119899

119873997904rArr 1 ge 1

(A6)




1199091015840

= 119891 (119909) (B1)

where 119891 isin 1198621


0) be




120597119909) lt 0 120583(minus120597119891[2]





Acknowledgments


References







































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 01 02 03 04 05 06 07 08 09 1

0

2


minus10

minus8

minus6

minus4

minus2

log(Rna)

(a)


0

2

minus10

minus8

minus6

minus4

minus2

log(Rna)

(b)


0

2

minus10

minus8

minus6

minus4

minus2

log(Rna)

(c)





119899

119878119886 119868119899 and 119868



119869 (1199061 1199062)

= int

119905119891

0


1+ 11986021199062

2)] 119889119905

(38)


120579 = 02

120579 = 05

120579 = 08

0 10 20 30 40 50 600

500

1000

1500

2000

2500

3000

3500

Time (years)

Cum

ulat

ive H

SV-2

case

s

(a)

120579 = 02

120579 = 05

120579 = 08

0 10 20 30 40 50 60100

200

300

400

500

600

Time (years)

Cum

ulat

ive H

SV-2

case

s

(b)



0 10 20 30 40 50 600

500

1000

1500

2000

2500

3000

3500

Time (years)

Cum

ulat

ive H

SV-2

case

s

(a)

0 10 20 30 40 50 60100

200

300

400

500

600

700

800

900

Time (years)

Cum

ulat

ive H

SV-2

case

s

(b)



subject to

1198781015840

119899= 1205870Λ minus 120572 (1 minus 119906

1) 120582119878119899+ 120601119878119886minus (1 minus 119906

2) 120588119878119899minus 120583119878119899

1198781015840

119886= (1 minus 119906

2) 1205871Λ minus (1 minus 119906

1) 120582119878119886

+ (1 minus 1199062) 120588119878119899minus (120583 + V + 120601) 119878

119886

1198681015840

119899= 120572 (1 minus 119906

1) 120582119878119899+ 120601119868119886minus (1 minus 119906

2) 120588119868119899minus 120583119868119899

1198681015840

119886= (1 minus 119906

1) 120582119878119886+ (1 minus 119906

2) 120588119868119899minus (120583 + V + 120601) 119868

119886

(39)








2






1and 119906lowast

2such that

119869 (119906lowast

1 119906lowast

2) = max 119869 (119906

1 1199062) | 1199061 1199062isin U (40)

whereU = 1199061(119905) 1199062(119905) | 119906

1(119905) 1199062(119905) is measurable 0 le 119886

11le

1199061(119905) le 119887

11le 1 0 le 119886

22le 1199062(119905) le 119887

22le 1 119905 isin [0 119905

119891]





1and 119906





1and 119906

2with



+ |1199062|2

) where 1198881 1198882gt 0



1198613119868119886minus (11986011199062

1+11986021199062



1 1198612 1198613 1198601 1198602gt 1 hence

satisfying


1+ 11986021199062

2)

le 1198882minus 1198881(10038161003816100381610038161199061

1003816100381610038161003816

2

+10038161003816100381610038161199062

1003816100381610038161003816

2

)

(41)




1+ 11986021199062

2)

+ 1205821[1205870Λ minus

120572 (1 minus 1199061) 120573119868119886119878119899

119873

minus120572 (1 minus 119906

1) (1 minus 120578) 120573119868

119899119878119899

119873

+120601119878119886minus (1 minus 119906

2) 120588119878119899minus 120583119878119899]

+ 1205822[(1 minus 119906

2) 1205871Λ minus

(1 minus 1199061) 120573119868119886119878119886

119873


1) (1 minus 120578) 120573119868

119899119878119886

119873

+ (1 minus 1199062) 120588119878119899minus (120583 + V + 120601) 119878

119886]

+ 1205823[120572 (1 minus 119906

1) 120573119868119886119878119899

119873

+120572 (1 minus 119906

1) (1 minus 120578) 120573119868

119899119878119899

119873

+120601119868119886minus (1 minus 119906

2) 120588119868119899minus 120583119868119899]

+ 1205824[(1 minus 119906

1) 120573119868119886119878119886

119873+(1 minus 119906

1) (1 minus 120578) 120573119868

119899119878119886

119873

+ (1 minus 1199062) 120588119868119899minus (120583 + V + 120601) 119868

119886]

+ 12059611(119905) (1198871minus 1199061(119905)) + 120596

12(119905) (1199061(119905) minus 119886

1)

+ 12059621(119905) (1198872minus 1199062(119905)) + 120596

22(119905) (1199062(119905) minus 119886

2)

(42)

where 12059611(119905) 12059612(119905) ge 0 120596

21(119905) and 120596



12(119905)(1199061(119905) minus

1198861) = 0 120596

21(119905)(1198872minus 1199062(119905)) = 0 and 120596

22(119905)(1199062(119905) minus 119886

2) = 0 at


2



119889119905= minus

120597119871

120597119878119899

= minus1 +120572 (1 minus 119906

1) 120573119868119886[1205821minus 1205823]

119873

+120572 (1 minus 119906

1) (1 minus 120578) 120573119868

119899[1205821minus 1205823]

119873

+ (1 minus 1199062) 120588 [1205821minus 1205822] + 120583120582

1


1198891205822

119889119905= minus

120597119871

120597119878119886

= 1198611+ 120601 [120582

2minus 1205821] +

(1 minus 1199061) 120573119868119886[1205822minus 1205824]

119873

+(1 minus 119906

1) (1 minus 120578) 120573119868

119899[1205822minus 1205824]

119873+ (120583 + V) 120582

2

1198891205822

119889119905= minus

120597119871

120597119868119899

= 1198612+120572 (1 minus 119906

1) (1 minus 120578) 120573119878

119899[1205821minus 1205823]

119873

+(1 minus 119906

1) (1 minus 120578) 120573119878

119886[1205822minus 1205824]

119873

+ (1 minus 1199062) 120588 [1205823minus 1205824] + 120583120582

3

1198891205822

119889119905= minus

120597119871

120597119868119899

= 1198613+120572 (1 minus 119906

1) 120573119878119899[1205821minus 1205823]

119873

+(1 minus 119906

1) 120573119878119886[1205822minus 1205824]

119873+ 120601 [120582

4minus 1205823] + (120583 + V) 120582

4

(43)




1199061(119905)lowast

=120573 (119868119886+ (1 minus 120578) 119868

119899)

21198601119873

[120572119878119899(1205821minus 1205823) + 119878119886(1205822minus 1205824)]

+12059612(119905) minus 120596

11(119905)

21198601

(44)


1199062(119905)lowast

=120588

21198602


1205871Λ1205822

21198602

+12059622minus 12059621

21198602

(45)


1(119905) and 120596




1199061(119905)lowast

= minmax 1198861120573 (119868119886+ (1 minus 120578) 119868

119899)

21198601119873


1

1199062(119905)lowast

= minmax1198862120588

21198602

[119878119899(1205821minus 1205822) + 119868119899(1205823minus 1205824)]

minus1205871Λ1205822

21198602

1198872

(46)


1 and 119906lowast

2for 1199061(119905) and 119906

2(119905)


119891[60]



1= 0005 119860

2=

0005 1198611= 006 119861

2= 005 and 119861

3= 005 The assumed


= 041198781198860

= 03 1198681198990

= 02 and 1198681198860





036

038

04

042

044

046

048

0 2 4 6 8 10 12 14 16 18 20Time (years)

S n(t)

(a)

024

026

028

03

032

034

036

0 2 4 6 8 10 12 14 16 18 20Time (years)

S a(t)

(b)

005

01

015

02

025

03

035


00 2 4 6 8 10 12 14 16 18 20

Time (years)

I n(t)

(c)


0

005

01

015

02

025

03

035

0 2 4 6 8 10 12 14 16 18 20Time (years)

I a(t)


119899 susceptible




of 20 years






0 2 4 6 8 10 12 14 16 18 200

02

04

06

08

1

Time (years)

u1

(a)

0 2 4 6 8 10 12 14 16 18 200

02

04

06

08

1

Time (years)

u2

(b)


2)



2 The condom






2= 0



(1199062)

6 Discussion




Appendices



1205870(120583 + V) + 120601

120583 + 120601 + 120588 + V1205870

ge119878119899

119873

1205831205871+ 120588

120583 + 120601 + 120588 + V1205870

ge119878119886

119873

(A1)



(i)1205870(120583 + V) + 120601

120583 + 120601 + 120588 + V1205870

lt119878119899

119873

(ii)1205831205871+ 120588

120583 + 120601 + 120588 + V1205870

lt119878119886

119873

(A2)


120583 (1205870+ 1205871) + 120601 + 120588 + 120587

0

120583 + 120601 + 120588 + V1205870

lt119878119886+ 119878119899

119873997904rArr 1 lt 1 (A3)



(i) 1198671ge 2

(ii) 1198672ge 1

(A4)



(i)1205870(120583 + V) + 120601

120583 + 120601 + 120588 + V1205870

ge119878119899

119873

(ii)1205831205871+ 120588

120583 + 120601 + 120588 + V1205870

ge119878119886

119873

(A5)


120583 (1205870+ 1205871) + 120601 + 120588 + V120587

0

120583 + 120601 + 120588 + V1205870

ge119878119886+ 119878119899

119873

997904rArr120583 + 120601 + 120588 + V120587

0

120583 + 120601 + 120588 + V1205870

ge119878119886+ 119878119899

119873997904rArr 1 ge 1

(A6)




1199091015840

= 119891 (119909) (B1)

where 119891 isin 1198621


0) be




120597119909) lt 0 120583(minus120597119891[2]





Acknowledgments


References







































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










120579 = 02

120579 = 05

120579 = 08

0 10 20 30 40 50 600

500

1000

1500

2000

2500

3000

3500

Time (years)

Cum

ulat

ive H

SV-2

case

s

(a)

120579 = 02

120579 = 05

120579 = 08

0 10 20 30 40 50 60100

200

300

400

500

600

Time (years)

Cum

ulat

ive H

SV-2

case

s

(b)



0 10 20 30 40 50 600

500

1000

1500

2000

2500

3000

3500

Time (years)

Cum

ulat

ive H

SV-2

case

s

(a)

0 10 20 30 40 50 60100

200

300

400

500

600

700

800

900

Time (years)

Cum

ulat

ive H

SV-2

case

s

(b)



subject to

1198781015840

119899= 1205870Λ minus 120572 (1 minus 119906

1) 120582119878119899+ 120601119878119886minus (1 minus 119906

2) 120588119878119899minus 120583119878119899

1198781015840

119886= (1 minus 119906

2) 1205871Λ minus (1 minus 119906

1) 120582119878119886

+ (1 minus 1199062) 120588119878119899minus (120583 + V + 120601) 119878

119886

1198681015840

119899= 120572 (1 minus 119906

1) 120582119878119899+ 120601119868119886minus (1 minus 119906

2) 120588119868119899minus 120583119868119899

1198681015840

119886= (1 minus 119906

1) 120582119878119886+ (1 minus 119906

2) 120588119868119899minus (120583 + V + 120601) 119868

119886

(39)








2






1and 119906lowast

2such that

119869 (119906lowast

1 119906lowast

2) = max 119869 (119906

1 1199062) | 1199061 1199062isin U (40)

whereU = 1199061(119905) 1199062(119905) | 119906

1(119905) 1199062(119905) is measurable 0 le 119886

11le

1199061(119905) le 119887

11le 1 0 le 119886

22le 1199062(119905) le 119887

22le 1 119905 isin [0 119905

119891]





1and 119906





1and 119906

2with



+ |1199062|2

) where 1198881 1198882gt 0



1198613119868119886minus (11986011199062

1+11986021199062



1 1198612 1198613 1198601 1198602gt 1 hence

satisfying


1+ 11986021199062

2)

le 1198882minus 1198881(10038161003816100381610038161199061

1003816100381610038161003816

2

+10038161003816100381610038161199062

1003816100381610038161003816

2

)

(41)




1+ 11986021199062

2)

+ 1205821[1205870Λ minus

120572 (1 minus 1199061) 120573119868119886119878119899

119873

minus120572 (1 minus 119906

1) (1 minus 120578) 120573119868

119899119878119899

119873

+120601119878119886minus (1 minus 119906

2) 120588119878119899minus 120583119878119899]

+ 1205822[(1 minus 119906

2) 1205871Λ minus

(1 minus 1199061) 120573119868119886119878119886

119873


1) (1 minus 120578) 120573119868

119899119878119886

119873

+ (1 minus 1199062) 120588119878119899minus (120583 + V + 120601) 119878

119886]

+ 1205823[120572 (1 minus 119906

1) 120573119868119886119878119899

119873

+120572 (1 minus 119906

1) (1 minus 120578) 120573119868

119899119878119899

119873

+120601119868119886minus (1 minus 119906

2) 120588119868119899minus 120583119868119899]

+ 1205824[(1 minus 119906

1) 120573119868119886119878119886

119873+(1 minus 119906

1) (1 minus 120578) 120573119868

119899119878119886

119873

+ (1 minus 1199062) 120588119868119899minus (120583 + V + 120601) 119868

119886]

+ 12059611(119905) (1198871minus 1199061(119905)) + 120596

12(119905) (1199061(119905) minus 119886

1)

+ 12059621(119905) (1198872minus 1199062(119905)) + 120596

22(119905) (1199062(119905) minus 119886

2)

(42)

where 12059611(119905) 12059612(119905) ge 0 120596

21(119905) and 120596



12(119905)(1199061(119905) minus

1198861) = 0 120596

21(119905)(1198872minus 1199062(119905)) = 0 and 120596

22(119905)(1199062(119905) minus 119886

2) = 0 at


2



119889119905= minus

120597119871

120597119878119899

= minus1 +120572 (1 minus 119906

1) 120573119868119886[1205821minus 1205823]

119873

+120572 (1 minus 119906

1) (1 minus 120578) 120573119868

119899[1205821minus 1205823]

119873

+ (1 minus 1199062) 120588 [1205821minus 1205822] + 120583120582

1


1198891205822

119889119905= minus

120597119871

120597119878119886

= 1198611+ 120601 [120582

2minus 1205821] +

(1 minus 1199061) 120573119868119886[1205822minus 1205824]

119873

+(1 minus 119906

1) (1 minus 120578) 120573119868

119899[1205822minus 1205824]

119873+ (120583 + V) 120582

2

1198891205822

119889119905= minus

120597119871

120597119868119899

= 1198612+120572 (1 minus 119906

1) (1 minus 120578) 120573119878

119899[1205821minus 1205823]

119873

+(1 minus 119906

1) (1 minus 120578) 120573119878

119886[1205822minus 1205824]

119873

+ (1 minus 1199062) 120588 [1205823minus 1205824] + 120583120582

3

1198891205822

119889119905= minus

120597119871

120597119868119899

= 1198613+120572 (1 minus 119906

1) 120573119878119899[1205821minus 1205823]

119873

+(1 minus 119906

1) 120573119878119886[1205822minus 1205824]

119873+ 120601 [120582

4minus 1205823] + (120583 + V) 120582

4

(43)




1199061(119905)lowast

=120573 (119868119886+ (1 minus 120578) 119868

119899)

21198601119873

[120572119878119899(1205821minus 1205823) + 119878119886(1205822minus 1205824)]

+12059612(119905) minus 120596

11(119905)

21198601

(44)


1199062(119905)lowast

=120588

21198602


1205871Λ1205822

21198602

+12059622minus 12059621

21198602

(45)


1(119905) and 120596




1199061(119905)lowast

= minmax 1198861120573 (119868119886+ (1 minus 120578) 119868

119899)

21198601119873


1

1199062(119905)lowast

= minmax1198862120588

21198602

[119878119899(1205821minus 1205822) + 119868119899(1205823minus 1205824)]

minus1205871Λ1205822

21198602

1198872

(46)


1 and 119906lowast

2for 1199061(119905) and 119906

2(119905)


119891[60]



1= 0005 119860

2=

0005 1198611= 006 119861

2= 005 and 119861

3= 005 The assumed


= 041198781198860

= 03 1198681198990

= 02 and 1198681198860





036

038

04

042

044

046

048

0 2 4 6 8 10 12 14 16 18 20Time (years)

S n(t)

(a)

024

026

028

03

032

034

036

0 2 4 6 8 10 12 14 16 18 20Time (years)

S a(t)

(b)

005

01

015

02

025

03

035


00 2 4 6 8 10 12 14 16 18 20

Time (years)

I n(t)

(c)


0

005

01

015

02

025

03

035

0 2 4 6 8 10 12 14 16 18 20Time (years)

I a(t)


119899 susceptible




of 20 years






0 2 4 6 8 10 12 14 16 18 200

02

04

06

08

1

Time (years)

u1

(a)

0 2 4 6 8 10 12 14 16 18 200

02

04

06

08

1

Time (years)

u2

(b)


2)



2 The condom






2= 0



(1199062)

6 Discussion




Appendices



1205870(120583 + V) + 120601

120583 + 120601 + 120588 + V1205870

ge119878119899

119873

1205831205871+ 120588

120583 + 120601 + 120588 + V1205870

ge119878119886

119873

(A1)



(i)1205870(120583 + V) + 120601

120583 + 120601 + 120588 + V1205870

lt119878119899

119873

(ii)1205831205871+ 120588

120583 + 120601 + 120588 + V1205870

lt119878119886

119873

(A2)


120583 (1205870+ 1205871) + 120601 + 120588 + 120587

0

120583 + 120601 + 120588 + V1205870

lt119878119886+ 119878119899

119873997904rArr 1 lt 1 (A3)



(i) 1198671ge 2

(ii) 1198672ge 1

(A4)



(i)1205870(120583 + V) + 120601

120583 + 120601 + 120588 + V1205870

ge119878119899

119873

(ii)1205831205871+ 120588

120583 + 120601 + 120588 + V1205870

ge119878119886

119873

(A5)


120583 (1205870+ 1205871) + 120601 + 120588 + V120587

0

120583 + 120601 + 120588 + V1205870

ge119878119886+ 119878119899

119873

997904rArr120583 + 120601 + 120588 + V120587

0

120583 + 120601 + 120588 + V1205870

ge119878119886+ 119878119899

119873997904rArr 1 ge 1

(A6)




1199091015840

= 119891 (119909) (B1)

where 119891 isin 1198621


0) be




120597119909) lt 0 120583(minus120597119891[2]





Acknowledgments


References







































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













2






1and 119906lowast

2such that

119869 (119906lowast

1 119906lowast

2) = max 119869 (119906

1 1199062) | 1199061 1199062isin U (40)

whereU = 1199061(119905) 1199062(119905) | 119906

1(119905) 1199062(119905) is measurable 0 le 119886

11le

1199061(119905) le 119887

11le 1 0 le 119886

22le 1199062(119905) le 119887

22le 1 119905 isin [0 119905

119891]





1and 119906





1and 119906

2with



+ |1199062|2

) where 1198881 1198882gt 0



1198613119868119886minus (11986011199062

1+11986021199062



1 1198612 1198613 1198601 1198602gt 1 hence

satisfying


1+ 11986021199062

2)

le 1198882minus 1198881(10038161003816100381610038161199061

1003816100381610038161003816

2

+10038161003816100381610038161199062

1003816100381610038161003816

2

)

(41)




1+ 11986021199062

2)

+ 1205821[1205870Λ minus

120572 (1 minus 1199061) 120573119868119886119878119899

119873

minus120572 (1 minus 119906

1) (1 minus 120578) 120573119868

119899119878119899

119873

+120601119878119886minus (1 minus 119906

2) 120588119878119899minus 120583119878119899]

+ 1205822[(1 minus 119906

2) 1205871Λ minus

(1 minus 1199061) 120573119868119886119878119886

119873


1) (1 minus 120578) 120573119868

119899119878119886

119873

+ (1 minus 1199062) 120588119878119899minus (120583 + V + 120601) 119878

119886]

+ 1205823[120572 (1 minus 119906

1) 120573119868119886119878119899

119873

+120572 (1 minus 119906

1) (1 minus 120578) 120573119868

119899119878119899

119873

+120601119868119886minus (1 minus 119906

2) 120588119868119899minus 120583119868119899]

+ 1205824[(1 minus 119906

1) 120573119868119886119878119886

119873+(1 minus 119906

1) (1 minus 120578) 120573119868

119899119878119886

119873

+ (1 minus 1199062) 120588119868119899minus (120583 + V + 120601) 119868

119886]

+ 12059611(119905) (1198871minus 1199061(119905)) + 120596

12(119905) (1199061(119905) minus 119886

1)

+ 12059621(119905) (1198872minus 1199062(119905)) + 120596

22(119905) (1199062(119905) minus 119886

2)

(42)

where 12059611(119905) 12059612(119905) ge 0 120596

21(119905) and 120596



12(119905)(1199061(119905) minus

1198861) = 0 120596

21(119905)(1198872minus 1199062(119905)) = 0 and 120596

22(119905)(1199062(119905) minus 119886

2) = 0 at


2



119889119905= minus

120597119871

120597119878119899

= minus1 +120572 (1 minus 119906

1) 120573119868119886[1205821minus 1205823]

119873

+120572 (1 minus 119906

1) (1 minus 120578) 120573119868

119899[1205821minus 1205823]

119873

+ (1 minus 1199062) 120588 [1205821minus 1205822] + 120583120582

1


1198891205822

119889119905= minus

120597119871

120597119878119886

= 1198611+ 120601 [120582

2minus 1205821] +

(1 minus 1199061) 120573119868119886[1205822minus 1205824]

119873

+(1 minus 119906

1) (1 minus 120578) 120573119868

119899[1205822minus 1205824]

119873+ (120583 + V) 120582

2

1198891205822

119889119905= minus

120597119871

120597119868119899

= 1198612+120572 (1 minus 119906

1) (1 minus 120578) 120573119878

119899[1205821minus 1205823]

119873

+(1 minus 119906

1) (1 minus 120578) 120573119878

119886[1205822minus 1205824]

119873

+ (1 minus 1199062) 120588 [1205823minus 1205824] + 120583120582

3

1198891205822

119889119905= minus

120597119871

120597119868119899

= 1198613+120572 (1 minus 119906

1) 120573119878119899[1205821minus 1205823]

119873

+(1 minus 119906

1) 120573119878119886[1205822minus 1205824]

119873+ 120601 [120582

4minus 1205823] + (120583 + V) 120582

4

(43)




1199061(119905)lowast

=120573 (119868119886+ (1 minus 120578) 119868

119899)

21198601119873

[120572119878119899(1205821minus 1205823) + 119878119886(1205822minus 1205824)]

+12059612(119905) minus 120596

11(119905)

21198601

(44)


1199062(119905)lowast

=120588

21198602


1205871Λ1205822

21198602

+12059622minus 12059621

21198602

(45)


1(119905) and 120596




1199061(119905)lowast

= minmax 1198861120573 (119868119886+ (1 minus 120578) 119868

119899)

21198601119873


1

1199062(119905)lowast

= minmax1198862120588

21198602

[119878119899(1205821minus 1205822) + 119868119899(1205823minus 1205824)]

minus1205871Λ1205822

21198602

1198872

(46)


1 and 119906lowast

2for 1199061(119905) and 119906

2(119905)


119891[60]



1= 0005 119860

2=

0005 1198611= 006 119861

2= 005 and 119861

3= 005 The assumed


= 041198781198860

= 03 1198681198990

= 02 and 1198681198860





036

038

04

042

044

046

048

0 2 4 6 8 10 12 14 16 18 20Time (years)

S n(t)

(a)

024

026

028

03

032

034

036

0 2 4 6 8 10 12 14 16 18 20Time (years)

S a(t)

(b)

005

01

015

02

025

03

035


00 2 4 6 8 10 12 14 16 18 20

Time (years)

I n(t)

(c)


0

005

01

015

02

025

03

035

0 2 4 6 8 10 12 14 16 18 20Time (years)

I a(t)


119899 susceptible




of 20 years






0 2 4 6 8 10 12 14 16 18 200

02

04

06

08

1

Time (years)

u1

(a)

0 2 4 6 8 10 12 14 16 18 200

02

04

06

08

1

Time (years)

u2

(b)


2)



2 The condom






2= 0



(1199062)

6 Discussion




Appendices



1205870(120583 + V) + 120601

120583 + 120601 + 120588 + V1205870

ge119878119899

119873

1205831205871+ 120588

120583 + 120601 + 120588 + V1205870

ge119878119886

119873

(A1)



(i)1205870(120583 + V) + 120601

120583 + 120601 + 120588 + V1205870

lt119878119899

119873

(ii)1205831205871+ 120588

120583 + 120601 + 120588 + V1205870

lt119878119886

119873

(A2)


120583 (1205870+ 1205871) + 120601 + 120588 + 120587

0

120583 + 120601 + 120588 + V1205870

lt119878119886+ 119878119899

119873997904rArr 1 lt 1 (A3)



(i) 1198671ge 2

(ii) 1198672ge 1

(A4)



(i)1205870(120583 + V) + 120601

120583 + 120601 + 120588 + V1205870

ge119878119899

119873

(ii)1205831205871+ 120588

120583 + 120601 + 120588 + V1205870

ge119878119886

119873

(A5)


120583 (1205870+ 1205871) + 120601 + 120588 + V120587

0

120583 + 120601 + 120588 + V1205870

ge119878119886+ 119878119899

119873

997904rArr120583 + 120601 + 120588 + V120587

0

120583 + 120601 + 120588 + V1205870

ge119878119886+ 119878119899

119873997904rArr 1 ge 1

(A6)




1199091015840

= 119891 (119909) (B1)

where 119891 isin 1198621


0) be




120597119909) lt 0 120583(minus120597119891[2]





Acknowledgments


References







































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1198891205822

119889119905= minus

120597119871

120597119878119886

= 1198611+ 120601 [120582

2minus 1205821] +

(1 minus 1199061) 120573119868119886[1205822minus 1205824]

119873

+(1 minus 119906

1) (1 minus 120578) 120573119868

119899[1205822minus 1205824]

119873+ (120583 + V) 120582

2

1198891205822

119889119905= minus

120597119871

120597119868119899

= 1198612+120572 (1 minus 119906

1) (1 minus 120578) 120573119878

119899[1205821minus 1205823]

119873

+(1 minus 119906

1) (1 minus 120578) 120573119878

119886[1205822minus 1205824]

119873

+ (1 minus 1199062) 120588 [1205823minus 1205824] + 120583120582

3

1198891205822

119889119905= minus

120597119871

120597119868119899

= 1198613+120572 (1 minus 119906

1) 120573119878119899[1205821minus 1205823]

119873

+(1 minus 119906

1) 120573119878119886[1205822minus 1205824]

119873+ 120601 [120582

4minus 1205823] + (120583 + V) 120582

4

(43)




1199061(119905)lowast

=120573 (119868119886+ (1 minus 120578) 119868

119899)

21198601119873

[120572119878119899(1205821minus 1205823) + 119878119886(1205822minus 1205824)]

+12059612(119905) minus 120596

11(119905)

21198601

(44)


1199062(119905)lowast

=120588

21198602


1205871Λ1205822

21198602

+12059622minus 12059621

21198602

(45)


1(119905) and 120596




1199061(119905)lowast

= minmax 1198861120573 (119868119886+ (1 minus 120578) 119868

119899)

21198601119873


1

1199062(119905)lowast

= minmax1198862120588

21198602

[119878119899(1205821minus 1205822) + 119868119899(1205823minus 1205824)]

minus1205871Λ1205822

21198602

1198872

(46)


1 and 119906lowast

2for 1199061(119905) and 119906

2(119905)


119891[60]



1= 0005 119860

2=

0005 1198611= 006 119861

2= 005 and 119861

3= 005 The assumed


= 041198781198860

= 03 1198681198990

= 02 and 1198681198860





036

038

04

042

044

046

048

0 2 4 6 8 10 12 14 16 18 20Time (years)

S n(t)

(a)

024

026

028

03

032

034

036

0 2 4 6 8 10 12 14 16 18 20Time (years)

S a(t)

(b)

005

01

015

02

025

03

035


00 2 4 6 8 10 12 14 16 18 20

Time (years)

I n(t)

(c)


0

005

01

015

02

025

03

035

0 2 4 6 8 10 12 14 16 18 20Time (years)

I a(t)


119899 susceptible




of 20 years






0 2 4 6 8 10 12 14 16 18 200

02

04

06

08

1

Time (years)

u1

(a)

0 2 4 6 8 10 12 14 16 18 200

02

04

06

08

1

Time (years)

u2

(b)


2)



2 The condom






2= 0



(1199062)

6 Discussion




Appendices



1205870(120583 + V) + 120601

120583 + 120601 + 120588 + V1205870

ge119878119899

119873

1205831205871+ 120588

120583 + 120601 + 120588 + V1205870

ge119878119886

119873

(A1)



(i)1205870(120583 + V) + 120601

120583 + 120601 + 120588 + V1205870

lt119878119899

119873

(ii)1205831205871+ 120588

120583 + 120601 + 120588 + V1205870

lt119878119886

119873

(A2)


120583 (1205870+ 1205871) + 120601 + 120588 + 120587

0

120583 + 120601 + 120588 + V1205870

lt119878119886+ 119878119899

119873997904rArr 1 lt 1 (A3)



(i) 1198671ge 2

(ii) 1198672ge 1

(A4)



(i)1205870(120583 + V) + 120601

120583 + 120601 + 120588 + V1205870

ge119878119899

119873

(ii)1205831205871+ 120588

120583 + 120601 + 120588 + V1205870

ge119878119886

119873

(A5)


120583 (1205870+ 1205871) + 120601 + 120588 + V120587

0

120583 + 120601 + 120588 + V1205870

ge119878119886+ 119878119899

119873

997904rArr120583 + 120601 + 120588 + V120587

0

120583 + 120601 + 120588 + V1205870

ge119878119886+ 119878119899

119873997904rArr 1 ge 1

(A6)




1199091015840

= 119891 (119909) (B1)

where 119891 isin 1198621


0) be




120597119909) lt 0 120583(minus120597119891[2]





Acknowledgments


References







































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










036

038

04

042

044

046

048

0 2 4 6 8 10 12 14 16 18 20Time (years)

S n(t)

(a)

024

026

028

03

032

034

036

0 2 4 6 8 10 12 14 16 18 20Time (years)

S a(t)

(b)

005

01

015

02

025

03

035


00 2 4 6 8 10 12 14 16 18 20

Time (years)

I n(t)

(c)


0

005

01

015

02

025

03

035

0 2 4 6 8 10 12 14 16 18 20Time (years)

I a(t)


119899 susceptible




of 20 years






0 2 4 6 8 10 12 14 16 18 200

02

04

06

08

1

Time (years)

u1

(a)

0 2 4 6 8 10 12 14 16 18 200

02

04

06

08

1

Time (years)

u2

(b)


2)



2 The condom






2= 0



(1199062)

6 Discussion




Appendices



1205870(120583 + V) + 120601

120583 + 120601 + 120588 + V1205870

ge119878119899

119873

1205831205871+ 120588

120583 + 120601 + 120588 + V1205870

ge119878119886

119873

(A1)



(i)1205870(120583 + V) + 120601

120583 + 120601 + 120588 + V1205870

lt119878119899

119873

(ii)1205831205871+ 120588

120583 + 120601 + 120588 + V1205870

lt119878119886

119873

(A2)


120583 (1205870+ 1205871) + 120601 + 120588 + 120587

0

120583 + 120601 + 120588 + V1205870

lt119878119886+ 119878119899

119873997904rArr 1 lt 1 (A3)



(i) 1198671ge 2

(ii) 1198672ge 1

(A4)



(i)1205870(120583 + V) + 120601

120583 + 120601 + 120588 + V1205870

ge119878119899

119873

(ii)1205831205871+ 120588

120583 + 120601 + 120588 + V1205870

ge119878119886

119873

(A5)


120583 (1205870+ 1205871) + 120601 + 120588 + V120587

0

120583 + 120601 + 120588 + V1205870

ge119878119886+ 119878119899

119873

997904rArr120583 + 120601 + 120588 + V120587

0

120583 + 120601 + 120588 + V1205870

ge119878119886+ 119878119899

119873997904rArr 1 ge 1

(A6)




1199091015840

= 119891 (119909) (B1)

where 119891 isin 1198621


0) be




120597119909) lt 0 120583(minus120597119891[2]





Acknowledgments


References







































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 2 4 6 8 10 12 14 16 18 200

02

04

06

08

1

Time (years)

u1

(a)

0 2 4 6 8 10 12 14 16 18 200

02

04

06

08

1

Time (years)

u2

(b)


2)



2 The condom






2= 0



(1199062)

6 Discussion




Appendices



1205870(120583 + V) + 120601

120583 + 120601 + 120588 + V1205870

ge119878119899

119873

1205831205871+ 120588

120583 + 120601 + 120588 + V1205870

ge119878119886

119873

(A1)



(i)1205870(120583 + V) + 120601

120583 + 120601 + 120588 + V1205870

lt119878119899

119873

(ii)1205831205871+ 120588

120583 + 120601 + 120588 + V1205870

lt119878119886

119873

(A2)


120583 (1205870+ 1205871) + 120601 + 120588 + 120587

0

120583 + 120601 + 120588 + V1205870

lt119878119886+ 119878119899

119873997904rArr 1 lt 1 (A3)



(i) 1198671ge 2

(ii) 1198672ge 1

(A4)



(i)1205870(120583 + V) + 120601

120583 + 120601 + 120588 + V1205870

ge119878119899

119873

(ii)1205831205871+ 120588

120583 + 120601 + 120588 + V1205870

ge119878119886

119873

(A5)


120583 (1205870+ 1205871) + 120601 + 120588 + V120587

0

120583 + 120601 + 120588 + V1205870

ge119878119886+ 119878119899

119873

997904rArr120583 + 120601 + 120588 + V120587

0

120583 + 120601 + 120588 + V1205870

ge119878119886+ 119878119899

119873997904rArr 1 ge 1

(A6)




1199091015840

= 119891 (119909) (B1)

where 119891 isin 1198621


0) be




120597119909) lt 0 120583(minus120597119891[2]





Acknowledgments


References







































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











(i)1205870(120583 + V) + 120601

120583 + 120601 + 120588 + V1205870

lt119878119899

119873

(ii)1205831205871+ 120588

120583 + 120601 + 120588 + V1205870

lt119878119886

119873

(A2)


120583 (1205870+ 1205871) + 120601 + 120588 + 120587

0

120583 + 120601 + 120588 + V1205870

lt119878119886+ 119878119899

119873997904rArr 1 lt 1 (A3)



(i) 1198671ge 2

(ii) 1198672ge 1

(A4)



(i)1205870(120583 + V) + 120601

120583 + 120601 + 120588 + V1205870

ge119878119899

119873

(ii)1205831205871+ 120588

120583 + 120601 + 120588 + V1205870

ge119878119886

119873

(A5)


120583 (1205870+ 1205871) + 120601 + 120588 + V120587

0

120583 + 120601 + 120588 + V1205870

ge119878119886+ 119878119899

119873

997904rArr120583 + 120601 + 120588 + V120587

0

120583 + 120601 + 120588 + V1205870

ge119878119886+ 119878119899

119873997904rArr 1 ge 1

(A6)




1199091015840

= 119891 (119909) (B1)

where 119891 isin 1198621


0) be




120597119909) lt 0 120583(minus120597119891[2]





Acknowledgments


References







































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra



































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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