Mathematical Modeling of Ebola Virus Disease in Bat...

Research ArticleMathematical Modeling of Ebola Virus Disease inBat Population

Zineb EL Rhoubari1 Hajar Besbassi 1 Khalid Hattaf 12 and Noura Yousfi 1

1Laboratory of Analysis Modeling and Simulation (LAMS) Faculty of Sciences Ben Mrsquosik Hassan II UniversityPO Box 7955 Sidi Othman Casablanca Morocco2Centre Regional des Metiers de lrsquoEducation et de la Formation (CRMEF) 20340 Derb Ghalef Casablanca Morocco

Correspondence should be addressed to Khalid Hattaf khattafyahoofr

Received 4 September 2018 Accepted 14 November 2018 Published 2 December 2018

Academic Editor J R Torregrosa

Copyright copy 2018 Zineb EL Rhoubari et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Many scientific researches have demonstrated that bats can survive Ebola virus and they represent natural reservoirs of this zoonoticvirus In this study we develop a generalized epizootic model for the transmission dynamics of Ebola virus disease (EVD) in batpopulation by taking into account the environment contaminationThe transmission process is modeled by two general incidencefunctions that includemany incidence rates used in infectious diseasesmodelingWe first prove that themodel is epidemiologicallyand mathematically well-posed by showing the existence positivity and boundedness of solutions By analyzing the characteristicequations and constructing suitable Lyapunov functionals the stability analysis of equilibria is investigated

1 Introduction

Ebola virus (EBOV) is a filovirus that belongs to the Filoviri-dae family and it causes a severe hemorrhagic disease inhuman and nonhuman primates EBOV is transmitted tothe human population through close contact with the bloodsecretions organs or other bodily fluids of infected animalssuch as fruit bats chimpanzees gorillas monkeys antelopesand porcupines found ill or dead in the rainforest [1] Thefirst outbreak of Ebola virus disease (EVD) appeared in1976 near the Ebola River in the Democratic Republic ofCongo formerly Zaire So far five species of Ebola have beenidentified which are Zaire Sudan Bundibugyo Reston andTaı Forest The first three species have been associated withlarge outbreaks in Africa For example the virus causingthe 2014-2016 West African outbreak belongs to the Zaireebolavirus species [1] The EVD dominated internationalnews in 2014 and the World Health Organization (WHO)reported more than 28000 cases worldwide and over 11000deaths [2]

Bats are recognized as a major reservoir of many viralinfections including Ebola virus and other human viralinfections such as measles and mumps [3ndash5] These species

are long-lived mammals with typically highly synchronousbirthing [6 7] Other studies have shown that bats cansurvive Ebola infection without necessarily becoming ill Forinstance Swanpoel et al [8] experienced that the Ebola virusreplicates in bats without being affected Also a seropositivebat for Ebola has demonstrated a long-term survival livinghealthy over 13months after sampling [9]On the other handthe mechanism of Ebola transmission to nonhuman primatesis still unknown but Leroy et al [3] think that it occurswith consumption of contaminated fruits by bodily fluids ofinfected bats

Over the past years several mathematical models havebeen proposed and developed to describe the dynamicsof EVD [10ndash20] However these models have traditionallytreated the transmission of the disease in human populationand ignored the initial source of Ebola in bats In additionthey do not take into account the contaminated environmentby bodily fluids of infected bats For these mathematical andbiological considerations we propose a generalized epizooticmodel for Ebola that is given by the following nonlinearsystem

119889119878119889119905 = 119860 minus 120583119878 minus 119891 (119878 119868) 119868 minus 119892 (119878 119875) 119875

HindawiDiscrete Dynamics in Nature and SocietyVolume 2018 Article ID 5104524 7 pageshttpsdoiorg10115520185104524

2 Discrete Dynamics in Nature and Society

119889119868119889119905 = 119891 (119878 119868) 119868 + 119892 (119878 119875) 119875 minus (120583 + 119903) 119868119889119877119889119905 = 119903119868 minus 120583119877119889119875119889119905 = 120590119868 minus 120578119875

(1)

where 119878(119905) 119868(119905) and 119877(119905) represent the numbers of sus-ceptible infected and recovered bats at time 119905 respectivelyThen the total population of bats is 119873(119905) = 119878(119905) + 119868(119905) +119877(119905) Further 119875(119905) represents the concentration of EBOVin the environment at time 119905 The susceptible populationincreases at recruitment rate 119860 by births or immigration anddecreases at the natural mortality rate 120583 It also decreasesand converts into the infected subpopulation by direct con-tact with infected bats at rate 119891(119878 119868)119868 or by contact withcontaminated environment at rate 119892(119878 119875)119875 Thus the term119891(119878 119868)119868 + 119892(119878 119875)119875 is the total infection rate of susceptiblepopulation Moreover the infected bats recover from Ebolaat rate 119903 and die only at the natural mortality rate 120583 Finallythe parameter 120590 denotes the deposition rate of EBOV in theenvironment by infected bats and 120578 is the decay rate of EBOVin the environment

As in [21] we suppose that the general incidence func-tions 119891(119878 119868) and 119892(119878 119875) are continuously differentiable in theinterior of R2+ and satisfy the following hypotheses

f (0 I) = 0120597119891120597119878 (119878 119868) gt 0120597119891120597119868 (119878 119868) le 0

for all 119878 119868 ge 0

(1198671)

g (0 P) = 0120597119892120597119878 (119878 119875) gt 0120597119892120597119875 (119878 119875) le 0

for all 119878 119875 ge 0

(1198672)

Epidemiologically the above hypotheses are reasonable andconsistent with the reality In fact the first assumption (1198671)on the function 119891(119878 119868) means that the incidence rate bydirect contact with infected bats is equal to zero if there areno susceptible bats This incidence rate is increasing whenthe number of infected bats is constant and the number ofsusceptible bats increases Also it is decreasing when thenumber of susceptible bats is constant and the number ofinfected bats increases Similarly the second assumption (1198672 )on the function 119892(119878 119875) means that the incidence rate bycontaminated environment is equal to zero if there are nosusceptible bats Besides this incidence rate is increasing

when the concentration of EBOV is constant in the envi-ronment and the number of susceptible bats increases Alsoit is decreasing when the number of susceptible is constantand the concentration of EBOV in the environment increasesTherefore the more susceptible bats are the more infectiousevents will occur However the higher the number of infectedbats or the concentration of EBOV in the environment is theless infectious events will be Further the biological meaningof (1198671) for other diseases is given in [22 23] On the otherhand the schematic representation of model (1) is illustratedin Figure 1

Since the third equation of system (1) is decoupled fromthe other equations it suffices to study the following reducedsystem

119889119878119889119905 = 119860 minus 120583119878 minus 119891 (119878 119868) 119868 minus 119892 (119878 119875) 119875119889119868119889119905 = 119891 (119878 119868) 119868 + 119892 (119878 119875) 119875 minus (120583 + 119903) 119868119889119875119889119905 = 120590119868 minus 120578119875

(2)

It important to note the simple epizootic model for Ebolaproposed by Berge et al [24] is a special case of (2) it sufficesto take 119891(119878 119868) = 1205731119878 119892(119878 119868) = 1205732119878 and 119903 = 0 This simplemodel used a bilinear incidence rate that is based on theprinciple of mass action However such bilinear incidencerate is not perfect to describe minutely the dynamics ofthe infectious diseases Also Berge et al [24] assumed thatinfected bats did not recover during Ebola outbreaksThis lasthypothesis has been rejected by many researchers [25ndash27]

Our main purpose of this study is to study the dynamicsof our generalized epizootic model presented by system (2)To do this we organize the rest of the paper as follows Thenext section deals with the well-posedness of the model andthe existence of equilibria The stability analysis of the cor-responding equilibria is established in Section 3 by applyingdirect and indirect Lyapunov method An application andsome numerical simulations are given in Section 4 in orderto confirm and illustrate our analytical results Finally somebiological and mathematical conclusions are summarized inSection 5

2 Well-Posedness and Equilibria

For system (2) to be biologically meaningful it is necessaryto show the nonnegativity and boundedness of solutions

Theorem 1 The set R3+ is positively invariant with respectto system (2) Furthermore all solutions of (2) are uniformlybounded in the compact subset

Γ = (119878 119868 119875) isin R3+ 119878 + 119868 le 119860

120583 119875 le120590119860120583120578 (3)

Proof We have

11988911987811988911990510038161003816100381610038161003816100381610038161003816119878=0 = 119860 gt 0

Discrete Dynamics in Nature and Society 3

f(SI)I

g(SP)０

A

rS I

P

R

Figure 1 Schematic representation of model (1)

11988911986811988911990510038161003816100381610038161003816100381610038161003816119868=0 = 0

11988911987511988911990510038161003816100381610038161003816100381610038161003816119875=0 = 120590119868 ge 0 for all 119868 ge 0

(4)

This proves the positively invariant property of R3+ withrespect to system (2) Let 119876(119905) = 119878(119905) + 119868(119905) then

119889119876119889119905 = 119860 minus 120583119878 minus (120583 + 119903) 119868 le 119860 minus 120583119876 (5)

Hence

lim sup119905997888rarrinfin

119876 (119905) le 119860120583 (6)

This implies that 119878 and 119868 are uniformly bounded in the regionΓ Furthermore from the bound for 119868 and the last equation of(2) it follows that


119875 (119905) le 120590119860120583120578 (7)

This guarantees the boundedness of 119875 This completes theproof

Now we compute the basic reproduction number andprove the existence of two equilibria Obviously model (2)always has a disease-free equilibrium 119864119891(119860120583 0 0) Then thebasic reproduction number of (2) is given by

1198770 = 120578119891 (119860120583 0) + 120590119892 (119860119906 0)120578 (120583 + 119903) (8)

The other equilibrium of system (2) satisfies the followingequations

119860 minus 120583119878 minus 119891 (119878 119868) 119868 minus 119892 (119878 119875) 119875 = 0 (9)

119891 (119878 119868) 119868 + 119892 (119878 119875) 119875 minus (120583 + 119903) 119868 = 0 (10)

120590119868 minus 120578119875 = 0 (11)

By (9) to (11) we have

119891(119878 119860 minus 120583119878120583 + 119903 ) +120590120578119892(119878

120590 (119860 minus 120583119878)120578 (120583 + 119903) ) = 120583 + 119903 (12)

Since 119868 = (119860 minus 120583119878)(120583 + 119903) ge 0 we have 119878 le 119860120583 So there isno equilibrium when 119878 gt 119860120583 Hence we define a function120595 on the interval [0 119860120583] by

120595 (119878) = 119891(119878 119860 minus 120583119878119903 + 120583 ) + 120590120578119892(119878

120590 (119860 minus 120583119878)120578 (119903 + 120583) ) minus 120583

minus 119903(13)

We have 120595(0) = minus119903 minus 120583 lt 0 120595(119860120583) = (119903 + 120583)(1198770 minus 1) and1205951015840 (119878) = 120597119891

120597119878 minus120583

119903 + 120583120597119891120597119868 +

120590120578 (

120597119892120597119878 minus

120583120590120578 (119903 + 120583)

120597119892120597119875)

gt 0(14)

Thus for 1198770 gt 1 there is one endemic equilibrium 119864lowast(119878lowast119868lowast 119875lowast) with 119878lowast isin (0 119860120583) and 119868lowast 119875lowast gt 0Therefore we summarize the above discussions in the

following result

Theorem 2 Let 1198770 be defined by (8)(i) If 1198770 le 1 then model (2) admits only one disease-free

equilibrium 119864119891(1198780 0 0) where 1198780 = 119860120583(ii) If 1198770 gt 1 in addition to 119864119891 model (2) has a unique

endemic equilibrium 119864lowast(119878lowast 119868lowast 119875lowast) with 119878lowast isin (0 119860120583) 119868lowast = (119860minus120583119878lowast)(120583+119903) and119875lowast = 120590(119860minus120583119878lowast)120578(120583+119903)3 Stability Analysis

The aim of this section is to analyze the stability of the twoequilibria 119864119891 and 119864lowast First we have the following resultTheorem3 Thedisease-free equilibrium119864119891 is globally asymp-totically stable when 1198770 le 1 and it becomes unstable when1198770 gt 1


Proof To show the first part of this theorem we consider thefollowing Lyapunov functional

119881(119905) = 120578119868 + 119892 (119860120583 0)119875 (15)

Differentiating 119881 with respect to 119905 along the solutions of (2)we get

(119905)10038161003816100381610038161003816(2) = (120578119891 (119878 119868) minus 120578 (120583 + 119903) + 120590119892(119860120583 0)) 119868

+ 120578(119892 (119878 119875) minus 119892(119860120583 0))119875(16)

We have lim sup119905997888rarrinfin119878(119905) le 119860120583 which implies that each120596-limit point satisfies 119878(119905) le 119860120583 Hence it is sufficient toconsider solutions for which 119878(119905) le 119860120583 From (8) and (1198671)-(1198672) we get

(119905)10038161003816100381610038161003816(2) le (120578119891(119860120583 0) minus 120578 (120583 + 119903) + 120590119892 (119860120583 0)) 119868

le 120578 (120583 + 119903) (1198770 minus 1) 119868(17)

Consequently (119905)|(2) le 0 for 1198770 le 1 Furthermore it is easyto check that the largest invariant subset of (119878 119868 119875) | (119905) =0 is 119864119891 By LaSallersquos invariance principle [28] we get that119864119891 is globally asymptotically stable when 1198770 le 1

In order to show the remaining part we determine thecharacteristic equation at the disease-free equilibrium119864119891 thatis given by

(120583 + 120582) (1205822 + 120582(120578 + 120583 + 119903 minus 119891(119860120583 0))

+ 120578 (120583 + 119903) (1 minus 1198770)) = 0(18)

Let

Φ (120582) = 1205822 + 120582(120578 + 120583 + 119903 minus 119891(119860120583 0))+ 120578 (120583 + 119903) (1 minus 1198770)

(19)

We have lim120582997888rarr+infinΦ(120582) = +infin and Φ(0) = 120578(120583 + 119903)(1 minus 1198770)If1198770 gt 1 thenΦ(0) lt 0Thus there exists a positive real rootof the characteristic equation (18) and hence the disease-freeequilibrium 119864119891 is unstable whenever 1198770 gt 1

For the global stability of the endemic equilibrium 119864lowast weassume that 1198770 gt 1 and the functions 119891 and 119892 satisfy for all119878 119868 119875 gt 0 the following hypothesis

(1 minus 119891 (119878 119868)119891 (119878 119868lowast))(

119891 (119878 119868lowast)119891 (119878 119868) minus 119868

119868lowast) le 0

(1 minus 119891 (119878lowast 119868lowast) 119892 (119878 119875)119891 (119878 119868lowast) 119892 (119878lowast 119875lowast))

sdot (119891 (119878 119868lowast) 119892 (119878lowast 119875lowast)119891 (119878lowast 119868lowast) 119892 (119878 119875) minus 119875119875lowast) le 0

(1198673)

Theorem 4 If 1198770 gt 1 and (1198673) holds then the endemic equi-librium 119864lowast is globally asymptotically stable

Proof Define a Lyapunov functional as follows

119882(119905) = 119878 (119905) minus 119878lowast minus int119878(119905)119878lowast

119891 (119878lowast 119868lowast)119891 (119883 119868lowast) 119889119883

+ 119868lowastΦ(119868 (119905)119868lowast ) +119892 (119878lowast 119875lowast)

120578 119875lowastΦ(119875 (119905)119875lowast ) (20)

whereΦ(119883) = 119883minus1minus ln119883119883 gt 0 It is obvious thatΦ attainsits strict global minimum at 1 and Φ(1) = 0 Then Φ(119883) ge 0and the functional119882 is nonnegative

For convenience we let 120593 = 120593(119905) for any 120593 isin 119878 119868 119875Differentiating119882 with respect to 119905 along the solutions of (2)we obtain

(119905)10038161003816100381610038161003816(2) = (1 minus 119891 (119878lowast 119868lowast)119891 (119878 119868lowast) )

119889119878119889119905 + (1 minus

119868lowast119868 )

119889119868119889119905

+ 119892 (119878lowast 119875lowast)120578 (1 minus 119875lowast119875 ) 119889119875119889119905

(21)

Using 119860 = 120583119878lowast +119891(119878lowast 119868lowast)119868lowast +119892(119878lowast 119875lowast)119875lowast = 120583119878lowast + (120583 + 119903)119868lowastand 120590119868lowast = 120578119875lowast we get

(119905)10038161003816100381610038161003816(2) = 120583119878lowast (1 minus 119878119878lowast )(1 minus

119891 (119878lowast 119868lowast)119891 (119878 119868lowast) )

+ 119891 (119878lowast 119868lowast) 119868lowast (2 minus 119891 (119878lowast 119868lowast)119891 (119878 119868lowast) minus 119891 (119878 119868)

119891 (119878lowast 119868lowast)+ 119891 (119878 119868) 119868119891 (119878 119868lowast) 119868lowast minus

119868119868lowast) + 119892 (119878lowast 119875lowast) 119875lowast (3

minus 119891 (119878lowast 119868lowast)119891 (119878 119868lowast) + 119891 (119878lowast 119868lowast) 119892 (119878 119875) 119875119891 (119878 119868lowast) 119892 (119878lowast 119875lowast) 119875lowast minus

119875119875lowast

minus 119892 (119878 119875) 119875119868lowast119892 (119878lowast 119875lowast) 119875lowast119868 minus

119875lowast119868119875119868lowast )

(22)

Thus



+ 119891 (119878lowast 119868lowast) 119868lowast (minus1 minus 119868119868lowast +

119891 (119878 119868lowast)119891 (119878 119868)

+ 119891 (119878 119868) 119868119891 (119878 119868lowast) 119868lowast) + 119892 (119878lowast 119875lowast) 119875lowast (minus1 minus

119875119875lowast

+ 119891 (119878 119868lowast) 119892 (119878lowast 119875lowast)119891 (119878lowast 119868lowast) 119892 (119878 119875) + 119891 (119878lowast 119868lowast) 119892 (119878 119875) 119875

119891 (119878 119868lowast) 119892 (119878lowast 119875lowast) 119875lowast)

minus 119891 (119878lowast 119868lowast) 119868lowast [Φ(119891 (119878lowast 119868lowast)119891 (119878 119868lowast) )

+ Φ( 119891 (119878 119868)119891 (119878lowast 119868lowast)) + Φ(

119891 (119878 119868lowast)119891 (119878 119868) )] minus 119892 (119878lowast 119875lowast)


sdot 119875lowast [Φ(119891 (119878lowast 119868lowast)119891 (119878 119868lowast) ) + Φ(119875lowast119868119875119868lowast )

+ Φ( 119892 (119878 119875) 119875119868lowast119892 (119878lowast 119875lowast) 119875lowast119868)

+ Φ(119891 (119878 119868lowast) 119892 (119878lowast 119875lowast)119891 (119878lowast 119868lowast) 119892 (119878 119875) )] (23)

By using (1198671) we easily obtain the following inequality

(1 minus 119878119878lowast)(1 minus

119891 (119878lowast 119868lowast)119891 (119878 119868lowast) ) le 0 (24)

From (1198673) we obtainminus 1 minus 119868

119868lowast +119891 (119878 119868lowast)119891 (119878 119868) + 119891 (119878 119868) 119868

119891 (119878 119868lowast) 119868lowast= (1 minus 119891 (119878 119868)

119891 (119878 119868lowast)) (119891 (119878 119868lowast)119891 (119878 119868) minus 119868

119868lowast) le 0(25)

and

minus 1 minus 119875119875lowast +

119891 (119878 119868lowast) 119892 (119878lowast 119875lowast)119891 (119878lowast 119868lowast) 119892 (119878 119875)

+ 119891 (119878lowast 119868lowast) 119892 (119878 119875) 119875119891 (119878 119868lowast) 119892 (119878lowast 119875lowast) 119875lowast

= (1 minus 119891 (119878lowast 119868lowast) 119892 (119878 119875)119891 (119878 119868lowast) 119892 (119878lowast 119875lowast))


(26)

Since Φ(119883) ge 0 for 119883 gt 0 we deduce that |(2) le 0 and theequality occurs at 119864lowast Consequently the global asymptoticstability of 119864lowast follows fromLaSallersquos invariance principle

4 Application

The main purpose of this section is to apply our previousresults to the following model

119889119878119889119905 = 119860 minus 120583119878 minus

12057311198781198681 + 1205721119868 minus12057321198781198751 + 1205722119875

119889119868119889119905 =

12057311198781198681 + 1205721119868 +12057321198781198751 + 1205722119875 minus (120583 + 119903) 119868

119889119875119889119905 = 120590119868 minus 120578119875

(27)

where1205731 and1205732 are the infection rates caused by infected batsand contaminated environment respectively The nonnega-tive constants 1205721 and 1205722 measure the saturation effect Theremaining parameters have the same biological significanceas those in system (2)

Clearly system (27) is a particular case of (2) with119891(119878 119868) = 1205731119878(1 + 1205721119868) and 119892(119878 119875) = 1205732119878(1 + 1205722119875) Basedon the previous sections system (27) has one disease-freeequilibrium 119864119891(119860120583 0 0) and a unique endemic equilibrium119864lowast(119878lowast 119868lowast 119875lowast) when 1198770 = 1198601205731120583(120583 + 119903) + 1205901198601205732120583120578(120583 +119903) gt 1 In addition it is clear that the hypotheses (1198671)-(1198673)are checked for both incidence functions 119891 and 119892 ApplyingTheorems 3 and 4 we deduce the following corollary

Corollary 5

(i) When 1198770 le 1 the disease-free equilibrium119864119891 of model(27) is globally asymptotically stable

(ii) When 1198770 gt 1 119864119891 becomes unstable and the endemicequilibrium 119864lowast of model (27) is globally asymptoticallystable

Now we simulate system (27) with the following param-eter values 119860 = 15 120583 = 00003 1205731 = 45 times 10minus6 1205732 =13 times 10minus4 1205721 = 001 1205722 = 001 119903 = 0041 120590 = 002 and120578 = 08 By a simple calculation we get 1198770 = 09383 Thensystem (27) has one disease-free equilibrium 119864119891(5000 0 0) Itfollows from Corollary 5(i) that 119864119891 is globally asymptoticallystable and the solution of (27) converges to 119864119891 as shown inFigure 2 In this case the disease dies out

Next we simulate the case when the basic reproductionnumber is bigger than one For this end we choose 1205731 =25 times 10minus5 and do not change the other parameter valuesWe get 1198770 = 34201 gt 1 Hence system (27) has a uniqueendemic equilibrium 119864lowast(17598457 235361 05887) From(ii) of Corollary 5 119864lowast is globally asymptotically stable whichmeans that the Ebola virus persists in the bat populationand the disease becomes endemic This result is illustrated byFigure 3

5 Conclusions

In this article we have proposed and investigated a newgeneralized epizootic model that describes the transmis-sion dynamics of Ebola virus disease in bat populationThe transmission process from bat-to-bat and contaminatedenvironment-to-bat is modeled by two general nonlinearfunctions which cover many special cases using in theprevious studies such as the saturated incidence the classi-cal bilinear incidence the Beddington-DeAngelis functionalresponse the Hattaf-Yousfi functional response and theCrowley-Martin functional response The well-posedness ofthe proposed model and the stability analysis of equilibriaare rigorously studied More precisely we have establishedthe existence uniqueness nonnegativity and boundednessof solutions By using appropriate Lyapunov functionalsand linearization technique we have proved the first steadystate 119864119891 is globally asymptotically stable when 1198770 le 1which means that the disease dies out in the bat populationHowever when 1198770 gt 1 119864119891 becomes unstable and themodel has an endemic steady state 119864lowast which is globallyasymptotically stable This leads to the persistence of diseasein the bat population when 1198770 gt 1


1000

2000

3000

4000

5000

Susc

eptib

le S

0

1

2

3

4

5

Conc

entr

atio

n P

0 2000 4000 6000

minus1000

100minus5

0

5

SI

P

0

10

20

30

40

50

Infe

cted

I

5000 10000 150000Days

5000 10000 150000Days

5000 10000 150000Days

Figure 2 Demonstration of the global stability of the disease-free equilibrium 119864119891 for 1198770 = 09383 le 1

0 2000 4000 6000

0100

2000

5

SI

P

0

50

100

150

200

Infe

cted

I

1000 2000 3000 40000Days

1000 2000 3000 40000Days

1000

2000

3000

4000

5000

Susc

eptib

le S

1000 2000 3000 40000Days

0

2

4

6

Con

cent

ratio

n P

Figure 3 Demonstration of the global stability of the endemic equilibrium 119864lowast for 1198770 = 34201 gt 1

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] WHO ldquoEbola virus diseaserdquo httpwwwwhointennews-roomfact-sheetsdetailebola-virus-disease

[2] WHO ldquoEbola situation reportrdquo httpwwwwhointcsrdis-easeebolasituation-reportsarchiveen

[3] E M Leroy B Kumulungui X Pourrut et al ldquoFruit bats asreservoirs of Ebola virusrdquo Nature vol 438 no 7068 pp 575-576 2005

[4] X Pourrut M Souris J S Towner et al ldquoLarge serologicalsurvey showing cocirculation of Ebola and Marburg viruses inGabonese bat populations and a high seroprevalence of bothviruses in Rousettus aegyptiacusrdquo BMC Infectious Diseases vol9 p 159 2009

[5] D T S Hayman ldquoBats as Viral Reservoirsrdquo Annual Review ofVirology vol 3 pp 77ndash99 2016

[6] D T S Hayman R McCrea O Restif et al ldquoDemography ofstraw-colored fruit bats in Ghanardquo Journal of Mammalogy vol93 no 5 pp 1393ndash1404 2012

[7] E G Crichton and P H Krutzsch Krutzsch Reproductivebiology of bats Academic Press 2000


[8] R Swanepoel P A Leman F J Burt et al ldquoExperimentalinoculation of plants and animals with Ebola virusrdquo EmergingInfectious Diseases vol 2 no 4 pp 321ndash325 1996

[9] D T S Hayman P EmmerichM Yu et al ldquoLong-term survivalof anurban fruit bat seropositive for ebola and lagos bat virusesrdquoPLoS ONE vol 5 no 8 Article ID e11978 2010

[10] X-S Wang and L Zhong ldquoEbola outbreak inWest Africa real-time estimation and multiple-wave predictionrdquo MathematicalBiosciences and Engineering vol 12 no 5 pp 1055ndash1063 2015

[11] A Rachah and D F M Torres ldquoMathematical modellingsimulation and optimal control of the 2014 Ebola outbreak inWest AfricardquoDiscrete Dynamics in Nature and Society vol 2015Article ID 842792 9 pages 2015

[12] A Rachah and D F Torres ldquoDynamics and optimal control ofEbola transmissionrdquo Mathematics in Computer Science vol 10no 3 pp 331ndash342 2016

[13] A Rachah and D F Torres ldquoPredicting and controlling theEbola infectionrdquoMathematical Methods in the Applied Sciencesvol 40 no 17 pp 6155ndash6164 2017

[14] C Althaus ldquoEstimating the reproduction number of Ebolavirus (EBOV) during the 2014 outbreak in West Africardquo PLOSCurrents Outbreaks 2014

[15] G Chowell N W Hengartner C Castillo-Chavez P WFenimore and J M Hyman ldquoThe basic reproductive numberof Ebola and the effects of public health measures the cases ofCongo and Ugandardquo Journal of Theoretical Biology vol 229 no1 pp 119ndash126 2004

[16] E V Grigorieva and E N Khailov ldquoOptimal vaccination treat-ment and preventive campaigns in regard to the SIR epidemicmodelrdquo Mathematical Modelling of Natural Phenomena vol 9no 4 pp 105ndash121 2014

[17] I Area H Batarfi J Losada J J Nieto W Shammakh and ATorres ldquoOn a fractional order Ebola epidemic modelrdquoAdvancesin Difference Equations vol 2015 article 278 2015

[18] A K Jones ldquoGreen computing new challenges and opportuni-tiesrdquo in Proceedings of the on Great Lakes Symposium on VLSIp 3 ACM 2017

[19] Z Xia S Wang S Li et al ldquoModeling the transmissiondynamics of Ebola virus disease in Liberiardquo Scientific Reportsvol 5 no 1 2015

[20] Jia-Ming Zhu Lu Wang and Jia-Bao Liu ldquoEradication of EbolaBased on Dynamic Programmingrdquo Computational and Math-ematical Methods in Medicine vol 2016 Article ID 1580917 9pages 2016

[21] K Hattaf A A Lashari Y Louartassi and N Yousfi ldquoA delayedSIR epidemic model with general incidence raterdquo ElectronicJournal of Qualitative Theory of Differential Equations No 3 9pages 2013

[22] X-Y Wang K Hattaf H-F Huo and H Xiang ldquoStabilityanalysis of a delayed social epidemics model with generalcontact rate and its optimal controlrdquo Journal of Industrial andManagement Optimization vol 12 no 4 pp 1267ndash1285 2016

[23] K Hattaf andN Yousfi ldquoA numericalmethod for a delayed viralinfection model with general incidence raterdquo Journal of KingSaud University - Science vol 28 no 4 pp 368ndash374 2016

[24] T Berge J Lubuma A J O Tasse andHM Tenkam ldquoDynam-ics of host-reservoir transmission of Ebola with spilloverpotential to humansrdquo Electronic Journal of Qualitative Theoryof Differential Equations vol 14 pp 1ndash32 2018

[25] B R Amman S A Carroll Z D Reed et al ldquoSeasonalPulses of Marburg Virus Circulation in Juvenile Rousettus

aegyptiacus Bats Coincide with Periods of Increased Risk ofHuman Infectionrdquo PLoS Pathogens vol 8 no 10 p e10028772012

[26] J Buceta andK Johnson ldquoModeling the Ebola zoonotic dynam-ics Interplay between enviroclimatic factors and bat ecologyrdquoPLoS ONE vol 12 no 6 2017

[27] G Fiorillo P Bocchini and J Buceta ldquoA Predictive SpatialDistribution Framework for Filovirus-Infected Batsrdquo ScientificReports vol 8 no 1 2018

[28] J P LaSalleThe Stability of Dynamical Systems SIAM Philadel-phia Pa USA 1976

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119889119868119889119905 = 119891 (119878 119868) 119868 + 119892 (119878 119875) 119875 minus (120583 + 119903) 119868119889119877119889119905 = 119903119868 minus 120583119877119889119875119889119905 = 120590119868 minus 120578119875

(1)

where 119878(119905) 119868(119905) and 119877(119905) represent the numbers of sus-ceptible infected and recovered bats at time 119905 respectivelyThen the total population of bats is 119873(119905) = 119878(119905) + 119868(119905) +119877(119905) Further 119875(119905) represents the concentration of EBOVin the environment at time 119905 The susceptible populationincreases at recruitment rate 119860 by births or immigration anddecreases at the natural mortality rate 120583 It also decreasesand converts into the infected subpopulation by direct con-tact with infected bats at rate 119891(119878 119868)119868 or by contact withcontaminated environment at rate 119892(119878 119875)119875 Thus the term119891(119878 119868)119868 + 119892(119878 119875)119875 is the total infection rate of susceptiblepopulation Moreover the infected bats recover from Ebolaat rate 119903 and die only at the natural mortality rate 120583 Finallythe parameter 120590 denotes the deposition rate of EBOV in theenvironment by infected bats and 120578 is the decay rate of EBOVin the environment

As in [21] we suppose that the general incidence func-tions 119891(119878 119868) and 119892(119878 119875) are continuously differentiable in theinterior of R2+ and satisfy the following hypotheses

f (0 I) = 0120597119891120597119878 (119878 119868) gt 0120597119891120597119868 (119878 119868) le 0

for all 119878 119868 ge 0

(1198671)

g (0 P) = 0120597119892120597119878 (119878 119875) gt 0120597119892120597119875 (119878 119875) le 0

for all 119878 119875 ge 0

(1198672)

Epidemiologically the above hypotheses are reasonable andconsistent with the reality In fact the first assumption (1198671)on the function 119891(119878 119868) means that the incidence rate bydirect contact with infected bats is equal to zero if there areno susceptible bats This incidence rate is increasing whenthe number of infected bats is constant and the number ofsusceptible bats increases Also it is decreasing when thenumber of susceptible bats is constant and the number ofinfected bats increases Similarly the second assumption (1198672 )on the function 119892(119878 119875) means that the incidence rate bycontaminated environment is equal to zero if there are nosusceptible bats Besides this incidence rate is increasing

when the concentration of EBOV is constant in the envi-ronment and the number of susceptible bats increases Alsoit is decreasing when the number of susceptible is constantand the concentration of EBOV in the environment increasesTherefore the more susceptible bats are the more infectiousevents will occur However the higher the number of infectedbats or the concentration of EBOV in the environment is theless infectious events will be Further the biological meaningof (1198671) for other diseases is given in [22 23] On the otherhand the schematic representation of model (1) is illustratedin Figure 1

Since the third equation of system (1) is decoupled fromthe other equations it suffices to study the following reducedsystem

119889119878119889119905 = 119860 minus 120583119878 minus 119891 (119878 119868) 119868 minus 119892 (119878 119875) 119875119889119868119889119905 = 119891 (119878 119868) 119868 + 119892 (119878 119875) 119875 minus (120583 + 119903) 119868119889119875119889119905 = 120590119868 minus 120578119875

(2)

It important to note the simple epizootic model for Ebolaproposed by Berge et al [24] is a special case of (2) it sufficesto take 119891(119878 119868) = 1205731119878 119892(119878 119868) = 1205732119878 and 119903 = 0 This simplemodel used a bilinear incidence rate that is based on theprinciple of mass action However such bilinear incidencerate is not perfect to describe minutely the dynamics ofthe infectious diseases Also Berge et al [24] assumed thatinfected bats did not recover during Ebola outbreaksThis lasthypothesis has been rejected by many researchers [25ndash27]

Our main purpose of this study is to study the dynamicsof our generalized epizootic model presented by system (2)To do this we organize the rest of the paper as follows Thenext section deals with the well-posedness of the model andthe existence of equilibria The stability analysis of the cor-responding equilibria is established in Section 3 by applyingdirect and indirect Lyapunov method An application andsome numerical simulations are given in Section 4 in orderto confirm and illustrate our analytical results Finally somebiological and mathematical conclusions are summarized inSection 5

2 Well-Posedness and Equilibria

For system (2) to be biologically meaningful it is necessaryto show the nonnegativity and boundedness of solutions

Theorem 1 The set R3+ is positively invariant with respectto system (2) Furthermore all solutions of (2) are uniformlybounded in the compact subset

Γ = (119878 119868 119875) isin R3+ 119878 + 119868 le 119860

120583 119875 le120590119860120583120578 (3)

Proof We have

11988911987811988911990510038161003816100381610038161003816100381610038161003816119878=0 = 119860 gt 0


f(SI)I

g(SP)０

A

rS I

P

R


11988911986811988911990510038161003816100381610038161003816100381610038161003816119868=0 = 0

11988911987511988911990510038161003816100381610038161003816100381610038161003816119875=0 = 120590119868 ge 0 for all 119868 ge 0

(4)



Hence


119876 (119905) le 119860120583 (6)



119875 (119905) le 120590119860120583120578 (7)



1198770 = 120578119891 (119860120583 0) + 120590119892 (119860119906 0)120578 (120583 + 119903) (8)



119891 (119878 119868) 119868 + 119892 (119878 119875) 119875 minus (120583 + 119903) 119868 = 0 (10)

120590119868 minus 120578119875 = 0 (11)


119891(119878 119860 minus 120583119878120583 + 119903 ) +120590120578119892(119878

120590 (119860 minus 120583119878)120578 (120583 + 119903) ) = 120583 + 119903 (12)


120595 (119878) = 119891(119878 119860 minus 120583119878119903 + 120583 ) + 120590120578119892(119878

120590 (119860 minus 120583119878)120578 (119903 + 120583) ) minus 120583

minus 119903(13)


120597119878 minus120583

119903 + 120583120597119891120597119868 +

120590120578 (

120597119892120597119878 minus

120583120590120578 (119903 + 120583)

120597119892120597119875)

gt 0(14)


following result







119881(119905) = 120578119868 + 119892 (119860120583 0)119875 (15)


(119905)10038161003816100381610038161003816(2) = (120578119891 (119878 119868) minus 120578 (120583 + 119903) + 120590119892(119860120583 0)) 119868

+ 120578(119892 (119878 119875) minus 119892(119860120583 0))119875(16)


(119905)10038161003816100381610038161003816(2) le (120578119891(119860120583 0) minus 120578 (120583 + 119903) + 120590119892 (119860120583 0)) 119868

le 120578 (120583 + 119903) (1198770 minus 1) 119868(17)



(120583 + 120582) (1205822 + 120582(120578 + 120583 + 119903 minus 119891(119860120583 0))

+ 120578 (120583 + 119903) (1 minus 1198770)) = 0(18)

Let

Φ (120582) = 1205822 + 120582(120578 + 120583 + 119903 minus 119891(119860120583 0))+ 120578 (120583 + 119903) (1 minus 1198770)

(19)



(1 minus 119891 (119878 119868)119891 (119878 119868lowast))(

119891 (119878 119868lowast)119891 (119878 119868) minus 119868

119868lowast) le 0



(1198673)






120578 119875lowastΦ(119875 (119905)119875lowast ) (20)




119889119878119889119905 + (1 minus

119868lowast119868 )

119889119868119889119905


(21)








119875119875lowast


119875lowast119868119875119868lowast )

(22)

Thus




119891 (119878 119868lowast)119891 (119878 119868)


119875119875lowast




+ Φ( 119891 (119878 119868)119891 (119878lowast 119868lowast)) + Φ(










119868lowast +119891 (119878 119868lowast)119891 (119878 119868) + 119891 (119878 119868) 119868



119868lowast) le 0(25)

and






(26)


4 Application


119889119878119889119905 = 119860 minus 120583119878 minus

12057311198781198681 + 1205721119868 minus12057321198781198751 + 1205722119875

119889119868119889119905 =

12057311198781198681 + 1205721119868 +12057321198781198751 + 1205722119875 minus (120583 + 119903) 119868

119889119875119889119905 = 120590119868 minus 120578119875

(27)



Corollary 5





5 Conclusions



1000

2000

3000

4000

5000

Susc

eptib

le S

0

1

2

3

4

5

Conc

entr

atio

n P

0 2000 4000 6000

minus1000

100minus5

0

5

SI

P

0

10

20

30

40

50

Infe

cted

I

5000 10000 150000Days

5000 10000 150000Days

5000 10000 150000Days


0 2000 4000 6000

0100

2000

5

SI

P

0

50

100

150

200

Infe

cted

I

1000 2000 3000 40000Days

1000 2000 3000 40000Days

1000

2000

3000

4000

5000

Susc

eptib

le S

1000 2000 3000 40000Days

0

2

4

6

Con

cent

ratio

n P


Data Availability




References






































Journal of












Journal of







Volume 2018






Volume 2018









f(SI)I

g(SP)０

A

rS I

P

R


11988911986811988911990510038161003816100381610038161003816100381610038161003816119868=0 = 0

11988911987511988911990510038161003816100381610038161003816100381610038161003816119875=0 = 120590119868 ge 0 for all 119868 ge 0

(4)



Hence


119876 (119905) le 119860120583 (6)



119875 (119905) le 120590119860120583120578 (7)



1198770 = 120578119891 (119860120583 0) + 120590119892 (119860119906 0)120578 (120583 + 119903) (8)



119891 (119878 119868) 119868 + 119892 (119878 119875) 119875 minus (120583 + 119903) 119868 = 0 (10)

120590119868 minus 120578119875 = 0 (11)


119891(119878 119860 minus 120583119878120583 + 119903 ) +120590120578119892(119878

120590 (119860 minus 120583119878)120578 (120583 + 119903) ) = 120583 + 119903 (12)


120595 (119878) = 119891(119878 119860 minus 120583119878119903 + 120583 ) + 120590120578119892(119878

120590 (119860 minus 120583119878)120578 (119903 + 120583) ) minus 120583

minus 119903(13)


120597119878 minus120583

119903 + 120583120597119891120597119868 +

120590120578 (

120597119892120597119878 minus

120583120590120578 (119903 + 120583)

120597119892120597119875)

gt 0(14)


following result







119881(119905) = 120578119868 + 119892 (119860120583 0)119875 (15)


(119905)10038161003816100381610038161003816(2) = (120578119891 (119878 119868) minus 120578 (120583 + 119903) + 120590119892(119860120583 0)) 119868

+ 120578(119892 (119878 119875) minus 119892(119860120583 0))119875(16)


(119905)10038161003816100381610038161003816(2) le (120578119891(119860120583 0) minus 120578 (120583 + 119903) + 120590119892 (119860120583 0)) 119868

le 120578 (120583 + 119903) (1198770 minus 1) 119868(17)



(120583 + 120582) (1205822 + 120582(120578 + 120583 + 119903 minus 119891(119860120583 0))

+ 120578 (120583 + 119903) (1 minus 1198770)) = 0(18)

Let

Φ (120582) = 1205822 + 120582(120578 + 120583 + 119903 minus 119891(119860120583 0))+ 120578 (120583 + 119903) (1 minus 1198770)

(19)



(1 minus 119891 (119878 119868)119891 (119878 119868lowast))(

119891 (119878 119868lowast)119891 (119878 119868) minus 119868

119868lowast) le 0



(1198673)






120578 119875lowastΦ(119875 (119905)119875lowast ) (20)




119889119878119889119905 + (1 minus

119868lowast119868 )

119889119868119889119905


(21)








119875119875lowast


119875lowast119868119875119868lowast )

(22)

Thus




119891 (119878 119868lowast)119891 (119878 119868)


119875119875lowast




+ Φ( 119891 (119878 119868)119891 (119878lowast 119868lowast)) + Φ(










119868lowast +119891 (119878 119868lowast)119891 (119878 119868) + 119891 (119878 119868) 119868



119868lowast) le 0(25)

and






(26)


4 Application


119889119878119889119905 = 119860 minus 120583119878 minus

12057311198781198681 + 1205721119868 minus12057321198781198751 + 1205722119875

119889119868119889119905 =

12057311198781198681 + 1205721119868 +12057321198781198751 + 1205722119875 minus (120583 + 119903) 119868

119889119875119889119905 = 120590119868 minus 120578119875

(27)



Corollary 5





5 Conclusions



1000

2000

3000

4000

5000

Susc

eptib

le S

0

1

2

3

4

5

Conc

entr

atio

n P

0 2000 4000 6000

minus1000

100minus5

0

5

SI

P

0

10

20

30

40

50

Infe

cted

I

5000 10000 150000Days

5000 10000 150000Days

5000 10000 150000Days


0 2000 4000 6000

0100

2000

5

SI

P

0

50

100

150

200

Infe

cted

I

1000 2000 3000 40000Days

1000 2000 3000 40000Days

1000

2000

3000

4000

5000

Susc

eptib

le S

1000 2000 3000 40000Days

0

2

4

6

Con

cent

ratio

n P


Data Availability




References






































Journal of












Journal of







Volume 2018






Volume 2018










119881(119905) = 120578119868 + 119892 (119860120583 0)119875 (15)


(119905)10038161003816100381610038161003816(2) = (120578119891 (119878 119868) minus 120578 (120583 + 119903) + 120590119892(119860120583 0)) 119868

+ 120578(119892 (119878 119875) minus 119892(119860120583 0))119875(16)


(119905)10038161003816100381610038161003816(2) le (120578119891(119860120583 0) minus 120578 (120583 + 119903) + 120590119892 (119860120583 0)) 119868

le 120578 (120583 + 119903) (1198770 minus 1) 119868(17)



(120583 + 120582) (1205822 + 120582(120578 + 120583 + 119903 minus 119891(119860120583 0))

+ 120578 (120583 + 119903) (1 minus 1198770)) = 0(18)

Let

Φ (120582) = 1205822 + 120582(120578 + 120583 + 119903 minus 119891(119860120583 0))+ 120578 (120583 + 119903) (1 minus 1198770)

(19)



(1 minus 119891 (119878 119868)119891 (119878 119868lowast))(

119891 (119878 119868lowast)119891 (119878 119868) minus 119868

119868lowast) le 0



(1198673)






120578 119875lowastΦ(119875 (119905)119875lowast ) (20)




119889119878119889119905 + (1 minus

119868lowast119868 )

119889119868119889119905


(21)








119875119875lowast


119875lowast119868119875119868lowast )

(22)

Thus




119891 (119878 119868lowast)119891 (119878 119868)


119875119875lowast




+ Φ( 119891 (119878 119868)119891 (119878lowast 119868lowast)) + Φ(










119868lowast +119891 (119878 119868lowast)119891 (119878 119868) + 119891 (119878 119868) 119868



119868lowast) le 0(25)

and






(26)


4 Application


119889119878119889119905 = 119860 minus 120583119878 minus

12057311198781198681 + 1205721119868 minus12057321198781198751 + 1205722119875

119889119868119889119905 =

12057311198781198681 + 1205721119868 +12057321198781198751 + 1205722119875 minus (120583 + 119903) 119868

119889119875119889119905 = 120590119868 minus 120578119875

(27)



Corollary 5





5 Conclusions



1000

2000

3000

4000

5000

Susc

eptib

le S

0

1

2

3

4

5

Conc

entr

atio

n P

0 2000 4000 6000

minus1000

100minus5

0

5

SI

P

0

10

20

30

40

50

Infe

cted

I

5000 10000 150000Days

5000 10000 150000Days

5000 10000 150000Days


0 2000 4000 6000

0100

2000

5

SI

P

0

50

100

150

200

Infe

cted

I

1000 2000 3000 40000Days

1000 2000 3000 40000Days

1000

2000

3000

4000

5000

Susc

eptib

le S

1000 2000 3000 40000Days

0

2

4

6

Con

cent

ratio

n P


Data Availability




References






































Journal of












Journal of







Volume 2018






Volume 2018
















119868lowast +119891 (119878 119868lowast)119891 (119878 119868) + 119891 (119878 119868) 119868



119868lowast) le 0(25)

and






(26)


4 Application


119889119878119889119905 = 119860 minus 120583119878 minus

12057311198781198681 + 1205721119868 minus12057321198781198751 + 1205722119875

119889119868119889119905 =

12057311198781198681 + 1205721119868 +12057321198781198751 + 1205722119875 minus (120583 + 119903) 119868

119889119875119889119905 = 120590119868 minus 120578119875

(27)



Corollary 5





5 Conclusions



1000

2000

3000

4000

5000

Susc

eptib

le S

0

1

2

3

4

5

Conc

entr

atio

n P

0 2000 4000 6000

minus1000

100minus5

0

5

SI

P

0

10

20

30

40

50

Infe

cted

I

5000 10000 150000Days

5000 10000 150000Days

5000 10000 150000Days


0 2000 4000 6000

0100

2000

5

SI

P

0

50

100

150

200

Infe

cted

I

1000 2000 3000 40000Days

1000 2000 3000 40000Days

1000

2000

3000

4000

5000

Susc

eptib

le S

1000 2000 3000 40000Days

0

2

4

6

Con

cent

ratio

n P


Data Availability




References






































Journal of












Journal of







Volume 2018






Volume 2018









1000

2000

3000

4000

5000

Susc

eptib

le S

0

1

2

3

4

5

Conc

entr

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n P

0 2000 4000 6000

minus1000

100minus5

0

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SI

P

0

10

20

30

40

50

Infe

cted

I

5000 10000 150000Days

5000 10000 150000Days

5000 10000 150000Days


0 2000 4000 6000

0100

2000

5

SI

P

0

50

100

150

200

Infe

cted

I

1000 2000 3000 40000Days

1000 2000 3000 40000Days

1000

2000

3000

4000

5000

Susc

eptib

le S

1000 2000 3000 40000Days

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Con

cent

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n P


Data Availability




References






































Journal of












Journal of







Volume 2018






Volume 2018






































Journal of












Journal of







Volume 2018






Volume 2018








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