Optimal controlapp lication to an Ebolamodel

Ebenezer Bonyah,Kingsley Badu,Samuel Kwesi Asiedu-AddoDepartmentofMathematics and Statistics,Kumasi Polytechnic,Kumasi,GhanaNoguchiMemorial Institute for Medical Research,College of Health Science,University of Ghana,Legon,Accra,GhanaFaculty of Health Sciences,Garden City University College,Kenyase,Kumasi,GhanaDepartmentofMathematics Education,University of Education,Winneba,Ghana



Optimal controlapp lication to an Ebolamodel

Ebenezer Bonyah1*,Kingsley Badu2,3,Samuel Kwesi Asiedu-Addo41DepartmentofMathematics and Statistics,Kumasi Polytechnic,Kumasi,Ghana
2NoguchiMemorial Institute for Medical Research,College of Health Science,University of Ghana,Legon,Accra,Ghana3Faculty of Health Sciences,Garden City University College,Kenyase,Kumasi,Ghana
4DepartmentofMathematics Education,University of Education,Winneba,Ghana

Review article http://dx.doi.org/10.1016/j.apjtb.2016.01.012

ARTICLE INFO

Article history:

Received in revised form 17Dec,2nd revised form 23 Dec 2015

Accepted 4 Jan 2016

Availableonline8Mar2016

Ebola

Optimal control

Pontryaginsmaximum principle Case fi nding

Case holding

ABSTRACT

Ebola virus is a severe,frequently fatal illness,w ith a case fatality rate up to 90%.The outbreak of the disease has been acknow ledged by W orld Health Organization as Public Health Emergency of International Concern.The threatof Ebola inW est A frica is still a major setback to the socioeconom ic development.Optimal control theory is applied to a system of ordinary differentialequationswhich ismodeling Ebola infection through three different routes including contact between humans and a dead body.In an attempt to reduce infection in susceptible population,a preventive control is put in the form of education and campaign and two treatmentcontrols are applied to infected and late-stage infected(super)human population.The Pontryaginsmaximum principle is employed to characterize optimality control,which is then solved numerically.It isobserved that time optimal control is existed in themodel.The activation of each control showed a positive reduction of infection.The overall effect of activation of all the controls simultaneously reduced the effort required for the reduction of the infection quickly.Theobtained results present a good framework for planning and designing cost-effective strategies for good interventions in dealing w ith Ebola disease.It is established that in order to reduce Ebola threatall the three controlsmust be taken into consideration concurrently.

1.Introduction

The principal aim of modeling infectious diseases is to be able to make judicious decisions in the application of control interventions of the infection to elim inate and ideally to eradicate it from the human population.Simulations and modeling can optim ize control efforts such that lim ited resources are targeted to achieve the highest impact[1].The aim of this paper is to review the epidem iology of the Ebola pandem ic and discuss the optimal controlmodel that governs the spread of the virus in human population and suggest the optimum control strategies to control and curb the spread in the future.The world w itnessed an unprecedented Ebola outbreak in West A frica which in the end was reported in some parts of Europe and North America[2].By December 13th 2015 there had been con fi rmed reported cases in excess of 28600 in total w ith over 11000 people losing their lives particularly in West A frica and to a lesser extent elsewhere in the world.Outside A frica;Italy,Spain,the United Kingdom and United States of America[2]were also affected albeit w ith no case fatalities except for one in the USA(Table 1).

The outbreak has been acknow ledged by the World Health Organization as a Public Health Emergency of International Concern.The three West African countries(Guinea,Sierra Leoneand Liberia)by farbeen hardesthitnow account forabout 99.9%of all infectionsand deaths.These countriesare known to have only recently emerged from long periods of con fl ict and instability and thus have weak health systems,human and infrastructural resources[3].

Ebola is known to be transm itted to humans via contactw ith bodily fl uids and secretion of infected animalsmainly fruitbats,monkeys,porcupines,forestantelopeand the like.It is thereafter spread from person to person through direct contact w ith infected persons[2,4,5].The incubation period beyond which infected people become symptomatic has been estimated to range between 2 and 21 days[6].The Ebola virus is a unique virus having a fi lamentous,enveloped non-segmented negative-sense RNA.Itbelongs to the genus Ebolavirus,w ithin the family of Filoviridae.Its envelope glycoprotein facilitates the entry of the virus to living cells[7,8].Till date fi ve strains of the virus are known:Zaire,Sudan,Tai Forest,Bundibugyo,and Reston w ith Zaire stain being the most virulent w ith up to 90%fatalities[9].The Pasteur Institute(Lyon,France)has sequenced the viral strain currently under circulation in West A frica and reports a strong homology of 98%with Zaire ebolavirus(the most virulent).The origin of the Ebola virus has been somewhat unclear[10,11].However,in 2005,Leroy and co-workers reported in the Nature,evidence of asymptomatic infection by Ebola virus in three speciesof fruitbat[12].This indicated that fruitbats belonging to the fam ily Pteropodidae are the natural host of the Ebola virus.Ebola viral disease has no effective treatment or vaccines,currently only supportive care can be given to patients.

Table1 Ebola situation report13th December 2015[2].

Emerging tropical infectious diseases have been persistent in causing untold econom ic hardships to relatively poor countries w ithweak health systems.Theoverarching goalof public health is to reduce disease burden by curtailing transm ission orm itigating its severity.There are at least two fundamental public health guiding principles that exist to manage the spread of an infectious disease like Ebola viral disease that has no effective vaccinesor treatment.These are(i)effective isolation of persons w ith symptoms and(ii)tracing the contacts of symptomatic cases for clusters of exposed persons and quarantining them for monitoring[13].

Mathematical models have played a vital role in the dynam ics and controlofmany epidem ics includingmalaria,severe acute respiratory syndromes and Ebola[14].

Some previous models of Ebola virus,especially the predictivemodels endeavour to calculate a threshold called basic reproductive number R0.The dynam ics of transm ission of the disease has been analyzed in terms of the reduction of the basic reproductive number[15–18].

However,all these models fail to take into account time dependent control strategies and all their discussions have been concentrated on prevalence of the disease at equilibra.Time dependent control has been employed in the study of dynam ics of diseases.For example,Rachah and Torres,2015 investigated the effect of vaccination on a proportion of susceptible population and observed that the rate of infection of Ebola reduced due to this intervention[19].Chowell and Nishiura 2014 also applied time optimal control to study Ebola epidem ic reduction[14].Other studies on using time optimal control to provide appropriate interventions to m inim ize the spread of diseases have also been carried out successfully[20–23].This technique of studying control strategies present an enviable theoretical results that can assist in providing tools for designing epidem ic control programmes.

In thiswork,time dependentoptimal control is explored and considered which deals w ith both“case holding”and“case fi nding”on Ebola model proposed by Rivers et al.[24].The model assumes that there is a difference between fi rst-stage infections and late-stage infection called super infection.Their model further assumes individuals in the latent stage develop active Ebola infection at a given rate.It also assumes that a proportion of both fi rst and late stage-infection(super infection)recover and othersmove to death compartment.We present three control mechanisms which comprise two“case fi nding”and a“case holding”in the model.The“case fi nding”is usually made up of activities that lead to preventivemeasures including screening,public education and others.The“case holding”also has to do w ith designed activities that ensure patients take their drugs within stipulated times so that they are cured.The fi rst case fi nding is incorporated by adding a control term that characterizes the contact between susceptible and infectious individuals so that the rate of infection w ill be reduced.The second case fi nding is instituted in the model by adding a control term that identi fi es proportions of those individuals in the latent stage or exposed to the disease and cure them so that the rate of getting the disease w ill be reduced.The holding is incorporated in the model by adding a control term thatmay m inim ize the treatment failure rate of individuals with Ebola disease.We choose the reduction of infected individuals of Ebola to be our main objective having a lower cost of the controls.

The paper is arranged as follows:section 2 is devoted to describing an Ebolamodel w ith three control term s incorporated.In addition,the objective function is introduced in this section.In section 3,the analysis of optimal control is discussed.Section 4 contains some numerical studies of optimal controls.Finally,section 5 deals w ith the conclusions of the studies.

2.Ebola diseasem odel

Figure 1.Stage-structured compartmentalmodel of Ebola virus disease, which splits the population into susceptible(S),exposed(E),fi rst-stage infected(I1),late-stage infected(I2),recovered(R),and funeral transm issible(F).Red compartments are transm issible,and recovery rates are greater from I1than from I2[25].

Themodel structure is presented in Figure 1 and the state system of the Ebolamodel is the follow ing six nonlinear ordinary differentialequationsproposed by Pontryagin etal.[25]and slightly modi fi ed:w ith S(0)≥0,E(0)≥0,I1(0)≥0,I2,F(0)≥0,R(0)≥0 given,where the model partitions the total population,denoted by into the follow ing epidem iologicalsub-population ofsusceptible(S)those exposed to Ebola virus(E)those individuals with fi rst stage of infection(I1),those individualsw ith second stage infection also known as super infection(I2),fraction of the populationwho are recovered is denoted by(R),fraction of the population who have died and being processed for burial(F).Coef fi cientβ1,β2andβFare the rates atwhich the susceptible become infected by an infectious individual in the fi rst,second and burialstage per unitof time respectively.An exposed individualmoves to the fi rst stage of infectious class ata rateα.The average length of fi rststage of illness isdenoted byγ1−1and the average length of second stage of illnessisγ2−1.Theaverage time from death tillone isburied is denoted byγF−1.Average duration of Ebola treatment unit bed occupancy after recovery isγR−1.

δ1Denotes the fraction of infected who progress to second stage of infectious andδ2is the fraction who subsequently progress to death.The natural per capita mortality of Ebola disease rate is denoted byμ.

The control functions being employed,u1(t),u2(t)and u3(t) are bounded Lebesques integrable functions.The“case fi nding”control u1(t),deals w ith efforts that facilitate the keeping of a distance between susceptible and infectious individuals including education and public campaigns.The“case holding”control,u2(t)dealsw ith effortneeded to identify the proportion of typical Ebola exposed individuals that is known and w ill be putunder treatment in order to reduce thenumberof individuals thatmay turn to be infectious.The term u3(t)deals w ith the effort that ensures those thatare infectious both in the fi rstand super infection stages are given treatmentandmonitored to take their drugs in order to m inim ize the number of individuals developing and dying of Ebola.

Our goal,therefore,is tom inimize the number of infected individualsw ith the Ebola viruswhile at the same time keeping the costof treatmentvery low.Inmathematicalperspective,fora fi xed term inal time tf,the problem is tom inim ize theobjective functional

It is assumed that costof treatments is of nonlinearand takes a quadratic nature.The coef fi cients,A1,A2and A3,are representing thebalancing cost factorswhich have to do w ith the size and importance of the three segments of theobjective functional. Thus,we seek to determ ine an optimal control,,andsuch that

whereΩ={（u1;u2;u3）∈L1（0;tf）|ci≤ui≤di;i=1;2;3}and ci,di,i=1,2,3,are denoted as fi xed positive constants.In the entire work,we have assumed that the total population N to be constant.In order to achieve this,we chooseΛ=μN.

3.Analysis of op timal controls

The indispensable conditions that an optimal pair must satisfy emanate from Pontryagin'smaximum principle[25].This principle actually does convert(1)–(3)into a problem of m inim izing pointw ise,H,w ith respect to u1,u2and u3:

giDenotes the righthand side of the ith differential equation of the state variable.By employing Pontryagin's maximum principle[25]and the existence results obtained for the optimal from [2 6],we arrive at Theorem 1.There existan optimal control u*1, ,and associated solution,S*,E*,I*1,I*2,R*and F*,that m inim izes J(u1,u2,u3)overΩ.Furthermore,there exist adjoint functions,λ1(t),…,λ6(t),such that

Proof.Corollary 4.1 Flem ing and Rishel(1975)present the existence of an optimal control ow ing to the convexity of integrand of J w ith respect to(u1,u2,u3),a priori boundedness of the state solutions,and the Lipschitz property of the state system w ith respect to the state variables[26].By employing Pontryagin'smaximum principle,we have

evaluated at theoptimal control and corresponding states,which result in the stated adjoin system(5)and(6)[27].By considering the optimality condition,

subject to the constraints,the characterization(7)are determ ined aton the set{t|c＜（t）for i=1;2;3}.On this set,

Ow ing to the priori boundedness of the state and adjoint functions and the resulting Lipschitz structure of the ordinary differential equations,we get the uniqueness of the optimal control for small.The uniqueness of the optimal control does movew ith theuniquenessof the optimality system,thus(1)and (5),(6)w ith characterizations(7).There is a restraint on the length of the time interval to ensure that uniqueness of the optimality system is obtained.This smallest restrictions on the length on the time interval has to do w ith the opposite time orientations of(1)and(5),(6).The state problem presents initial values and the adjointproblem also dealsw ith fi nal values.This restriction is frequently observed in control problem[28,29].

4.Num erical sim ulation and discussion

In this section,we investigate the effect of optimal strategy on Ebola transm ission applying some numerical techniques.The optimal strategy is achieved by obtaining a solution for the state system(1)and co-state system(4).An iterative scheme is explored and used to determ ine the solution for the optimality system.

The state equations are initially solved by guessing for the controlsover the simulated time applying a forward fourth order Runge-Kutta scheme.

In addition,the co-state equations are at the same time computed by employing a backward fourth order Runge-Kutta scheme w ith the transversality conditions.This is then followed by the controls being updated by employing a convex combination of the precedingcontrols and the value obtained from the characterizations of ,,.This process is allowed to go on and iteration is ended if the values of unknowns at the previous iteration are almost the same as the value obtained at present iteration[30].For numerical simulation,the state system solution is determ ined based on forward in time w ith initial conditions x(0)=(10000,300,100,40,60,70),whereas the co-state system is also dealt w ith backward in time.In the light of numerical simulation,we employed the follow ing parameters:δ1=0.6/days;α=8/days;β1=0.9/days;δ2=0.7/ days;γF=1.2/days;μ=0.000 054 79/days;γ1=5.7/days; γ2=1.4/days;β2=0.67/days and the weighting control B1=20,B2=40,B3=50.The weight factor B1associated w ith control u1is less or equal to weight factor B2associated w ith control,however,weight factor B3associated w ith control u3could be higher than all due to cost associated w ith itbecause of the cost implications.

4.1.Controlwith prevention

W ith this approach,only the control u1which deals w ith prevention is applied to optim ize the objective function J, whereas the control u2and u3on treatment are set to zero as shown in Figure2.In order to reduce infection control,u1should be sustained intensively for the fi rst seven days.Figure 2 shows the optimal solution for fi rst stage infection I1and super infection I2which depict substantial difference w ith and w ithout control in both fi gures.In Figures 2 and 3,it is observed that effective preventivemechanisms such as proper education and campaign w ill help reduce the rate of infection of Ebola in communities.

4.2.Controlwith treatment and prevention

W ith this scenario,we activate controls u1and u2on prevention and treatment to optim ize the objective function J, whereas the control u3which is the treatment of fi rst stage of infection and super infection is set to zero as shown in Figure 4. W e can observe in Figure 4 that itw ill require about 12 days intensive intervention in the form of education and campaign and at the same time giving intensive treatment for those exposed to the disease through screening for about 5 days. Figure 5 shows the optimal solutions for both fi rst stage infection and super infection.Clearly there is a vast difference between w ithout treatment and treatment.Therefore,it can be deduced that giving the right education and campaign to susceptible individuals and providing good treatment to those exposed individualsw ill help reduce epidem ic signi fi cantly.

Figure 2.The pro fi le of the optimal control.

Figure 3.The optimal solutions for fi rst stage of infection(I1)and super infection(I2)via prevention.

4.3.Controlwith prevention,treatment of exposed and treatment of infectious populations

Figure 4.The pro fi le of the optimal control u1and u2.

Figure 5.The optimal solutions for fi rst stage of infection(I1)and super infection(I2)via treatment.

W ith this approach,all the controls u1,u2and u3treatments and prevention are activated to optim ize theobjective function J, and noneof the controls isset to zero asobserved in Figure 6.In Figure 6,it can be seen thatw ith all the controls activated one requires less effort and time to reduce infection w ith respect to control u1and u3.However,one needs about17 days'intensive treatment for those exposed to Ebola.Figure 7 depicts optimal solutions for fi rst stage I1infection and super infection I2.It can be observed that there is a clear signi fi cant difference in Figure 7,both fi rst stage infection I1and super infection I2in terms of w ith controls and w ithout controls when all the three controls activated.It is therefore envisaged that in order to design a costeffectivemechanism foreffective interventions,thenecessary attention should be given to all the three scenariosat the same time.

Figure 6.The pro fi le of the optimal control u1,u2and u3.

The optimal control strategy applied to this model for the“case holding”is sim ilar to previous studies by Rachah and Torres[31],who used vaccination as a strategy to reduce the number of infection in the susceptible population in susceptible-infected-recovered(SIR)Ebola model.This“case holding”strategy again is further supported by Rachah and Torres in theirextension of previouswork by comparing optimal effect of vaccination on SIR and susceptible-exposed-infectedrecovered Ebolamodels[31].The numerical results from their comparison studies indicated that vaccination programme is more effective in susceptible-exposed-infected-recovered Ebola model than SIR Ebolamodel.In this study,the“case fi nding”activities includemass education and vigorous campaign which also reduced Ebola infections on“case fi nding”scenario.The numerical results obtained based on“case holding”strategy that is treatment is sim ilar to the study of Jung etal.[23]on optimal treatment of tuberculosis.This therefore,shows that proper medication on right time bases w ill ensure that those affected w ith Ebola can be cured.The numerical result obtained in this study is new and is not contrary to any previous studies on Ebolamodels.

Figure 7.The optimal solutions for fi rst stage of infection(I1)and super infection(I2)via treatment.

5.Conclusions

In thiswork,mathematicalmodelof Ebola diseasew ith three possible routes of transm ission that include prevention and two treatmentmeasures as optimal control has been exam ined.By exploring and applying Pontryagin maximum principle,condition foroptimalcontrolwhich addressedminimizing the disease in a fi nite time are derived and analyzed.It could therefore be concluded that the combination of education,screening and giving treatment for exposed aswell as treatment for infectious population w ill be a more effective way of reducing Ebola disease in a community.

Con fl ict of interest statement

We declare thatwe have no con fl ict of interest.

Acknow ledgments

Ebenezer Bonyah acknow ledges the support of the Department of Mathematics and Statistics,Kumasi Polytechnic Institute.Samuel Kwesi Asiedu acknow ledges,w ith gratitude,the support from Department of Mathematics Education,W inneba, Ghana for the production of this paper.Dr.Kingsley Badu acknow ledges the support from the Post-Doctoral Fellowship Program at the Noguchi Memorial Institute for Medical Research,University of Ghana.Dr.Bonyah is supported by the foundation project which is funded by Government of Ghana Annual University Lecturers Research Grant(Grant No.01/ 2015).Dr.Badu is supported by Postdoctoral Grant from the NoguchiMemorial Institute for Medical Research.

References

[1]Bakare EA,Nwagwo A,Danso-Addo E.Optimal control analysis of an SIR epidem ic model w ith constant recruitment.Int J Appl Math Res 2014;3(3):273-85.

[2]World Health Organization.Ebola situation reports.Geneva: World Health Organization;2015.[Online]Available from:http:// apps.who.int/ebola/current-situation/ebola-situation-report-16-december-2015[Accessed on 18th December,2015]

[3]World Health Organization.Ebola virus disease,West Africa–update.Geneva:World Health Organization;2014.[Online] Available from:http://www.w ho.int/csr/don/2014_07_27_ebola/ en/[Accessed on 23rd August,2014]

[4]Centers for Disease Control and Prevention.Ebola(Ebola virus disease).Atlanta:Centers for Disease Control and Prevention; 2015.[Online]Available from:http://www.cdc.gov/vhf/ebola/ transm ission/index.htm l[Accessed on 24th August,2015]

[5]World Health Organization.(WHO)Geneva.Ebola viral disease. Fact sheet No.13.Geneva:World Health Organization;2016. [Online]Available from:http://www.who.int/mediacentre/ factsheets/fs103/en/[Accessed on 17th January,2016]

[6]Feldmann H,Geisbert TW.Ebola haemorrhagic fever.Lancet 2011;377:849-62.

[7]Takada A,Robison C,Goto H,Sanchez A,Murti KG,W hittMA, etal.A system for functional analysis of Ebola virus glycoprotein. Proc Natl Acad SciU SA 1997;94:14764-9.

[8]Wool-Lew is RJ,Bates P.Characterization of Ebola virus entry by using pseudotyped viruses:identi fi cation of receptor-de fi cient cell lines.JVirol 1998;72:3155-60.

[9]Peters CJ.Marburg and Ebola viral hemorrhagic fevers.In: M andellGL,Bennett JE,Dolin R,editors.Principles and practice of infectious diseases.7th ed.Philadelphia:Churchill Livingstone; 2010.

[10]Leendertz SAJ,Gogarten JF,D¨ux A,Calvignac-Spencer S, Leendertz FH.Assessing the evidence supporting fruitbats as the primary reservoirs for Ebola viruses.Ecohealth 2015;http:// dx.doi.org/10.1007/s10393-015-1053-0.

[11]Darren M aganga GD,Rougeron V,Leroy EM.Bat fi loviruses.In: Wang LF,Cow led C,editors.Bats and viruses:a new frontier of emerging infectious diseases.New York:JohnW iley&Sons,Inc.; 2015,p.157-75.

[12]Leroy EM,Kumulungui B,Pourrut X,Rouquet P,Hassanin A, Yaba P,etal.Fruitbats as reservoirs of Ebola virus.Nature 2005; 438:575-6.

[13]Fraser C,Riley S,Anderson RM,Ferguson NM.Factors thatmake an infectious disease outbreak controllable.Proc Natl Acad SciU S A 2004;101(16):6146-51.

[14]Chowell G,Nishiura H.Transmission dynamics and control of Ebola virus disease(EVD):a review.BMC Med 2014;http:// dx.doi.org/10.1186/s12916-014-0196-0.

[15]Mamo DK,Koya DR.M athematical modeling and simulation study of SEIR disease and data fi tting of Ebola epidemic spread in West Africa.JMultidiscip Eng Sci Technol 2015;2:106-14.

[16]A lthaus CL.Estimating the rep roduction number o f Ebo la virus(EBOV)during the 2014 outbreak in W est A frica.PLoS Curr 2014;http://dx.doi.org/10.1371/ currents.outbreaks.91afb5e0f279e7 f29e7056095255b288.

[17]Nishiura H,Chowell G.Early transmission dynamics of Ebola virus disease(EVD),West A frica,March to August 2014.Euro Surveill 2014;19(36).pii:20894.

[18]Lewnard JA,Ndeffo Mbah ML,Alfaro-Murillo JA,Altice FL, Bawo L,Nyenswah TG,etal.Dynamicsand controlof Ebola virus transmission in M ontserrado,Liberia:a mathematical modelling analysis.Lancet Infect D is 2014;14(12):1189-95.

[19]Rachah A,Torres DFM.Mathematicalmodelling,simulation,and optimal control of the 2014 Ebola outbreak in West A frica. Discrete Dyn Nat Soc 2015;2015:842792.

[20]Agusto FB,Adekunle AI.Optimal control of a two-strain tuberculosis-HIV/AIDS co-infectionmodel.Biosystems 2014;119:20-44.

[21]Okosun KO,Makinde OD.Optimal control analysis ofmalaria in the presence of non-linear incidence rate.ApplComputMath 2013; 12(1):20-32.

[22]Bonyah E,Dontw i I,Nyabadza F.Department optimal control applied to the spread of buruliulcer disease.Am JComp ApplMath 2014;4(3):61-76.

[23]Jung E,Lenhart S,Feng Z.Optimal controlof treatments in a twostrain tuberculosis model.Disc Cont Dyn Syst Ser B 2002;2(4): 473-82.

[24]Rivers CM,Lofgren ET,Marathe M,Eubank S,Lew is BL. Modeling the im pact of interventions on an epidem ic of Ebola in Sierra Leone and Liberia.PLOS Curr 2014;http://dx.doi.org/ 10.1371/currents.outbreaks.4d41fe5d6c05e9df30ddce33c66d084c. [25]Pontryagin LS,BoltyanskiiVG,Gamkrelidze RV,M ishchenko EF. Themathematical theory of optimal processes.New York:Gordon &Breach Science Publishers Inc.;1986,p.360.

[26]Fleming WH,Rishel RW.Deterministic and stochastic optimal control.Berlin:Springer-Verlag;1979,p.494.

[27]Kamien M I,Schwartz NL.Dynamic optimization:the calculus of variationsand optimal control in economicsandmanagement.2nd ed.Amsterdam:Elsevier;1991,p.399.

[28]Hattaf K,Yousf N.Two optimal treatments of HIV infection model.World JModel Simulat2012;8(1):27-35.

[29]Kirschner D,Lenhart S,Serbin S.Optimal control of the chemotherapy of HIV.JMath Biol 1997;35:775-92.

[30]Lenhart S,Workman JT.Optimal control applied to biological models.Boca Raton:Chapman and Hall/CRC;2007,p.280.

[31]Rachah A,Torres DFM.Predicting and controlling the Ebola infection.Math Method Appl Sci 2015;http://dx.doi.org/10.1002/ mma.3841.

26 Nov 2015

*Corresponding author:Dr.Ebenezer Bonyah,Department of Mathematicsand Statistics,Kumasi Polytechnic,Kumasi,Ghana.

Tel:+233 243357651

E-mail:ebbonya@yahoo.com

Foundation Project:Supported by Governm ent o f Ghana Annual University Lecturers Research Grant(Grant No.01/2015)and Postdoctoral Grant from the Noguchi M emorial Institute for M edical Research(Grant ID:OP52155).

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