BioSystems 111 (2013) Contents lists available at SciVerse ScienceDirect. BioSystems

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1 BioSystems 111 (213) Contents lists available at SciVerse ScienceDirect BioSystems journa l o me pa g e: Optimal control strategies and cost-effectiveness analysis of a malaria model Kazeem O. Okosun a,, Ouifki Racid b, Nizar Marcus c a Department of Matematics, Vaal University of Tecnology, Vanderbijlpark, Sout Africa b DST/NRF Centre of Excellence in Epidemiological Modelling and Analysis (SACEMA), Stellenbosc University, Stellenbosc 76, Sout Africa c Department of Matematics and Applied Matematics, University of te Western Cape, Sout Africa a r t i c l e i n f o Article istory: Received 22 December 211 Received in revised form 15 June 212 Accepted 17 September 212 MSC: 92B5 93A3 93C15 Keywords: Malaria Cost-effectiveness Optimal control Stability Backward bifurcation Sensitivity index a b s t r a c t Te aim of tis paper is to investigate te effectiveness and cost-effectiveness of tree malaria preventive measures (use of treated bednets, spray of insecticides and a possible treatment of infective umans tat blocks transmission to mosquitoes). For tis, we consider a matematical model for te transmission dynamics of te disease tat includes tese measures. We first consider te constant control parameters case, we calculate te basic reproduction number and investigate te existence and stability of equilibria; te model is found to exibit backward bifurcation. We ten assess te relative impact of eac of te constant control parameters measures by calculating te sensitivity index of te basic reproductive number to te model s parameters. In te time-dependent constant control case, we use Pontryagin s Maximum Principle to derive necessary conditions for te optimal control of te disease. We also calculate te Infection Averted Ratio (IAR) and te Incremental Cost-Effectiveness Ratio (ICER) to investigate te cost-effectiveness of all possible combinations of te tree control measures. One of our findings is tat te most cost-effective strategy for malaria control, is te combination of te spray of insecticides and treatment of infective individuals. Tis strategy requires a 1% effort in bot treatment (for 2 days) and spray of insecticides (for 57 days). In practice, tis will be extremely difficult, if not impossible to acieve. Te second most cost-effective strategy wic consists of a 1% use of treated bednets and 87% treatment of infective individuals for 42 and 1 days, respectively, is sustainable and terefore preferable. 213 Elsevier Ireland Ltd. All rigts reserved. 1. Introduction Te main reason for taking protective and control measures against malaria is to reduce te prevalence of te disease and, if possible, eradicate it completely. Tat is, reducing te level of susceptibility of ealty individuals against te infection and te number of infectious individuals. As reported in World Healt Organisation (WHO) fact seet (29), malaria, is a life-treatening disease caused by parasites tat are transmitted to people troug te bites of infected mosquitoes, wic resulted in te deat of a cild from malaria every 3 s. Tere were 247 million cases of malaria in 26, causing nearly 1 million deats, tis is mostly among African cildren. It is estimated tat well over 2 young lives are lost daily across te globe. Tese estimates render malaria te pre-eminent tropical parasitic disease and one of te top tree killers among communicable disease (Sacs, 22). Malaria impedes development in so many ways; it affects fertility, population growt, saving and investment, worker productivity, absenteeism, premature mortality and medical costs (Sacs, 22). In areas were malaria is igly endemic, young cildren bear te larger burden in terms of te disease morbidity and mortality. Malaria also affects fetal development during early stage of pregnancy in women due to loss of immunity. However, malaria is preventable and curable wen treatment and prevention measures are sougt early. Despite relentless efforts of researcers, tere is no break troug yet for malaria vaccines and most control measures are at te level of protective ealt strategies against te disease. Te current strategies of controlling te disease include te use of cemoterapy, intermittent preventive treatment for cildren and pregnant women (preventive doses of sulfadoxine-pyrimetamine (IPT/ST)), use of insecticides treated bednets and insecticides against te vector. Te callenges posed by te resistance of parasites against drugs and resistance of mosquitoes against insecticides call for a better understanding of te disease transmission and development of effective and optimal strategies for prevention and control of te spread of malaria disease. Matematical modeling as become an important tool in understanding te dynamics of disease transmission and in decision making processes regarding intervention programs for disease control. Concerning malaria disease, Ross (1911) developed te first matematical Corresponding autor /$ see front matter 213 Elsevier Ireland Ltd. All rigts reserved. ttp://dx.doi.org/1.116/j.biosystems

2 84 K.O. Okosun et al. / BioSystems 111 (213) models of malaria transmission. He focused is study on mosquito control and sowed tat for te disease to be eliminated te mosquito population sould be brougt below a certain tresold. His work was later extended by Macdonald (195, 1957) to account for superinfection. Tese two works were furter extended by Ngwa and Su (2) wit te popular generalized SEIR malaria model, wic includes bot te uman and mosquito interactions. Oter furter studies include Koella and Anita (23) wo included a latent class for mosquitoes. Tey considered different strategies to reduce te spread of resistance and studied te sensitivity of teir results to te parameters. Anderson and May (1991) derived a malaria model wit te assumption tat acquired immunity in malaria is independent of exposure duration. Different control measures and role of transmission rate on te disease prevalence were furter examined. Hyun (2, 21) studied a malaria transmission model for different levels of acquired immunity and temperature dependent parameters, relating also to global warming and local socioeconomic conditions. In Isao et al. (24), Kawaguci et al. examined te combined use of insecticide spray and zoopropylaxis as a strategy for malaria control. Dietz et al. (1974) proposed a model tat accounts for acquired immunity. Ciyaka et al. (28) formulated a deterministic model wit two latent periods in te ost and vector populations to assess te impact of personal protection, treatment and possible vaccination strategies on te transmission dynamics of malaria. Tey also considered treatment and spread of drug resistance in an endemic population (Ciyaka et al., 29). Jia (28) formulated and examined a compartmental matematical model for malaria transmission tat includes incubation periods for bot infected umans and mosquitoes. Mukandavire et al. (29) proposed and examined a deterministic model for te co-infection of HIV and malaria in a community. Mwasa and Tcuence (211) examined a matematical model tat captures te dynamics of Colera transmission to study te impact of public ealt educational campaigns, vaccination and treatment in controlling te disease. Altoug some of tese studies considered different interventions for malaria control, tey did not take into consideration te optimality, costs and cost-effectiveness of tese interventions wic may sometimes be limited by availability of resources. Specifically, carrying out a comparative analysis, knowing costs and outcomes of alternative control strategies is important to decision makers wo are often faced wit te callenge of resource allocation. In view of tis, application of optimal control teory can be an important tool to estimate te efficacy of various policies and control measures vis-a-vis te cost of implementing tem. Since te development of te Pontryagin maximum principle by Pontryagin et al. (1962), te teory of optimal control as been successfully used in decision making in various applications. In epidemiology, applications of tis teory include te optimization of te costs of using active and passive immunization in controlling infectious diseases (Gupta and Rink, 1973). Wickwire (1977) applied optimal control to matematical models of pests and infectious diseases control. Wiemer (1987) studied Scistosomiasis using optimal control metods. Sures (1978) formulated and analyzed an optimal control problem wit a simple epidemic model to examine effect of a quarantine program. He also considered an optimal control problem to study te effect of te level of medical program effort in minimizing te social and medical costs (Sures, 1978). Marco and Takasi (21) used optimal control to study dengue disease transmission. Adams et al. (24) derived HIV terapeutic strategies by formulating and analyzing an optimal control problem using two types of dynamic treatments. Karrakcou et al. (26) used optimal control to examine te role of cemoterapy in controlling te virus reproduction in HIV patients. Xiefei et al. (27) applied optimal control metods to study te outbreak of SARS using Pontryagin s Maximum Principle and a genetic algoritm. Zaman et al. (28) used optimal control to determine te optimal vaccination strategy to reduce te susceptible and infective individuals for a general SIR epidemic model. More studies on te applications of optimal control to infectious diseases, mainly HIV/AIDS and Tuberculosis can be found in Felippe de Souza et al. (2), Josi (22), Josi et al. (26), Jung et al. (22), Kar and Batabyal (211), Kirscner et al. (1997), Lenart and Yong (1997) and Racik et al. (29), tese studies focused more on cost minimization analysis of te examined control strategies. However, very few studies ave been carried out on applying optimal control teory to malaria transmission models. Only recently, Kbenes et al. (29) used optimal control to study a model for vector-borne diseases wit treatment and prevention as control measures. Rafikov et al. (29), formulated a continuous model for malaria vector control wit te aim of studying ow genetically modified mosquitoes sould be introduced in te environment using optimal control problem strategies. Okosun et al. (211) derived and analyzed a malaria disease transmission matematical model tat includes treatment and vaccination wit waning immunity and applied optimal control to study te impact of a possible vaccination wit treatment strategies in controlling te spread of malaria. In tis paper, we use optimal control teory to study te effectiveness and cost effectiveness of all possible combinations of tree malaria preventive measures, namely (i) treated bednets, (ii) treatment of infective umans and (iii) spray of insecticides. For tis, we consider a standard model for malaria transmission similar to tose considered in Ngwa and Su (2), Ngwa (26) and Rafikov et al. (29) in wic we incorporate tree time dependent controls representing te interventions. Firstly, we analyze te model wit constant control parameters and investigate te stability properties. Secondly, we consider te control parameters to be time dependent controls and examine te impact of different combinations of tese measures in controlling te disease. We use Pontryagin s Maximum Principle to derive necessary conditions for te optimal control of te disease. By calculating IAR and ICER we investigate te cost-effectiveness of all possible combinations of te tree measures in order to determine te most effective strategy for eliminating malaria wit minimum costs. Te paper is organized as follows: Section 2 is devoted to te model description and te underlying assumptions. In Section 3, we analyze te model wit constant control parameters and perform a sensitivity analysis of te basic reproductive number. Te economic evaluation and te optimal control analysis are presented in Section 5, wile in Section 5 we presented te numerical simulations of te model. Section 6 is devoted to te cost-effectiveness analysis of te model wit time dependent controls. Te conclusion is presented in Section Model formulation Te model tat we consider ere is a sligt modification of models for malaria transmission considered in Ngwa and Su (2), Ngwa (26) and Rafikov et al. (29), it is not a generalization of tese ones; nor is it a special case of tem. It is a standard model of SEIRS type for umans and SEI for mosquitoes in wic we incorporated tree time dependent control measures simultaneously: (i) te use of treated bednets, (ii) treatment of infective umans and (iii) spray of insecticides. It is common knowledge tat malaria treatment reduces te risk of disease but as only a low or negligible transmission blocking effect. Here we consider a possible treatment tat blocks transmission from infective umans to mosquitoes.

3 K.O. Okosun et al. / BioSystems 111 (213) Te model sub-divides te total uman population, denoted by N, into te following sub-classes of: individuals wo are susceptible to infection wit malaria (S ), tose exposed to malaria parasite (E ), individuals wit malaria symptoms (I ) and recovered individuals (R ). So tat, N = S + E + I + R. Te total vector (mosquito) population, denoted by N v, is sub-divided into susceptible mosquitoes (S v ), mosquitoes exposed to te malaria parasite (E v ) and infectious mosquitoes (I v ). Tat is, N v = S v + E v + I v. Susceptible individuals are recruited at a rate. Tey eiter die from natural causes (at a rate ) or move to te exposed class by acquiring malaria troug contact wit infectious mosquitoes at a rate (1 )ˇ, were ˇ is te transmission probability per bite, is te per capita biting rate of mosquitoes, is te contact rate of vector per uman per unit time and [, 1] is te control on te use of treated bednets. Exposed individuals move to te infectious class at a rate 1. Infectious individuals are assumed to recover at a rate b + u 2, were b is te rate of spontaneous recovery, u 2 is te control on treatment of infected individuals and [, 1] is te efficacy of treatment. Infectious individuals wo do not recover die at a rate +. Susceptible mosquitoes are generated at a rate v. Tey eiter die from natural causes (at a rate v ) or move to te exposed class by acquiring malaria troug contacts wit infected umans at a rate (1 ), were is te probability for a vector to get infected by an infectious uman. Exposed mosquitoes are assumed to die at a rate v or move to te infected class at a rate 2. Infected mosquitoes die at a rate v. Te mosquito population is reduced, due to te use of insecticides spray, at a rate pu 3, were u 3 and p represent, respectively, te control and te efficacy of insecticides spray. Putting te above formulations and assumptions togeter gives te following vector ost model: ds = + R (1 (t))ˇms S de = (1 (t))ˇms ( 1 + )E di = 1E (b + u 2 (t))i ( + )I dr ds v de v = (b + u 2 (t))i ( + )R = v (1 (t)) v S v pu 3 (t)s v v S v = (1 (t)) v S v pu 3 (t)e v ( 2 + v )E v di v = 2E v pu 3 (t)i v v I v. Here, ˇm and v represent te force of infection of umans and mosquitoes, respectively, ˇm = ˇI v /N and v = I /N. Te SEIR malaria model (1) will be analyzed in a biologically feasible region for bot uman and mosquito populations. We ave te following result: Teorem 1. If S (), E (), I (), R (), S v (), E v () and I v () are non-negative, ten so are S (t), E (t), I (t), R (t), S v (t), E v (t) and I v (t) for all t >. Moreover, lim supn (t), lim supn v (t) t t v v. Furtermore, if in addition N () ( / ) (resp. N v () ( v / v )) ten N (t) ( / ) (resp. N v (t) ( v / v )). In particular, te region D = D D v wit { D = (S, E, I, R ) R 4 + : S + E + I + R } (1) and { D v = (S v, E v, I v ) R 3 + : S v + E v + I v } v v is positively invariant. Proof: Let (S (), E (), I (), R (), S v (), E v (), I v ()) be a positive initial condition and denote by [, t max [ te maximum interval of existence of te corresponding solution. To sow tat te solution is positive and bounded in [, + [, it is sufficient to sow te positivity and boundedness results in [, t max [. Let t 1 = sup{ < t < t max : S, E, I, R, S v, E v andi v arepositiveon[, t]}. Since S (), E (), I (), R (), S v (), E v () and I v () are non-negative ten t 1 >. If t 1 < t max ten, by using te variation of constants formula to te first equation of te system (1) we ave S (t 1 ) = U(t 1, )S () + were U(t, ) = e t t1 (ˇm+ )(s) ds. U(t 1, ) d.

4 86 K.O. Okosun et al. / BioSystems 111 (213) Tis implies tat S (t 1 ) >. It can be sown in te same manner tat tis is te case for te oter variables. Tis contradicts te fact tat t 1 is te supremum because at least one of te variables sould be equal to zero at t 1. Terefore t 1 = t max. On anoter and, by adding togeter te last tree equations of system (1) and using te fact tat te solution is positive in [, t max [, we obtain (dn v /)(t) v v N v (t). Tis implies tat N v (t) N v ()e vt + v (1 e vt )foreact < t max. (2) v Terefore, lim supn v (t) ( v / v ) wic is in contradiction wit te fact tat t max is te upper bound of te maximum interval of existence. t t max Tus t max =, wic implies tat te solution is positive for eac t >. If in addition N v () ( v / v ) ten N v (t) ( v / v ). Moreover, from te first four equations of (1), we ave (dn /)(t) = N (t) I (t). Since < I (t) N (t), ten ( + )N (t) dn (t) N (t). By using a standard comparison teorem (Laksmikantam et al., 1989), we obtain N ()e ( + )t + Tis implies tat + (1 e ( + )t ) N (t) N ()e t + (1 e t ). (3) + lim inf t N (t) lim supn (t). t Moreover, if N () ( / ), ten N (t) ( / ). Tis establises te invariance of D as required. wic implies tat S, E, I, S v, E v and I v are positive for all t >. From tis teorem we conclude tat system (1) is epidemiologically feasible and matematically well-posed in D. 3. Analysis of te model wit constant controls In tis section, we assume tat te control parameters are constant and determine te basic reproductive number, te steady states and teir stability as well as te bifurcation beavior of system (1) Steady states and stability Te disease free equilibrium (DFE) of te malaria model (1) exists and is given by E = ( /,,,, v /pu 3 + v,, ). Te basic reproduction number, R, is calculated by using te next generation matrix (Van den and Watmoug, 22). It is given by 1 2 v R = ((1 )) 2 ˇ (pu 3 + v ) 2 ( + 1)(pu 3 + v + 2)( + + b + u 2 ). (4) Te steady states of (1) are calculated by equating its rigt and side to zero. We obtain S = ( + )ˇ m K + (( + )( + + r)( + 1))) ( + ˇ m)( + )ˇ mk + (( + )( + + r)( + 1))) E ˇ m = ( ( + )ˇ m K + (( + )( + + r)( + 1)))) ( + 1)( + ˇ m)( + )ˇ mk + (( + )( + + r)( + 1))) I = 1 ˇ m ( + ) ˇ mk + (( + )( + + r)( + 1))) R = r 1 ˇ m ( + ) ( + )ˇ mk + (( + )( + + r)( + 1))) S v = v v + v Ev = v v ( v + 2)( v + v) Iv = 2 v v v ( v + 2)( v + v) were ˇ m = ((1 )ˇIv )/N, v = ((1 )ˇI )/N, r = b + u 2, v = pu 3 + v and K = r 1 + ( + + r)( + 1)( + ). Tus ˇ m wic corresponds to te DFE, or Aˇ 2 m + Bˇ m + C, (6) were A = ( )( 1 + v ( ))( + ) 2 + v (r 1) 2 + r 1( + )[ 1(1 ) + 2(r ) v ], B = v M( 1 + )(r + + )(K R 2 ), C = v ( + ) 2 ( 1 + ) 2 (r + + ) 2 (1 R 2 ), (5)

5 K.O. Okosun et al. / BioSystems 111 (213) wit and M = [ ( + ) 2 (r + ( + )( 1 + )) + r 1( + )] 2 K = 2r 1 v 2 + ( + )[2 v (r ) + 1(1 )]. r 1 v + v ( + )[r + ( + )( 1 + )] We obtain te following result: Proposition 1. 1 If K 1, ten systems (1) exibits transcritical bifurcation. 2 If K < 1, ten system (1) exibits backward bifurcation. Tat is, tere exists R c in (, 1) suc tat i. If 1 R ten (1) as one endemic equilibrium point. ii. If R c < R < 1 ten (1) as two endemic equilibrium points. iii. If R = R c ten (1) as one endemic equilibrium point. iv. If R < R c ten (1) as no endemic equilibrium points. Proof: 1 If K 1. In tis case, we ave te following i. If R > 1, ten C <. In tis case (6) as a unique positive solution. ii. If R 1, ten C and B (because R 1 K). Tis togeter wit A > imply tat (6) as no positive solution. 2 If K < 1. In tis case we ave i. If R 1, ten C wic implies tat (6) as a unique positive solution. ii. If R K, ten B and C >. Tis implies tat (6) as no positive solution. iii. If K < R, we consider te discriminant of (6) (R ) : = B 2 4AC. One can see tat ( ) K := 4AC < and (1) : = B 2 >. Terefore, tere exists R c ( ) K, 1 suc tat (R c ) and < for R ( ) K, R c and > for R (R c, 1). In tis case we ave a. If K < R < R c ten (6) as no positive solution. b. If R = R c ten and B <. Tis implies tat (6) as one positive solution. c. If R c < R < 1 ten (6) as two real solutions wic are positive since C > and B <. Proposition 1 establises te existence of two endemic equilibria for R in (R c, 1). To investigate te stability of tese equilibria we use te centre manifold metod by Castillo-Cavez and Song (24) Backward bifurcation analysis Using Center Manifold teory (Gumel and Song, 28; Castillo-Cavez and Song, 24), we carry out a bifurcation analysis of system (1) at R = 1. For tis, we consider te transmission rate ˇas a bifurcation parameter so tat R = 1 if and only if ˇ = ˇ := 2 v ( 1 + )( + + u 2 )( 2 + v ) 1 2((1 )) 2. v Te Jacobian matrix of (1) calculated at ˇ = ˇ is given by ˇ 1 ˇ 1 u 2 r J(E ) = (1 ) v. v v (1 ) v 2 v v 2 v J(E ) as a simple zero eigenvalue, wit all oter eigenvalues aving negative real part. We are now in a position to apply te center manifold approac (Castillo-Cavez and Song, 24). We start by calculating a rigt and a left eigenvector of J(E ) denoted, respectively, by w = [w 1, w 2, w 3, w 4, w 5, w 6, w 7 ] T, and v = [v 1, v 2, v 3, v 4, v 5, v 6, v 7 ]. We ave

6 88 K.O. Okosun et al. / BioSystems 111 (213) Force of Infection Force of Infection Basic Reproductive Number (R ) (a) Basic Reproductive Number (R ) (b) Fig. 1. Bifurcation diagram sowing transcritical bifurcation for K 1 and backward bifurcation for K < 1. Te red line represents te stable equilibrium and te blue line represents unstable equilibrium. (For interpretation of te references to color in tis figure legend, te reader is referred to te web version of te article.) and w 1 = w 4 w 2 ( 1 + ), w 2 = ˇ 1w 2, w 3 =, w 4 = u 2w 3, w 5 = (1 ) v ( + )w u u 2 2, v w 6 = v, w 7 = 1 2 v 1 = v 4 = v 5, v 2 = v 3 1, v 3 = v( 1 + ) 2, v 6 =, v 7 = ˇ v After computations, it can be sown tat a = 2v 6 w 3 (1 ) ( ) v w 5 w 1 v b = v 2 w 7. (7) Clearly b >. Using Matematica we obtained tat if K < 1 ten a > implying tat te SEIR malaria model exibits a backward bifurcation and tat one of two endemic steady states is unstable. Tis is illustrated in Fig. 1. Te implication of te occurrence of backward bifurcation in tis model is tat it for te disease to be eradicated, it is no longer enoug tat te basic reproductive number R is less tan one. In fact, to acieve eradication, additional efforts and costs are required to bring R bellow a critical value R c < 1. Tis type of bifurcation as been observed in many infectious disease models, particularly, in Castillo-Cavez and Song (24), Garba et al. (28), Gumel and Song (28), Mukandavire et al. (29) and Nakul et al. (26). Concerning malaria disease, te autors in Asrafi and Gumel (28) demonstrated te occurrence of suc bifurcation but te parameter values tey used were not realistic. In Zixing et al. (28), te autors used te bifurcation software program AUTO (Doedel et al., 22) to prove te existence of backward bifurcation for large values of te disease-induced deat rate. Here, we derive some conditions tat ensure te occurrence of backward bifurcation. From Proposition 1 we deduce tat te model exibit suc bifurcation wen K < Sensitivity analysis Due to uncertainties associated wit te estimation of certain parameter values, it is useful to carry out a sensitivity analysis to investigate ow sensitive te basic reproductive number is wit respect to tese parameters. Tis will also allow us to determine wic of te (constant) controls causes te most reduction in R and terefore determine te control measure tat is te most effective in controlling malaria transmission. For tis we compute te normalized forward sensitivity index of te reproduction number wit respect to tese parameters. Tis index measures te relative cange in a variable wit respect to relative canges in its parameters (see Nakul et al., 28 for anoter application of tis index to a malaria transmission model). Definition. Te normalized forward sensitivity index of a variable,, tat depends differentially on a parameter, l, is defined as: l := l l. Te sensitivity index of R wit respect to and is equal to 1. It is equal to.5 wit respect to v, ˇ and and to.5 wit respect to. For te oter parameters we obtain

7 K.O. Okosun et al. / BioSystems 111 (213) Table 1 Sensitivity indices of R. Parameter Parameter description Sensitivity index Mosquito contact rate wit uman +1 Mosquito biting rate +1 ˇ Probability of uman getting infected +.5 Probability of a mosquito getting infected +.5 v Mosquitoes birt rate +.5 Human birt rate.5 Human natural deat rate b Spontaneous recovery.4 1 Humans progression rate from exposed to infected +.34 Table 2 Description of variables and parameters of te malaria model (1). Parameter Description Estimated value Ref Mosquito contact rate wit uman.52 day 1 Kbenes et al. (29) Mosquito biting rate.2 Kbenes et al. (29) ˇ Probability of uman getting infected.8333 Nakul et al. (26) Probability of a mosquito getting infected.9 Kbenes et al. (29) Natural deat rate in umans.4 day 1 Hyun (21) v Natural deat rate in mosquitoes.1429 day 1 Nakul et al. (26) Recovered individuals loss of immunity.792, 1/(2*365) day 1 Nakul et al. (26) and Kbenes et al. (29) 1 Humans progression rate from exposed to infected 1/17 day 1 Kbenes et al. (29) 2 Mosquitoes progression rate from exposed to infected 1/18 day 1 Kbenes et al. (29) Human birt rate 1 day 1 Kbenes et al. (29) v Mosquitoes birt rate 1 day 1 Kbenes et al. (29) Proportion of effectively treated individuals.1.7 Assumed Disease induced deat.5 day 1 Robert and Hove-Musekwa (28) b Spontaneous recovery.5 day 1 Ciyaka et al. (28) c b Per person unit cost of bednets $ Division of parasitic diseases (in press) c tr Per person unit cost of malaria treatment $ 2. or more Division of parasitic diseases (in press) c v Per area cost of insecticides spray $ 1.5 Assumed p Insecticide s efficacy.25 Assumed Discount rate 3/365 to 5/365% Varied 1 = 2( 1 + ), 2 = pu 3 + v 2( 2 + pu 3 + v ), v = v( ( v + pu 3 )) 2( 2 + v + pu 3 )( v + pu 3 ), = 2(b + u ), b b = 2(b + u ), = b + 1 u 2 2( + + b + u 2 )( + 1). Using parameter values from Table 2, we calculate te sensitivity indices of,, ˇ,, v, and 1. For and b, we find tat te sensitivity indices are always constant (even toug teir expressions depend on u 2 ). Teir values are given in te table below: Te oter indices are given in Fig. 2. (a) (b) Fig. 2. Sensitivity index of te basic reproductive number.

8 9 K.O. Okosun et al. / BioSystems 111 (213) (a) (b) (c) Fig. 3. Sensitivity index of te basic reproductive number wit respect to te control parameters. Te interpretation of te sensitivity index values of R given in Table 1 is tat an increase/decrease of 1% in any of te parameter values in te first column of Table 1 results in a percentage increase/decrease given by te corresponding value in te tird column of Table 1. From Table 1 and Fig. 2 we observe tat wen u 3 te igest (in absolute values) sensitivity index of R is wit respect to v and is equal to Tis means tat in te absence of te spray of insecticides intervention te transmission of te disease is most sensitive to mosquito natural deat rate, v. Te explanation of tis is tat increasing te mortality of mosquitoes as two effects: (i) it decreases te total population of mosquitoes and (ii) it leads to more infected mosquitoes dying before tey become infectious. However, as u 3 increases, it is important to note tat wen u 3 >.22, te sensitivity index of R wit respect to v is in absolute value less tan one. In tis case, R is now most sensitive to te mosquitoes biting rate () and contact rate wit uman (), suggesting tat te effort sould sift from te spray of insecticides to te use of treated bed nets. Te sensitivity indices of R wit respect to te constant controls are given by u1 = u 2, u2 = 1 2( + + b + u 2 ), u3 = p(3 pu v + 2 2)u 3 2(pu 3 + v )(pu 3 + v + 2) Using parameter values from Table 2, we plot tese indices as functions of te control parameters (Fig. 3). Our sensitivity analysis suggests tat te use of treated bednets ( ) as te most effect on te disease transmission followed by spray of insecticides (u 3 ). In tis section, we studied te beaviour of te model and performed a sensitivity analysis wen te controls are constant. Te occurrence of backward bifurcation in te constant control case suggests tat eradication of malaria is acievable only wen te (constant) controls are greater tan a critical value less tan one. Moreover, te sensitivity analysis indicates wic (constant) control or combination of controls is most effective (optimal) in acieving tis. However, for te disease not to become endemic again, interventions controls must be maintained at tis optimal level for all time, oterwise te system will re-stabilize at its previous endemic steady state. In reality, it is difficult, if not impossible, to keep te controls constant at all times. Tis, owever, does not make te model wit constant controls inadequate for tis study. Te fact of te matter is tat constant controls can be regarded as an approximation of time dependent ones, making te model wit constant controls be very useful in understanding te average dynamics of te disease. For a more specific

9 K.O. Okosun et al. / BioSystems 111 (213) analysis, we consider in te next section te model wit time dependent controls and proceed by investigating te most cost-effective strategy by using optimal control teory. 4. Model wit time-dependent controls: cost-effectiveness analysis In tis section, we carry out a cost-effectiveness study to determine te most cost-effective combination of te tree strategies considered in tis paper. We start by performing an economic evaluation of tese strategies ten use optimal control to study te optimality of tese interventions Economic evaluation In tis section, we consider te economic evaluation involved in te use of treated bednets, treatment of infective individuals and spray of insecticides. Te goal is to compare te costs of tese interventions and teir effectiveness in te control of malaria. For tis, we use te objective function tf W = min [c b (t)s(t) + c tr u 2 (t)i (t) + pc v u 3 (t)(s v (t) + E v (t) + I v (t))]e t (8),u 2,u 3 subject to (1). Here c b being te per unit cost of bednets, c tr te per unit cost of malaria treatment and c v te per unit area cost of insecticides wit being a discount rate of 3 5%. Te corresponding Hamiltonian is given by } H c = [c b (t)s + c tr u 2 (t)i + pc v u 3 (t)(s v + E v + I v )]e t + S { + R (1 )ˇI v S N S { } (1 )ˇI v S + } { } E ( 1 + N )E + I { 1E (b + u 2 )I ( + )I + R (b + u 2 )I ( + )R { + S v v (1 } { )I S v (1 } )I u 3 ps N v v S v + S v } E v u 3 pe N v ( 2 + v )E v + I v { 2E v u 3 pi v v I v (9) were S, E, I, R, S v, E v and I v represent te sadow prices associated wit teir respective classes. Tey represent te canges in te objective value of te optimal solution of an optimization problem obtained by relaxing te constraint by one unit. Tey are obtained by using te Pontryagin s Maximum Principle: d S = H c S, d E = H c E, d I = H c I, d R = H c R, d S v = H c S v, d E v = H c E v, d I v = H c I v. Tus d S d E d I d R d Sv ( ) (1 = c b e t )ˇI v + + N (1 )ˇI v S N 2 ( ) = (1 )ˇI v S N 2 E 1 I (1 )ˇI v S N 2 = u 2 c tr e t + (b + u ) I (b + u 2 ) R + ( ) = = S (1 ( )ˇI v 1 S ) N N S (1 )I S v N 2 ( S v E v ) ) ( (1 )S v N (1 )I S v N 2 E (1 )ˇI S v N 2 ( S v E v ) ( S v E v ) (1 )ˇI v S N 2 S + ( + ) R + (1 )ˇI v S N 2 E (1 )I S v N 2 ( S v E v ) ( (1 ) )I + u 3 p + v S v N + (1 )I E v N pu 3 c v e t d Ev d Iv = (u 3 p v ) E v 2 I v pu 3 c v e t = (1 )ˇS N ( S E ) + (u 3 p + v ) I v pu 3 c v e t, (1) Economic evaluation of treated bednets By differentiating H c wit respect to te control parameter of te use of treated bednets,, we obtain H c = c b e t S + [ˇS I v ( S E ) + S v I ( S E )] N. (11)

10 92 K.O. Okosun et al. / BioSystems 111 (213) Te expression [ˇS I v ( S E ) + S v I ( S E )]/N is te total marginal benefit of te use of treated bednets and c b S is te marginal cost. Te optimal policy is acieved wen te marginal cost of treated bednets is equal to te marginal benefit. (t) if c b S e t > [ˇS I v ( E S ) + S v I ( E v S v )] N (t) (, 1) if c b S e t = [ˇS I v ( E S ) + S v I ( E v S v )] N (t) = 1 if c b S e t < [ˇS I v ( E S ) + S v I ( E v S v )] N Te interpretation is tat malaria prevention troug te use of bednets will be optimal only wen te expected marginal benefit is larger tan te marginal cost of using treated bednets. Ten all susceptible umans best strategy is to use treated bednet. However, if te marginal benefit is less tan te marginal cost, ten only few susceptible umans will use treated bednets. Tis optimal policy is clear; increasing te use of treated bednets as two effects: firstly, it reduces te number of exposed umans and exposed mosquitoes, and secondly, it increases te numbers of susceptible (uninfected) umans and susceptible (uninfected) mosquitoes Economic evaluation of treatment of infective umans Differentiating H c wit respect to te control parameter of treatment of infective individuals, u 2, we get H c u 2 = c tr e t I I ( I R ). c tr I and I ( I R ) are, respectively, te marginal cost and marginal benefit of treatment. Te optimal policy is to ensure tat te marginal benefit for being treated is equal to te marginal costs for being treated. u 2 (t) if c tr I e t > ( I R )I u 2 (t) (, 1) if c tr I e t = ( I R )I u 2 (t) = 1 if c tr I e t < ( I R )I. If te marginal benefit of treatment is iger tan te marginal cost for being treated ten all infected umans will seek full treatment. If not, ten only few infected uman will seek treatment Economic evaluation of insecticide spray Differentiating H c wit respect to te control parameter of te spray of insecticides, u 3, we ave, H c u 3 = pc v e t N v p(s v S v + E v E v + I v I v ), (12) were pc v N v is te marginal cost for spray of insecticides against mosquitoes and p(s v S v + E v E v + I v I v ) is te marginal benefits. Te optimal policy is u 3 (t) if c v N v e t > S v S v + E v E v + I v I v u 3 (t) (, 1) if c v N v e t = S v S v + E v E v + I v I v u 3 (t) = 1 if c v N v e t < S v S v + E v E v + I v I v. If te marginal benefit for optimal spray of insecticides against mosquitoes is less tan te marginal cost of spray of insecticides ten te spray of insecticides is optimal. If te marginal cost of spray of insecticides is less tan te marginal benefits, ten it is optimal to spray insecticides against mosquitoes for malaria control. Next, we investigate te impact of te sadow prices and marginal benefits numerically, by evaluating te sadow price at te start of malaria epidemic as a function of te numbers of recovered or protected at te time of outbreak. Tese are sown in Fig. 4. In Fig. 4(a), we observed tat te marginal value (sadow price) of S is muc less damaging tan te marginal value of I. Tis is economically reasonable as infected umans represent a welfare cost in temselves and are also a source of infection for susceptibles. Te sadow price on infected umans starts by dropping, indicating a positive impact acieved by te intervention after time t = 12. Fig. 4(b) sows tat te sadow price on S is negative for small values of recovered susceptibles. As te number of protected and recovered umans increases, te sadow price increases to positive values to ultimately stabilize at zero. Te economic interpretation of tis is tat as more individuals are protected or recovered from te disease, te consequences of te diseases become negligible and Fig. 4(c) sows tat te marginal benefit of use of insecticides is muc smaller tan te marginal benefit of treated bednets. Te figure indicates tat te marginal benefit for furter reduction in disease prevalence falls as disease prevalence itself falls. Te figure furter sows tat a smaller amount of efforts on spray of insecticides is needed to eliminate te disease compared to treated bednets. For example wit te spray of insecticides, elimination of malaria will be optimal in time t = 17. Also wit te use of treated bednets, optimality will be acieved in t = 18.

11 K.O. Okosun et al. / BioSystems 111 (213) Sadow price sadow price on susceptibles sadow price on infected umans Sadowprice on susceptible umans (a) x Number of recovered umans Marginal benefit of treated bednets Marginal benefit of insecticides spray (b) 2.5 Marginal Benefit Fig. 4. Numerical simulations of te economic evaluations of te malaria model. (c) 4.2. Analysis of optimal control In case te elimination of malaria is not acievable weter due to costs or to social and environmental reasons, we need to investigate te optimal level of efforts tat would be needed to control te disease. For tis, we consider te objective functional J(, u 2, u 3 ) = tf [mi + nu cu du 2 3 ]e t (13) subject to te state equation (1) and te total cost at time t is given by t C T = [c b S + c tr u 2 I + c v u 3 (S v + E v + I v )] (14) were m, n, c and d are desired positive weigts on te benefits of preventing infection. Here, we assume tat tere is no linear relationsip between te coverage of tese interventions and teir corresponding costs, ence we coose a quadratic cost on te controls in keeping wit wat is in oter literature on cost of control of epidemics (Adams et al., 24; Felippe de Souza et al., 2; Josi, 22; Kirscner et al., 1997; Lenart and Yong, 1995). Our goal wit te given objective function is to minimize te number of infected umans I, wile minimizing te cost of control (t), u 2 (t) and u 3 (t). We seek an optimal control u 1, u 2 and u 3 suc tat J(u 1, u 2, u 3 ) = min J(, u 2, u 3 ),,u 2,u 3 U (15) were U is te set of measurable functions defined from [, t f ] onto [, 1]. Te necessary conditions tat an optimal control must satisfy come from te Pontryagin s Maximum Principle (Pontryagin et al., 1962). Tis consists in minimizing, wit respect to (, u 2, u 3 ), see Appendix A for detail optimal control analysis.

12 94 K.O. Okosun et al. / BioSystems 111 (213) Numerical simulations Due to te a priori boundedness of te solutions of bot te state and adjoint equations and te resulting Lipscitz structure of tese equations, we obtain te uniqueness of te optimality system ((17) (19)) for small t f. Te restriction on te lengt of time interval [, t f ] is common in control problems (see Felippe de Souza et al., 2; Josi, 22; Kirscner et al., 1997; Lenart and Yong, 1995), it guarantees te uniqueness of te optimality system. Tis is due to te opposite time orientations of (17) (19); te state problem as initial values wile te adjoint problem as final ones. Next, we investigate numerically te effect of te following optimal control strategies on te spread of malaria in a population. Strategy A: combination of use of treated bednets and treatment of infective individuals. Strategy B: combination of use of treated bednets and spray of insecticides. Strategy C: combination of treatment of infective individuals and spray of insecticides. Strategy D: combination of use of treated bednets, treatment of infective individuals and spray of insecticides. Te optimal control is obtained by solving te optimality system ((17) (19)). An iterative sceme is used for solving te optimality system. We start by solving te state equations wit a guess for te controls over te simulated time using fourt order Runge Kutta sceme. Because of te transversality conditions (19), te adjoint equations are solved by te backward fourt order Runge Kutta sceme using te current iterations solutions of te state equation. Ten te controls are updated by using a convex combination of te previous controls and te value from te caracterizations ((18)). Tis process is repeated and iterations stopped if te values of te unknowns at te previous iterations are very close to te ones at te present iterations (Lenart and Workman, 27). We assume tat c > n and c > d. Tis assumption is based on te facts tat te cost associated wit and u 3 will include te cost of spray of insecticides and te use of treated bed nets, and te cost associated wit treatment, u 2, will include te cost of antimalarial drugs, medical examinations and ospitalization. We coose c = 65, d = 1, m = 92 and n = 2 and use parameter values from Table 2. Te initial state variables are cosen as S () = 35, E () = 2, I (), R (), S v () = 95, E v () = 2 and I v () = Strategy A: optimal use of treated bednets and treatment In tis strategy, te treated bednet control and te treatment control u 2 are used to optimize te objective function J wile we set te spray of insecticides control u 3, to zero. We observed in Fig. 5(a) tat due to te control strategies, te number of infected umans I decreases, wile te population of infected mosquitoes increase wen tere is no control. A similar decrease is observed in Fig. 5(b) for infected mosquitoes I v in te presence of control strategy, wile an increased number is observed for te case witout control. Fig. 5(c) sows tat te igest cost recorded is $2,32, it is reaced witin 42 days. In Fig. 5(d), te control u 2 is at te upper bound of 87% and drops gradually until reacing te lower bound, wile control on treated bednets is at te maximum of 1% for 39 days before dropping gradually to te lower bound in 42nd day. Te result ere sows tat wit treatment coverage of 87% and a bednets coverage of 1% for 39 days, te disease will be eliminated witin 71 days in umans and 4 days in mosquitoes Strategy B: optimal use of treated bednets and insecticide spray Here under tis strategy te use of treated bednet control and te spray of insecticide control u 3 are used to optimize te objective function J wile we set treatment control u 2. In Fig. 6(a) and (b), we observed tat te numbers of infected umans I and infected mosquitoes I v are smaller in te control case tan in te case witout control. Te cost function is sown in Fig. 6(c), were te stable cost of $8117 is reaced in 3 days. Te control profile in Fig. 6(d) sows tat te control on treated bednets is at upper bound for 11 days of te intervention period before dropping slowly to te lower bound in te 16t day wile te spray of insecticides u 3 stays at upper bound for t = 32 days before dropping to te lower bound in te 34t day Strategy C: optimal treatment and spray of insecticides Under tis strategy we optimize te objective function J using te treatment control u 2 and te spray of insecticides control u 3 wile te treated bednet control is set to zero. Te results in Fig. 7(a) and (b) sow a significant difference in te numbers of infected umans I and infected mosquitoes I v wit optimal strategy compared to te numbers in te case witout control. In particular, te simulations sow a decrease in te number of infected umans and mosquitoes in te controlled case compared to te case witout control. Fig. 7(c) sows te cost function for using treatment and spray of insecticides, te cost reaces te stable state of $1148 in 52 days. Te control profile is sown in Fig. 7(d), we see tat te optimal treatment control u 2 rises to te upper bound till te time t = 22, wile te optimal spray of insecticides u 3 is at te maximum of 1% for 52 days before dropping slowly to te lower bound in te 55t day. Tis suggests tat a smaller effort is required on insecticide spray under tis strategy Strategy D: optimal use of treated bednets, treatment and insecticides Here under tis strategy we use all te tree controls, u 2 and u 3 to optimize te objective function J. We observed in Fig. 8(a) and (b) tat te control strategies resulted in a decrease in te numbers of infected umans I and infected mosquitoes I v, wile tere are increases in te numbers of I and I v in te cases witout control. Fig. 8(c) sows te cost function, were we observed tat te cost reaces te stable state of $899 in 25 days. Te control profile sown in Fig. 8(d) sows tat te control is at upper bound till time t = 13 before dropping slowly to te lower bound in te 15t day wile control u 2 decreases from te maximum of 87% to te lower bound in 8 days and te control u 3 is at upper bound for 33 days before dropping slowly to te lower bound in te 35t day.

13 K.O. Okosun et al. / BioSystems 111 (213) Infected Human , u 2,u 3, u 2, u 3 Infected Mosquitoes , u 2,u 3, u 2, u x (a) (b) , u 2, u u 2 Total Cost ($) Control Profile (c) (d) Fig. 5. Simulations of te malaria model sowing te effect of te optimal strategies: treated bednets and treatment on te spread of malaria. 6. Cost-effectiveness analysis In order to quantify te cost-effectiveness of te control measures, we consider an Infection Averted Ratio (IAR); IAR = Numberofinfectionaverted Numberofsuccessfullyrecovered. For tis, we assume tat te cost of te controls are directly proportional to te number of controls deployed. Te assumption is based on te understanding tat te primary goal of using treated bednets, treatment of infective individuals and spray of insecticides is to reduce infection. Te difference between te total infectious individuals witout control and te total infectious individuals wit control was used to determine te no of infection averted term in IAR formula. Using te parameter values as in Table 2, te combination of controls yielding maximum IAR was determined for eac intervention strategy. For clearer picture, we illustrate tis numerically in Fig. 9. From Figs. 7 and 9, one can see tat te most cost-effective strategy in-terms of IAR and total costs of interventions is te combination of treatment of infective individuals and spray of insecticides. However, for more clarity, we examine te cost effectiveness ratio of te strategies, so tat we can draw our conclusions. Tere are tree types of cost effectiveness ratios: (1) Average Cost-Effectiveness Ratio (ACER) wic deals wit a single intervention and evaluates tat intervention against its baseline option (e.g. no intervention or current practice). It is calculated by dividing te net cost of te intervention by te total number of ealt outcomes prevented by te intervention. (2) Marginal Cost-Effectiveness Ratio (MCER) for te assessment of te specific canges in cost and effect wen a program is expanded or contracted. (3) Incremental Cost-Effectiveness Ratio (ICER) used to compare te differences between te costs and ealt outcomes of two alternative intervention strategies tat compete for te same resources and is generally described as te additional cost per additional ealt outcome.

14 96 K.O. Okosun et al. / BioSystems 111 (213) Infected Human , u 2,u 3, u 3, u 2 Infected Mosquitoes , u 2,u 3, u 3, u (a) (b) 8 7 6, u 3, u u 3 Total Cost ($) Control Profile (c) (d) Fig. 6. Simulations of te malaria model sowing te effect of te optimal strategies: treated bednets and treatment on te spread of malaria. For te purpose of our study, we consider te Incremental Cost-Effectiveness Ratio (ICER). It allows us to compare te cost-effectiveness of combination of at least two of te control strategies, use of treated bednets, treatment of infective individuals and spray of insecticides. In ICER, wen comparing two competing intervention strategies incrementally, one intervention sould be compared wit te next-lesseffective alternative. Te ICER numerator includes te differences in intervention costs, averted disease costs, costs of prevented cases and averted productivity losses if applicable. Wile, ICER s denominator is te difference in ealt outcomes (e.g. total number of infection averted, number of susceptibility cases prevented). Based on te model simulation results, we rank te strategies in order of increasing effectiveness Strategies Total infection averted Total costs ($) ICER No strategy Strategy C Strategy A , Strategy B Strategy D Te ICER, is calculated as follows: ICER(C) = = ICER(A) = = ICER(B) = = ICER(D) = =

15 K.O. Okosun et al. / BioSystems 111 (213) , u 2,u 3, u 2, u 3 12, u 2,u 3, u 2, u Infected Human 8 6 Infected Mosquitoes (a) (b) , u 2, u u 2 u Total Cost ($) Control Profile (c) (d) Fig. 7. Simulations of te malaria model sowing te effect of te optimal strategies: treatment and spray of insecticides on te spread of malaria. Te comparison between strategies C and A sows a cost saving of $6.163 for strategy C over strategy A. Te lower ICER for strategy C indicates tat strategy A is strongly dominated. Tat is, strategy A is more costly and less effective tan strategy C. Terefore, strategy A is excluded from te set of alternatives so it does not consume limited resources. We recalculate ICER Strategies Total infection averted Total costs ($) ICER Strategy C Strategy B Strategy D Te comparison between strategies C and B sows a cost saving of $6.163 for strategy C over strategy B. Similarly, te ig ICER for strategy B indicates tat strategy B is strongly dominated. Tat is, strategy B is more costly and less effective tan strategy C. Terefore, strategy B is excluded from te set of alternatives so it does not consume limited resources. We recalculate ICER Strategies Total infection averted Total costs ($) ICER Strategy C Strategy D Wit tis result, we conclude tat strategy C (combination of treatment of infective individuals and spray of insecticides) as te least ICER and terefore is more cost-effective tan strategy D.