Research Article Mathematical Analysis of a Malaria Model with Partial Immunity to Reinfection

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1 Abstract and Applied Analysis Volume 2013, Article ID , 17 pages ttp://dx.doi.org/ /2013/ Researc Article Matematical Analysis of a Malaria Model wit Partial Immunity to Reinfection Li-Ming Cai, 1 Abid Ali Lasari, 2 Il Hyo Jung, 3 Kazeem Oare Okosun, 4 and Young Il Seo 5 1 Department of Matematics, Xinyang Normal University, Xinyang , Cina 2 CentreforAdvancedMatematicsandPysics,NationalUniversityofSciencesandTecnology,H-12,Islamabad44000,Pakistan 3 Department of Matematics, Pusan National University, Busan , Republic of Korea 4 Department of Matematics, Vaal University of Tecnology, Andries Potgieter Boulevard, X021, Vanderbijlpark 1900, Sout Africa 5 National Fiseries Researc and Development Institute, Busan , Republic of Korea Correspondence sould be addressed to Li-Ming Cai; lmcai06@yaoo.com.cn Received 5 October 2012; Accepted 7 December 2012 Academic Editor: Sanyi Tang Copyrigt 2013 Li-Ming Cai et al. Tis is an open access article distributed under te Creative Commons Attribution License, wic permits unrestricted use, distribution, and reproduction in any medium, provided te original work is properly cited. A deterministic model wit variable uman population for te transmission dynamics of malaria disease, wic allows transmission by te recovered umans, is first developed and rigorously analyzed. Te model reveals te presence of te penomenon of backward bifurcation, were a stable disease-free equilibrium coexists wit one or more stable endemic equilibria wen te associated reproduction number is less tan unity. Tis penomenon may arise due to te reinfection of ost individuals wo recovered from te disease. Te model in an asymptotical constant population is also investigated. Tis results in a model wit mass action incidence. A complete global analysis of te model wit mass action incidence is given, wic reveals tat te global dynamics of malaria disease wit reinfection is completely determined by te associated reproduction number. Moreover, it is sown tat te penomenon of backward bifurcation can be removed by replacing te standard incidence function wit a mass action incidence. Grapical representations are provided to study te effect of reinfection rate and to qualitatively support te analytical results on te transmission dynamics of malaria. 1. Introduction Malaria is a mosquito-borne disease caused by a parasite. It is endemic and widespread in tropical and subtropical regions, including muc of sub-saaran Africa, Asia, and te Americas. Malaria is still a public ealt problem today. Every year, tere are more tan 225 million cases of malaria, killing around 781,000 people according to te World Healt Organization s 2010 World Malaria Report [1]. In umans, malaria is caused due to infection by one of four Plasmodium species [2, 3]. Transmission from mosquito to uman occurs during a bite by an infectious mosquito. A mosquito becomes infected wen it takes a blood meal from an infected uman. Once ingested, te parasite gametocytes taken up in te blood will furter differentiate into gametes and ten fuse in te mosquito s gut. Gametocytes are responsible for transmission of te parasite from umans by mosquitoes bite. Fertilization of te parasite occurs in te mosquito gut, and after a sort period of replication and development, te cycle of transmission may begin anew. One of te most complex features of te epidemiology of malaria is te dynamic interaction between infection and immunity. A better understanding of tis interaction is important for evaluating te impact of malaria control activities. An important penomenon is noticed tat te canges wit age reflect te slow acquisition of an immunity tatreducesillnessbutdoesnotcompletelyblockinfection [4, 5]. In endemic areas, cildren younger tan five years ave repeated and often serious attacks of malaria. Te survivors

2 2 Abstract and Applied Analysis develop and maintain partial immunity tat reduces te severity of te disease but does not prevent subsequent infections. Tus, in tese areas older cildren and adults often ave become asymptomatic carriers of infection [6]. In areas of low malaria transmission, immunity develops slowly and may take years or decades and probably never results in sterile immunity [7]. Terefore, umans are susceptible to reinfections. Incomplete immunity to malaria complicates disease control strategies [8, 9] as te partially immune individuals suffer only mild infections, and migt not seek medical attention but continue to transmit te parasite in te community. Te enormous public ealt burden inflicted by malaria disease necessitates te use of matematical modeling and analysis to gain insigts into its transmission dynamics, and to determine effective control strategies. Te earliest malaria transmission models can be traced to te model formulated by Ross in 1911 [9]. He used a matematical model and sowed tat bringing a mosquito population below a certain tresold was sufficient to eliminate malaria. Tis tresold naturally depended on biological factors suc as te biting rate and vectorial capacity. To estimate infection and recovery rates, MacDonald extended te Ross model in 1957 [9]. Macdonald s model sows tat reducing te number of mosquitoes is an inefficient control strategy. Moreover, tis would ave little effect on te epidemiology of malaria in areas of intense transmission. Since ten, teemergenceandreemergenceofmalariadiseasesave promoted many autor s interest in matematical modeling to describe and to predict te transmission dynamics of malaria in te literature (see, e.g., [10 16] and te references terein). In paper [12], Dietz et al. applied te Garki model to sow tat te duration of acquired immunity in umans in malaria depends on repeated exposure. In paper [13], Niger and Gumel constructed a matematical model tat includes multiple infected and recovered classes, to assess te role of te partial immunity on te transmission dynamics of malaria in a uman population. Teir results reveals te presence of te penomenon of backward bifurcation in te standard incidence model wit te disease-induced deat in te uman population. Recently, a transmission model of uman malaria in a partially immune population is formulated in Wan and Cui s paper [14]. Tey establised te basic reproduction number and explicit subtresold conditions for te model, and sowed tat if te disease induced deat rate is large enoug, te model undergoes a backward bifurcation. Li [15] formulateda malariatransmissionmodelwitpartialimmunityinumansandsowedtat te establised model aving te same reproductive number but different numbers of progression stages can exibit different transient dynamics. Tus, te above mentioned models always let te recovered individuals return into te susceptible class to explore te transmission dynamics of diseases. But tis only takes states of complete immunity and full susceptibility in consideration. In addition, various vector-borne disease model concerning malaria transmission ave been establised and discussed [17 20]. For example, in paper [17], Yang et al ave investigated global stability of an epidemic model for vector-borne disease, owever, tey assumed tat te immunity of te recovered population ave never lose. Motivated by te recent work of [13, 15], in tis paper, we sall continue to construct a malaria transmission model wit partial immunity to reinfection in te recovered uman population. Our purpose is to explore te transmission dynamics of te malaria and to assess te role of partial immunity to reinfection on te transmission dynamics of malaria in a uman population. Te organization of tis paper is as follows: in te next section, te standard incidence malaria model, wic incorporates te partial immunity to reinfection, is formulated. Te existence and stability of te equilibria, and te penomena of te backward bifurcation are, respectively, explored in Sections 2.2 and 2.3. Grapical representations are provided to study te effect of reinfection rate in Section 2.4. In Section 3, te associated mass action incidence model is formulated, and matematical results suc as existence and local stability of equilibria are provided in Section 3.2. Our main teorems for te global stability of equilibria for te mass action model and te proofs are given in Section 3.3. Te paper ends wit a conclusion in Section Model Formulation We formulate a model for te spread of malaria in te uman and mosquito population, wit te total population size at time t given by N (t) and N v (t), respectively.te total uman population is divided into tree epidemiological classes: S (t), I (t),andr (t), wic denote, respectively, te number of te susceptible, infective, and immune class at time t. Tus,N (t) = S (t) + I (t) + R (t). Tesusceptible uman population is generated by te recruitment of umans (assumed susceptible) into te community at a rate Λ, μ, and γ are, respectively, te natural deat rate and recovery rate in uman osts population. Also, some disease-induced deat in uman population contributes to an additional population decrease at te constant rate δ. Due to its sort life, a mosquito never recovers from te infection,andwemaynotconsiderterecoveredclassintis population. Tus, te total vector population N v (t) is divided into te susceptible class, S v (t), andinfectiveclass, (t), so tat N v (t) = S v (t) + (t).susceptiblemosquitoesvectorsare generated at a rate Λ v by birt, μ v is te per capita mortality rate of mosquitoes. Let β be te transmission probability from vector to uman, and β v be te transmission probability from uman to vector. Te parameter b is te average number of bites per mosquito per day. Tis rate depends on a number of factors, in particular, climatic ones, but for simplicity in tis paper we assume b to be a constant. Te parameter σ(0 σ 1) determines te degree of partial protection for te recovered individuals given by a primary infection: σ = 0 implies complete protection, and σ = 1 implies no protection. Taking into account te assumptions made above, te interaction between uman osts and te mosquito vector population wit partial immunity to reinfection in

3 Abstract and Applied Analysis 3 ost population is described by te following system of equations: ds dt =Λ bβ S N μ S, 2.1. Basic Properties of te Model. Since te model (4) monitors umans ost and mosquitoes vector populations, it is plausible to assume tat all its state variables and parameters are nonnegative for all t 0.Furter,itcanbesowntatte region Ω given by di dt = bβ S + σbβ R (γ +δ +μ ) I, N N Ω= {(S,I,N, ) R 4 + : 0 S +I N 1, 0 1} (5) dr =γ dt I σbβ R μ N R, ds v dt =Λ v bβ vs v I μ v S v, N d dt = bβ vs v I N μ v. Te total umans ost and mosquitoes vector populations N = S +I +R and N v = S v + are governed, respectively, by dn =Λ dt μ N δ I, dn v =Λ dt v μ v N v. It is easily seen tat for te mosquitoes vector population te corresponding total population size is asymptotically constant: lim t t N v (t) = Λ v /μ v. Tis implies tat in our model we assume witout loss of generality tat N v (t) = Λ v /μ v,forallt 0,providedtatS v (0) + (0) = Λ v /μ v. Let S = S v = S Λ /μ, I = I Λ /μ, R = R Λ /μ, S v I, I Λ v /μ v = v, N v Λ v /μ = N. v Λ /μ Since N =S +I +R and S v + =1.UsingR =N S I and S v =1,system(1) is reduced to te following fourdimensional nonlinear system of ODEs: ds dt =μ bmβ S N μ S, di dt = bmβ S + σbmβ (N S I ) α N N 1 I, (1) (2) (3) is positively invariant wit respect to system (4). Tus, every solution of te model (4), wit initial conditions in Ω remains tere for t > 0. Terefore, it is sufficient to consider te dynamicsofteflowgeneratedby(4) inω. Intisregion, te model can be considered as been epidemiologically and matematically well posed Stability of Disease-Free Equilibria. Te stability of te disease-free equilibrium state can be obtained from studying te eigenvalues of te Jacobian matrix evaluated at te equilibriumpoint.ifallteeigenvaluesavenegativereal parts, ten te equilibrium point is stable. Te disease-free equilibrium for te system (4) ise 0 (1,0,1,0). Te Jacobian matrixattedisease-freeequilibriume 0 is μ 0 0 bmβ 0 α J=( 1 0 bmβ ). (6) 0 δ μ 0 0 bβ v 0 μ v Te caracteristic equation of te above matrix is (λ μ )(λ μ )(λ 2 +(α 1 +μ v )λ+α 1 μ v (1 R 0 ))= 0, (7) were R 0 = b 2 mβ β v /μ v α 1. Tere are four eigenvalues corresponding to (7). Two of te eigenvalues λ 1, λ 2 = μ ave negative real parts. Te oter two eigenvalues can be obtained from te equation λ 2 +(α 1 +μ v )λ+α 1 μ v (1 R 0 )=0. (8) Applying te Rout-Hurwitz criteria for a quadratic polynomial. It is easy to see tat bot te coefficients of (8) are positive if an only if R 0 < 1.Tus,allrootsof(8) are wit negative real parts if R 0 < 1,andoneofitsrootsis wit positive real part if R 0 >1. Terefore, te disease-free equilibrium (DFE) E 0 is locally asymptotically stable if R 0 <1 and unstable if R 0 >1.Tus,weavetefollowingresult. Teorem 1. Te uninfected equilibrium E 0 is locally asymptotically stable if R 0 <1and unstable if R 0 >1in Ω. d dt = bβ v (1 )I μ N v, dn =μ dt μ N δ I, were m=(λ v /μ v )/(Λ /μ ) and α 1 =γ +μ +δ. (4) From Teorem 1,tetresoldquantityR 0,iscalledte basic reproduction number of system (4). Te basic reproduction number, R 0 measures te average number of new malaria infections generated by a single infected individual in a completely susceptible population [21]. Teorem 1 also implies tat malaria can be eliminated from te community (wen R 0 < 1) if te initial sizes of te subpopulations of

4 4 Abstract and Applied Analysis te model are in te basin of attraction of te disease-free equilibrium (DFE) (E 0 ). To ensure tat disease elimination is independent of te initial sizes of te subpopulations, it is necessary to sow tat te DFE is globally asymptotically stable (GAS) if R 0 <1. Tis is explored for a special case in Section Te Existence of Endemic Equilibria and Backward Bifurcation. In tis section, conditions for te existence of endemic equilibria and penomenon of backward bifurcation in system (4) will be determined. In order to do tis, we let E =(S,I,N,I v ) (9) represent an arbitrary endemic equilibrium of te model (4). Setting te rigt-and sides of te equations in (4)tozeroand solving tem in terms of I gives te following expressions for te state variables of te model S = (μ δi )(α 2I +μ μ v ) b 2 mμ β β v I +(μ δi )(α 2I +μ μ v ), (10) N = μ δ I μ, I v = μ bβ v I α 2 I +μ μ v, (11) were α 2 = μ bβ v δ μ v,andi is determined from te following equation: were K 1 I 4 K 1 =δ 2 α 1α 2 2, +K 2I 3 +K 3I 2 +K 4I +K 5 =0, (12) K 2 =α 1 ( 2δ α 2 (μ α 2 δ μ μ v ) δ α 2 μ b 2 mβ β v ) μ δ α 2 α 3 σb 2 mβ β v, K 3 =α 1 ((μ α 2 δ μ μ v ) 2 2μ 2 μ vδ α 2 +μ b 2 mβ β v (μ α 2 δ μ μ v )) +μ 2 δ α 2 b 2 mβ β v +μ α 3 σb 2 mβ β v ((μ α 2 δ μ μ v )+μ b 2 mβ β v ), K 4 =α 1 (2μ 2 μ v (μ α 2 δ μ μ v )+μ 3 μ vb 2 mβ β v ) μ 2 b2 mβ β v (μ α 2 δ μ μ v ) σμ 3 b2 mβ β v (b 2 mβ β v α 3 μ v ) K 5 =α 1 μ 4 μ2 v (1 R 0), (13) and α 3 =μ +δ. It follows from (13)tatK 1 >0.Furter,K 5 >0wenever R 0 < 1. Tus, te number of possible positive real roots for (12) depends on te signs of K 2, K 3 and K 4.Tiscanbe analyzed using te Descartes Rule of Signs on te quartic f(i )=K 1I 4 +K 2 I 3 +K 3 I 2 +K 4 I +K 5.Tevarious possibilities for te roots of f(i ) are tabulated in Table 1. We ave te following result from te various possibilities enumerated in Table 1. Teorem 2. Te system (4) as a unique endemic equilibrium E if R 0 >1and Cases 1 3 and 6 are satisfied; it could ave more tan one endemic equilibrium if R 0 >1and Cases 4, 5, 7, and 8 are satisfied; it could ave 2 or more endemic equilibria if R 0 <1and Cases 2 8 are satisfied. Te existence of multiple endemic equilibria wen R 0 < 1 (is sown in Table 1). Table 1 suggests te possibility of backward bifurcation (see [22 24]), were te stable DFE coexists wit a stable endemic equilibrium, wen te reproduction number is less tan unity. Tus, te occurrence of a backward bifurcation as an important implications for epidemiological control measures, since an epidemic may persist at steady state even if R 0 <1. Tis is explored below by using Centre Manifold Teory (see, e.g., [25] andte references terein). Now, we sall establis te conditions on parameter values tat cause a backward bifurcation to occur in system (4),basedonteuseofCenterManifoldteory,oftepaper in Castillo-Cavez and Song [25]. Teorem 3. Let one consider te following general system of ordinary differential equations wit a parameter φ: dx dt =f(x, φ), f: Rn R R n, f C 2 (R R). (14) Witoutlossofgenerality,itisassumedtatx = 0 is an equilibrium for system (14) for all values of te parameter φ. Assume tat (A1) A = D x f(0, 0) is te linearized matrix of system (14) around te equilibrium x = 0 wit φ evaluated at 0. Zero is a simple eigenvalues of A and all oter eigenvalue of A ave negative real parts; (A2) Matrix A as a nonnegative rigt eigenvector w and a left eigenvector v corresponding to te zero eigenvalue. Let f k be te kt component of f and a 1 = b 1 = 5 2 f v k w i w k (0, 0) j, x i,j,k=1 i x j 5 i,k=1 2 f v k w k (0, 0) i. x i β (15) Te local dynamics of system (14) around 0 are totally determined by a 1 and b 1. (i) In te case were a 1 >0, b 1 >0, one as tat wen φ<0wit φ close to zero, x=0is unstable; wen 0<φ 1, x=0is unstable and tere exists a negative and locally asymptotically stable equilibrium; (ii) In te case were a 1 <0, b 1 <0, one as tat wen φ< 0 wit φ close to zero, x=0islocally asymptotically

5 Abstract and Applied Analysis 5 Table 1: Number of possible positive real roots of f(i ) for R 0 >1and R 0 <1. Cases K 1 K 2 K 3 K 4 K 5 R 0 Number of sign cange Number of positive real roots R 0 < R 0 > R 0 <1 2 0, 2 + R 0 > R 0 <1 2 0, R 0 > R 0 <1 4 0, 2, R 0 >1 3 1, R 0 <1 2 0, R 0 >1 3 1, R 0 <1 2 0, R 0 > R 0 <1 2 0, R 0 >1 3 1, R 0 <1 2 0, R 0 >1 3 1, 3 stable and tere exists a positive unstable equilibrium; wen 0<φ 1, x=0islocally asymptotically stable, and tere exists a positive unstable equilibrium; (iii) In te case were a 1 > 0, b 1 < 0, one as tat wen φ<0wit φ close to zero, x=0is unstable and tere exists a locally asymptotically stable negative equilibrium; wen 0<φ 1, x=0is stable and a positive unstable equilibrium appears; (iv) In te case were a 1 <0, b 1 >0, one as tat wen φ<0canges from negative to positive, x=0canges its stability from stable to unstable. Correspondingly a 1 negative unstable equilibrium becomes positive and locally asymptotically stable. Particularly, if a 1 >0and b 1 >0, ten a backward bifurcation occurs at φ=0. To apply te center manifold metod, te following simplification and cange of variables are made on te model (4). First of all, let x 1 =S, x 2 =I, x 3 =R, x 4 =S v,and x 5 =,sotatn =x 1 +x 2 +x 3 and N v =x 4 +x 5.Furter,by using te vector notation X=(x 1,x 2,x 3,x 4,x 5 ) T,tesystem (4)canbewritteninteform(dX/dt) = (f 1,f 2,f 3,f 4,f 5 ) T as follows: Coose β as a bifurcation parameter and solving R 0 =1 gives β =β = μ vα 1 b 2 mβ v. (17) TeJacobianmatrixevaluatedatdisease-freeequilibrium (1,0,0,1,0)wit β =β is μ bmβ 0 α bmβ J=( 0 γ μ 0 0 ). (18) 0 bβ v 0 μ v 0 0 bβ v 0 0 μ v It can be easily seen tat te Jacobian J of te linearized system as a simple zero eigenvalue and all oter eigenvalues ave negative real parts. Hence, te center manifold teory canbeusedtoanalyzetedynamicsoftesystem(16). For te case wen R 0 =1, it can be sown tat te Jacobian matrix J as a rigt eigenvector (corresponding to te zero eigenvalue) given by w =[w 1 w 2 w 3 w 4 w 5 ] T,were dx 1 dt =f 1 =μ bmβ x 1 x 5 x 1 +x 2 +x 3 μ x 1, w 1 = bmβ μ w 5, w 2 = bmβ α 1 w 5, w 3 = γ bmβ μ α 1 w 5, dx 2 dt =f 2 = bmβ x 1 x 5 + σbmβ x 3 x 5 α x 1 +x 2 +x 3 x 1 +x 2 +x 1 x 2, 3 dx 3 =f dt 3 =γ x 2 σbmβ x 3 x 5 μ x 1 +x 2 +x x 3, 3 dx 4 dt =f 4 =μ v bβ vx 4 x 2 x 1 +x 2 +x 3 μ v x 4, (16) w 4 = b2 mβ β v μ v α 1 w 5, w 5 =w 5 >0. (19) Similarly, te components of te left eigenvector of J (corresponding to te zero eigenvalue), denoted by v = [v 1 v 2 v 3 v 4 v 5 ],aregivenby dx 5 dt =f 5 = bβ v x 4 x 2 x 1 +x 2 +x 3 μ v x 5. v 1 =v 3 =v 4 =0, v 2 = bβ v α 1 v 5, v 5 =v 5 >0. (20)

6 6 Abstract and Applied Analysis Computation of a 1 :fortetransformedsystem(16), te associated non-zero partial derivatives of f (evaluated at te DFE) wic we need in te computation of a 1 are given by 2 f 2 x 2 x 5 = 2 f 2 x 5 x 2 = bmβ, I Stable endemic equilibrium 2 f 2 x 3 x 5 = 2 f 2 x 5 x 3 = bmβ +σbmβ, 2 f 5 x 1 x 2 = 2 f 5 x 2 x 1 = bβ v, 2 f 5 x 2 x 3 = 2 f 5 x 3 x 2 = bβ v, 2 f 5 x 2 2 = 2bβ v, 2 f 5 = 2 f 5 =bβ x 2 x 4 x 4 x v. 2 (21) 0.01 Unstable endemic Stable equilibrium Unstable disease disease free free equilibrium equilibrium Figure 1: Simulations of te model (4) illustrating te penomenon of backward bifurcation. R 0 Direct calculations sows tat a 1 =2b( mβ v 2 w 2 w 5 mβ v 2 w 3 w 5 +σmβ v 2 w 3 w 5 β v v 5 w 1 w 2 β v v 5 w 2 2 β v v 5 w 2 w 3 +β v v 5 w 2 w 4 ). (22) Computation of b 1 :Substitutingtevectorsv and w and te respective partial derivatives (evaluated at te DFE ) into te expression b 1 = 5 i,k=1 2 f v k w k (0, 0) i (23) x i β gives b 1 = bmv 2 w 5 > 0. Since te coefficient b 1 is automatically positive, it follows tat te sign of te coefficient a 1 decides te local dynamics around te disease-free equilibrium for β =β. Based on Teorem 3,system(4)will undergo backward bifurcation if te coefficient a 1 is positive. Te coefficient a 1 is positive if and only if σ> 1 (μ γ μ v (γ +μ )+bμ β v μ v δ ). (24) v Tus, we ave te following result. Teorem 4. Te system (4) exibits backward bifurcation wenever te condition (24) olds. Te backward bifurcation penomenon is illustrated by simulating te system (4) wit te following set of parameter values μ = , μ v = 0.015, Λ v =4, Λ =3, β v = 0.2, b = 0.4, δ = , γ = (so tat, a 1 >0and R 0 <1). Figure 1 depicts te associated backward bifurcation diagram Te Effect of te Reinfection. We furter investigate te effect of te reinfection parameter σ and te transmission probability from an infectious uman to a susceptible vector β on te associated backward bifurcation region, as a function of te average life span of mosquitoes (1/μ v ).Te backward bifurcation region is illustrated (Figures 2 4) by simulating te model (4) wit te following set of parameter values (note tat te parameters are cosen in order to illustrate te backward bifurcation region, and may not all be realistic epidemiologically), Λ = 30, Λ v = 24, β v = 0.09, δ = 0.2, γ = , μ = , b = 0.4, μ v = 0.2.Alsotobenotedis,teparametervaluesarecosensuc tat a 1 >0,b 1 >0and R 0 <1(so tat backward bifurcation occurs). Solving for a 1 >0in terms of 0 σ 1and β >0 (i.e., fixing all parameters in te expression for a 1 except β and σ) we obtained te backward bifurcation region for β.figure2 depicted te results obtained for σ = 0.5, it sows tat te region for backward bifurcation (for β ) increases as te average life span of vectors (1/μ v ) decreases. For instance, wen te average life span of vectors is 20 days (μ v = 0.05), te backward bifurcation region for β is β [ , ], assowninfigure2(a). Wen te average life span of vectors is decreased to 10 days (μ v = 0.1), te backward bifurcation region for β increases to β [0.2077, ] (Figure 2(b)). Furtermore, wen te average life span of vectors is decreased to 5 days (μ v =0.2), te backward bifurcation region for β increases to β [0.4153, ] (Figure 2(c)). Similar results are obtained for te cases σ = 0.6 (Figures 3(a), 3(b), and3(c)) andσ = 1 (Figures 4(a), 4(b), and4(c)), fromwicitisevidenttatte backward bifurcation regions for β increase wit increasing values ofte reinfectionrateσ. Tese results are tabulated in Table 2 (1/μ v represents average life span of vectors.) 3. Te Mass Action Model In tis section, we sall investigate te dynamics of system (1) if mass action incidence is used instead of te standard incidence function. Tus te resulting (mass action) model is S (t) =Λ bβ S (t) (t) μ S (t), I (t) =bβ S (t) (t) +σbβ R (t) (t) (μ +γ +δ )I (t),

7 Abstract and Applied Analysis 7 β a 1 >0, backward bifurcation region for β [0.1038, ] a 1 =0 a 1 <0, no backward bifurcation (a) σ β = β a 1 >0, backward bifurcation region for β [0.4153,1.4217] a 1 >0, backward bifurcation region for β [0.2077, ] a 1 =0 a 1 <0, no backward bifurcation β = σ (b) β a 1 = β = a 1 <0, no backward bifurcation σ (c) Figure 2: Backward bifurcation regions for te model (4) inteσ-β parameter space corresponding to σ = 0.5 and various ranges of β. Parametervaluesusedare:Λ = 30, Λ v = 24, δ = 0.2, γ = , μ = , b = 0.4, μ v = , β v = 0.09.In(a)μ v = 0.05, R 0 = (backward bifurcation region for β is β [0.1038, ]),(b)μ v = 0.1, R 0 = (backward bifurcation region for β is β [0.2077, ]),and(c)μ v = 0.2, R 0 = (backward bifurcation region for β is β [0.4153, ]).Witteabove set of parameter values, a 1 >0, b 1 >0,andR 0 <1. R (t) =γ I (t) σbβ R (t) (t) μ R (t), S v (t) =Λ v bβ v I (t) S v (t) μ v S v (t), I v (t) =bβ vi (t) S v (t) μ v (t), (25) were te prime ( ) stands for te derivative wit respect to time t and initial conditions S (0) 0, I (0) 0, R (0) 0, and S v (0) 0, (0) Basic Properties: Positivity and Invariant Regions. Te dynamics of te total uman population, obtained by adding first tree equations in te model (25), is given by N (t) =Λ μ N (t) δ I (t). (26) Tus, we ave Λ +(N μ +δ (0) Λ )e (μ +δ )t μ +δ N (t) Λ μ +(N (0) Λ μ )e μ t. (27) Tus, for a low level of disease induced deat rate (δ 0) total uman population could eventually assume a steadystate value. Motivated by tis, we consider a uman population wic assumes a steady-state value Λ /μ stationary. Similarly, te dynamics of te total mosquito population, obtained by adding last two equations in te model (25), is given by, N v (t) = Λ v μ v N v (t), sotat,n v (t) = Λ v /μ v as t.

8 8 Abstract and Applied Analysis β a 1 >0, backward bifurcationregion for β [0.1036, ] a 1 =0 β a 1 >0, backward bifurcation region for β [0.2071, ] a 1 = β = a 1 <0, no backward bifurcation (a) σ β = a 1 <0, no backward bifurcation σ (b) a 1 >0, backward bifurcation region for β [0.4142, ] β a 1 = a 1 <0, no backward bifurcation β = (c) Figure 3: Backward bifurcation regions for te model (4) inteσ-β parameter space corresponding to σ = 0.6 and various ranges of β. Parameter values used are: Λ = 30, Λ v = 24, δ = 0.2, γ = , μ = , b = 0.4, μ v = , β v = In(a)μ v = 0.05, R 0 = (backward bifurcation region for β is β [0.1036, ]),(b)μ v = 0.1, R 0 = (backward bifurcation region for β is β [0.2071, ]),and(c)μ v = 0.2, R 0 = (backward bifurcation region for β is β [0.4142, ]).Witteabove set of parameter values, a 1 >0, b 1 >0,andR 0 <1. σ Let H(t) = (S (t), I (t), R (t)) and V(t) = (S v (t), (t)). Based on te above discussion, we define a region Γ= {(H (t),v(t)) R 3 + R2 + 0 H(t) Λ μ, 0 V(t) Λ v μ v }. (28) It is easy to verify tat Γ is positively invariant wit respect to te system (25). In tis part, it is sufficient to consider te dynamics of te flow generated by (25)in Γ Equilibrium and Local Stability. In tis section, we investigate te existence and local stability of equilibria of system (25). Obviously, te system (25) always as adisease-freeequilibriume 0 mass (Λ /μ,0,0,λ v /μ v,0).let E mass (S,I,R,S v,i v ) represents any arbitrary endemic equilibrium of te model (25). Solving te equations in (25) at steady state gives S v = S = R = Λ μ v (bβ v I +μ v) b 2 Λ v β β v I +μ μ v (bβ v I +μ v), γ μ v (bβ v I +μ v)i b 2 Λ v σβ β v I +μ μ v (bβ v I +μ v), Λ v bβ v I +μ v, I v = bλ v β v I μ v (bβ v I +μ v), (29)

9 Abstract and Applied Analysis a 1 >0, backward bifurcation region for β [0.1025, ] a 1 >0, backward bifurcation region for β [0.2051, ] a 1 = a 1 =0 β β a 1 <0, no backward bifurcation σ (a) a 1 <0, no backward bifurcation σ (b) a 1 >0, backward bifurcation region for β [0.4101,1.4217] β a 1 = a 1 <0, no backward bifurcation σ (c) Figure 4: Backward bifurcation regions for te model (4) inteσ-β parameter space corresponding to σ=1and various ranges of β. Parameter values used are: Λ = 30, Λ v = 24, δ = 0.2, γ = , μ = , b = 0.4, μ v = , β v = In(a)μ v = 0.05, R 0 = (backward bifurcation region for β is β [0.1025, ]),(b)μ v = 0.1, R 0 = (backward bifurcation region for β is β [0.2051, ]),and(c)μ v = 0.2, R 0 = (backward bifurcation region for β is β [0.4101, ]).Witteabove set of parameter values, a 1 >0, b 1 >0,andR 0 <1. were I equation, wit is te positive root of te following quadratic d 1 I 2 +d 2I +d 3 =0, (30) d 1 =b 2 β 2 v (μ +δ )(σbλ v β +μ μ v )(bλ v β +μ μ v ) +γ b 2 β 2 v μ μ v (bλ v β +μ μ v )>0, d 2 =bβ v μ μ 2 v (μ +γ +δ )(bλ v β +μ μ v ) +bβ v μ μ 2 v (μ +δ )(σbλ v β +μ μ v )+γ bβ v μ 2 μ3 v b 3 Λ v Λ β β 2 v (bσλ vβ +μ v μ ), d 3 = (μ +γ +δ )μ 2 μ4 v (1 b 2 Λ Λ v β β v ). (μ +γ +δ )μ μv 2 (31) Te dynamics of te model (25)areanalyzedbyR mass given by b 2 Λ R mass = Λ v β β v. (32) (μ +γ +δ )μ μv 2 Te tresold quantity R mass is te basic reproduction number of te system (25). It can be derived from te Jacobian matrix of te system (25) at te disease-free equilibrium E 0 mass togeter wit te assumption of local asymptotical stability

10 10 Abstract and Applied Analysis Average life span of vectors (1/μ v ) Table 2: Backward Bifurcation Ranges for β for Various Values of 1/μ v and σ. σ = 0.5 σ = 0.6 σ =1 20 days β [0.1038, ] β [0.1036, ] β [0.1025, ] 10 days β [0.2077, ] β [0.2071, ] β [0.2051, ] 5days β [0.4153, ] β [0.4142, ] β [0.4101, ] of E 0 mass [21]. From (31), we see tat R mass >1if and only if, d 3 < 0.Sinced 1 > 0,(30) asauniquepositiveroot in feasible region Ω. IfR mass < 1,tend 3 > 0.Also b 2 Λ Λ v β β v <(μ +γ +δ )μ μ v,isequivalenttor mass <1. Hence, d 2 >0. Tus, by considering te sape of te grap f(i )=d 1 I 2 +d 2I +d 3 (and noting tat d 1 >0), we ave tat tere will be zero endemic equilibria in tis case. Terefore, we can conclude tat if R mass <1,(30)asnopositiverootin te feasible region Γ.If,R mass =1,(30)asauniquepositive root in te feasible region Γ. Tis result is summarized below. Teorem 5. System (25) always as te infection-free equilibrium E 0 mass = ((Λ /μ ), 0, 0, (Λ v /μ v ), 0). If R mass > 1, system(25) as a unique endemic equilibrium E mass = (S,I,R,S v,i v ) defined by (29) and (30). Linearizing te system (25) around te disease-free equilibrium E 0 mass yields te following caracteristic equation: (λ + μ ) 2 [λ 2 +(μ +γ +δ +μ v )λ +(μ +γ +δ )μ v b 2 Λ β β Λ v v ]=0. μ μ v (33) Twoofterootsoftecaracteristicequation(33) λ 1,2 = μ, ave negative real parts. Te oter two roots can be determine from te quadratic term in (33) and ave negative real parts if and only if R mass <1. Terefore, te disease-free equilibrium E 0 mass is locally asymptotically stable for R mass < 1.Wen R mass >1, E 0 mass becomes an unstable equilibrium point, and te endemic equilibrium E mass emerges in Γ. Tis result is summarized below. Teorem 6. Te disease-free equilibrium E 0 mass of system (25) is locally asymptotically stable if R mass < 1 and unstable if R mass >1. In order to discuss te stability of te endemic equilibrium E mass and to simplify our calculations, we assume bot umans and mosquitoes populations are at steady state. Tus, using N v =S +I +R =λ v /μ,andn v =S v + =λ v /μ v, system (25) inteinvariantspaceγ can be written as te following equivalent tree dimensional nonlinear system of ODEs: S (t) =Λ bβ S (t) (t) μ S (t), I (t) =bβ S (t) (t) +σbβ (N S (t) I (t)) (t) (μ +γ +δ ) I (t), I v (t) =bβ vi (t) (N v (t)) μ v (t). (34) Now, linearization of system (34) about an endemic equilibrium E mass gives te following caracteristic equation: λ+μ +bβ I v 0 bβ S bβ (σ 1) I v λ+α 1 +bσβ I v α 1I I v 0 μ vi v I λ+bβ v I +μ v were, α 1 =μ +γ +δ. Expanding (35)gives were =0, (35) λ 3 +Q 1 λ 2 +Q 2 λ+q 3 =0, (36) Q 1 =μ +bβ I v +α 1 +bσβ I v +μ v +bβ v I >0, Q 2 = (μ +bβ I v )(α 1 +bσβ I v +μ v +bβ v I ) +bβ v I (α 1 +bσβ I v )+μ vbσβ I v >0, Q 3 = (μ +bβ I v )[bβ vi (α 1 +bσβ I v )+μ vbσβ I v ] +b 2 β 2 (1 σ) μ v I I v S v I >0. (37) From te second equation of system (34) at steady state E, we ave α 1 = bβ S I v (1 σ) +bσn β I v I bσβ I v. (38)

11 Abstract and Applied Analysis 11 Using (37), direct calculations sow tat Q 1 Q 2 Q 3 = (μ +bβ I v +α 1 +bσβ I v +μ v +bβ v I ) [bβ v I (α 1 +bσβ I v )+μ vbσβ I v +(μ +bβ I v )(α 1 +bσβ I v +μ v +bβ v I )] [(μ +bβ I v )[bβ vi (α 1 +bσβ I v )+μ vbσβ I v ] +b 2 β 2 (1 σ) μ v I I v S v I ]. (39) Obviously, te term (μ +bβ I v ) in first bracket times te term (bβ v I (α 1+bσβ I v )+μ vbσβ I v ) intesecondbracketis (μ +bβ I v )[bβ vi (α 1 +bσβ I v )+μ vbσβ I v ]. Multiplying te terms under straigt line and using (30), we ave μ v bβ (α 1 + bσβ I v ) > b2 β 2 (1 σ)i v S (μ vi v /I ).Hence,Q 1Q 2 Q 3 > 0. Tus, by Rout Hurwitz criteria, te following result is establised. Teorem 7. Te endemic equilibrium E mass of te reduced model (34) is locally asymptotically stable if R mass > Global Stability of te Equilibria. In tis section, te global stability of te equilibria of system (34) will be explored. First, we claim te following teorem. Teorem 8. If R mass 1, ten te infection-free-equilibrium E 0 mass of system (34) is globally asymptotically stable in Γ. Proof. To establis te global stability of te disease-free equilibrium E 0 mass, we construct te following Lyapunov function L (t) =bβ v I (t) + (μ +γ +δ ) (t). (40) Calculating te derivative of L (were a dot represents differentiation wit respect to t) along te solutions of (34) we obtain L (t) =bβ v I (t) +(μ +γ +δ )I v (t), =bβ v [bβ S (t) (t)+bσβ (N S (t) I (t)) (t) (μ +γ +δ )I (t)] +(μ +γ +δ )[bβ v (N v (t))i (t) μ v (t)], =b 2 β v β (S (t) +σ(n S (t) I (t))) (t) (μ +γ +δ )μ v (t) b(μ +γ +δ )β v I (t) (t), =(μ +γ +δ )μ v b 2 β [ v β (μ +γ +δ )μ v (S (t) +σ(n S (t) I (t))) 1] (t) b(μ +γ +δ )β v I (t) (t), (μ +γ +δ )μ v (t) (R mass 1) b(μ +γ +δ )β v I (t) (t). (41) Tus, L (t) 0 if R mass 1 wit L (t) = 0 if and only if (t) = 0. Tus, from te second and te first equation of system (34), we ave lim t I (t) = 0, and lim t S (t) = Λ /μ. Terefore, te largest compact invariant set in {(S (t), I (t), (t)) α : L (t) = 0} is te singleton {(Λ /μ,0,0)} in Γ. Using te LaSalle s invariant principle [26], te infection-free equilibrium E 0 mass is globally asymptotically stable for R mass 1 in Γ. Te epidemiological implication of te above result is tat malaria will be eliminated from te population if R mass can be brougt to (and maintained at) a value less tan unity. Tus, te substitution of standard incidence wit mass action incidence in te model (1) removes te penomenon of backward bifurcation. Te result of Teorem 8 is illustrated numerically by simulating te model (30), for te case R mass <1,using various initial conditions. Te solution profiles obtained sows convergence to te DFE, as depicted in Figure 5. Now, we investigate te global stability of te endemic equilibrium E mass. We notice tat wen te incomplete immunity term 0 < σ < 1,system(30) isnolonger competitive. To investigate te global stability of E mass, we adopted a general approac of Li and Muldowney [27, 28], wic is developed for iger dimensional systems irrespective if tey are competitive. Wile te approac of Li and Muldowney as been successfully applied to many classes of epidemic models, we demonstrated in te present paper, for te first time, tat tis approac is also applicable to vector-ost model wic is non-competitive. We briefly state te approac developed recently in Li and Muldowney as follows: Let G R n be an open set and f:x f(x) R n be a C 1 function for x G. Consider te following differential equation: x =f(x). (42) Let x(t, x 0 ) denote te solution of (42) satisfyingx(0, x 0 )= x 0.Wemaketefollowingtwoassumptions. (H 1 ) Tere exists a compact absorbing set K G. (H 2 ) Eqution (42) as a unique equilibrium x in G.

12 12 Abstract and Applied Analysis Susceptible umans S Time (t) (a) Infected umans I Time (t) (b) Infected mosquito Time (t) (c) Figure 5: Simulations of te model (5) sowing (a) te number of susceptible umans, (b) te number of infected umans, (c) te number of infected mosquitoes, as a function of time using te parameters: b = 0.5, Λ =25, Λ V = 500, β 1 = , β 2 = 0.005, μ = 0.5 μ v = 0.06, σ = 0.8, δ = ,andγ = 0.6 (so tat R 0 = < 1). Let Q:x Q(x) be an ( n 2 ) ( n 2 ) matrix-valued function, tat is, it is C 1 and Q 1 (x) exists for x G.Letμbe a Lozinskii measure on R d d,wered=( n 2 ).Defineaquantityq 2 as t 1 q 2 = lim sup sup t x 0 E t 0 μ (M (x (s, x 0 ))) ds, (43) were M=Q f Q 1 +QJ [2] Q 1,tematrixQ f is obtained by replacing eac entry q ij of Q by its derivative in te direction of f, (q ij ) f,andj [2] is te second additive compound matrix of te Jacobian matrix J of system (42). Te following results ave been establised in Li and Muldowney [27, 28]. Teorem 9. For system (42), assume tat G is a simple connected and tat te assumptions (H 1 )and(h 2 )old.ten, teuniqueequilibriumx is globally asymptotically stable in G if tere exist a function Q(x) and a Lozinski i measure μ suc tat q 2 <0. From te above discussion, we know tat Γ is simply connect and E mass isteuniquepositiveequilibriumfor R mass >1in Γ. Toapply teresultofteorem9 to investigate te global stability of te infective equilibrium E mass, we first state and prove te following result. Teorem 10. If R mass > 1,tensystem(34) is uniformly persistent, tat is, tere exists ε>0(independent of initial conditions), suc tat lim inf t S (t) > ε, lim inf t I (t) > ε, and lim inf t (t) > ε. Proof. Similar to te proof of Teorem 3.4 in [29], we coose X = Γ,X 1 = int Γ, X 2 = bd(γ).itiseasytoobtaintat

13 Abstract and Applied Analysis 13 Y 2 = {(S,0,0) : 0 < S 1},Γ 2 = y Y2 Γ(y) = {E 0 },and {E 0 } is an isolated compact invariant set in X. Furtermore, let M={E 0 },tus,m is an acyclic isolated covering of Γ 2. Now, we only need to sow tat {(Λ /μ,0,0)}is a weak repeller for X 1. Suppose tat tere exists a positive orbit (S,I, ) of (34)suctat lim S t + (t) = Λ, μ lim t + I (t) =0, lim I t + v (t) =0. (44) Since R mass >1, tere exists a small enoug ε>0,suctat bλ Λ v β β v (1 ε) 2 >(μ +γ +δ )μ μ 2 v. (45) From (34), we coose t 0 >0large enoug suc tat wen t t 0,weave I (t) >bβ (1 ε) Λ μ (t) (μ +γ +δ )I (t), I v (t) >bβ v (1 ε) Λ v μ v I (t) μ v (t). Consider te following matrix M ε defined by (46) (μ +γ +δ ) bβ (1 ε) Λ μ M ε =( bβ v (1 ε) Λ ). (47) v μ μ v v Since M ε admits positive off-diagonal element, te Perron- Frobenius Teorem [26] implies tat tere is a positive eigenvector v=(v 1,v 2 ) for te maximum eigenvalue λ of M ε.from(45), we see tat te maximum eigenvalue λ is positive. Let us consider te following system: u 1 (t) =bβ (1 ε) Λ μ u 2 (t) (μ +γ +δ )u 1 (t), u 2 (t) =bβ v (1 ε) Λ v μ v u 1 (t) μ v u 2 (t). (48) Let u(t) = (u 1 (t), u 2 (t)) be a solution of (48)troug(lv 1,lv 2 ) at t=t 0,werel>0satisfies lv 1 <I (t 0 ), lv 2 < (t 0 ).Since te semiflow of (48) is monotone and M ε v>0,itfollowstat u i (t) are strictly increasing and u i (t) + as t +, contradicting te eventual boundedness of positive solutions of system (48). Tus, E 0 is weak repeller for X 1.Teproofis completed. From Teorem 10 and te boundedness of solutions, it follows tat a compact set M Γ exists in system (34). Terefore, in Teorem 9, bot assumptions (H 1 )and(h 2 )are satisfied for R mass >1. Now, we apply Teorem 9 to investigate te global stability of te endemic equilibrium E mass in te feasible region Γ. For te global stability of te endemic equilibrium E mass,weavetefollowingteorem. Teorem 11. If R mass >1, ten te infective equilibrium E mass of system (48) is globally asymptotically stable in int Γ. Proof. Te Jacobian matrix J evaluated at a general solution (S,I, ) of system (34)is J=( (μ +bβ ) 0 bβ S bβ (1 σ) (μ +γ +δ ) bσβ +σbβ (N I ) bβ S (1 σ) ), (49) ( 0 bβ v (N v ) (bβ v I +μ v ) ) and its corresponding second compound matrix J [2] takes te form (μ +bβ +μ +γ +bσβ ) J [2] = ( bβ v (N v ) ( bβ S (1 σ) +σbβ (N I ) (μ +bβ +bβ v I +μ v ) bβ S δ 0 bβ (1 σ) (m+μ v +bβ v I +bσβ ) ). (50) ) Set te function P(x) = P(S,I, )=diag(1, I /,I / ). Ten P f P 1 = diag(0, I /I I v /,I /I I v /).Moreover,

14 14 Abstract and Applied Analysis B=P f P 1 +PJ [2] P 1 = ( ( (bβ S (1 σ) +σbβ (N I )) I I v bβ S v I I I I v (μ I I +μ v v 0 +bβ +bβ v I ) I I v μ 0 bβ (1 σ) I I I v v γ δ μ v bβ v I bσβ (μ +bβ +μ +γ +δ +bσβ ) bβ v (N v ) I, (51) ) ) =( B 11 B 12 B 21 B 22 ), were B 11 = (μ +bβ +μ +γ +δ +bσβ ), B 12 = [(bβ S +σbβ (N S I )) /I, bβ S /I ], B 21 = [bβ v (N v )(I / ), 0] T,and B 22 According to paper [28], μ 1 (B 22 ) canbeevaluatedasfollows: μ 1 (B 22 )=max { I I I v (μ +μ v +bβ v I +σbβ ), = ( ( ( I I I v (μ +δ +μ v ) bβ bβ v I 0 bβ (1 σ), I I I v (+μ +γ +δ +μ v ) bθβ bβ v I ). ) (52) Let (u,v,w) be te vectors in R 3.Wecooseanormin R 3 as (u, v, w) = max{ u, v + w }, andletμ be te corresponding Lozinskii measure. From te paper [28], we ave te following estimate: were μ (B) sup {g 1,g 2 }, (53) g 1 =μ 1 (B 11 ) + B 12, g 2 =μ 1 (B 22 ) + B 21. (54) Tus, I I I v bσβ bβ v I μ δ γ μ v }, = I I I v (μ +μ v +bβ v I +σbβ ). g 1 = (μ +bβ +μ +δ +γ +bσβ ) +(bβ S +σbβ (N S I )) I, g 2 =bβ v (N v ) I + I I I v (56) (57) Here, B 12, B 21 are matrix norm wit respect to te l 1 vector norm, and μ 1 is te Lozinskii measure wit respect to l 1 norm. Tus, we ave From system (34), we ave (μ +μ v +bβ v I +σbβ ). μ 1 (B 11 )= (μ +bβ +μ +γ +δ +bσθβ ), B 12 =(bβ S +σbβ (N S I )) I, B 21 =bβ v (N v ) I. (55) I I = (bβ S +σbβ (N S I )) I (μ +δ +γ ), I v =bβ v (N v I V ) I μ v. (58)

15 Abstract and Applied Analysis 15 Susceptible umans S Time (t) (a) Infected umans I Time (t) (b) Infected mosquito Time (t) (c) Figure 6: Simulations of te model (5) sowing (a) te number of susceptible umans, (b) te number of infected umans, (c) te number of infected mosquitoes, as a function of time using te parameters: b = 0.5, Λ =25, Λ v = 500, β 1 = , β 2 = 0.005, μ = 0.03, μ v = 0.03, σ = 0.8, δ = ,andγ = 0.6 (so tat R 0 = > 1). Using (58)in(57)gives for t>t.tus,from(59)and(60), we get g 1 = I I (μ +bβ +bσβ ), g 2 = I I (μ +bβ v I +σbβ ). (59) g 1 I I (μ +bθβ ε+bσθβ ε), g 2 I I (μ +bβ v ε+σbβ ε). (61) From Teorem 9 we know tat for te uniform persistence constant ε > 0, tere exists a time T > 0 independent of x(0) Γ,tecompactabsorbingset,suctat Terefore, from (61), we ave μ(b) (I /I ) η for t>t, were I (t) >ε, (t) >ε (60) η=min {μ +bβ ε+bσθβ ε, μ +bβ v ε+σbβ ε}. (62)

16 16 Abstract and Applied Analysis Ten along eac solution (S (t), I (t), (t)) suc tat (S (0), I (0), (0)) Γ for t>t,givetefollowing: t 1 t μ (B) ds 1 T 0 t μ (B) ds t ln I (t) (t) ηt T, (63) t wic implies tat q 2 η/2 < 0. Tis proves tat te unique infective equilibrium E mass is globally asymptotically stable wenever it exists. Tis completes te proof. Remark 12. Te epidemiological implication of Teorem 11 is tat malaria would persist in te population if R mass >1. Teorem 11 is illustrated numerically by simulating te model (25), for te case R mass >1using various initial conditions. Te convergence of te solutions to E mass for te case R mass > 1, is depicted in Figure Conclusions Tis paper presents a deterministic model for te transmission dynamics of malaria wit partial immunity to reinfection. Te basic reproduction number of te model is obtained. Te proposed model wit standard incidence rate, undergoes te penomenon of backward bifurcation, were te stable disease-free equilibrium coexists wit one or more stable endemic equilibrium as te basic reproduction number (R 0 ) is less tan unity. In comparison wit te corresponding results of te model wit mass action incidence, we can conclude tat tis penomenon arises due to te use of standard incidence rate. Tis study suggests tat in some regions were malaria is inducing te varying total populations, it is difficult to control malaria due to te occurrence of backward bifurcation penomenon. If we ignore te disease-induced rate, and consider an asymptotical constant ost population, te standard incidence model results in a model wit mass action incidence. In tis case, te dynamics of te model are relatively simple. Tat is, te global dynamics of malaria disease wit reinfection is completely determined by te associated reproduction number R mass.ifr mass < 1,te disease-free equilibrium is globally asymptotically stable, so te disease always dies out. If R mass > 1, te disease-free equilibrium becomes unstable wile te endemic equilibrium emerges as te unique positive equilibrium and it is globally asymptotically stable in te interior of te feasible region, and once te disease appears, it eventually persists at te unique endemic equilibrium level. Terefore, we ave sown tat te backward bifurcation property can be removed by replacing te standard incidence function in te model wit a mass action incidence. Te numerical simulations suggest also tat te region of backward bifurcation increases wit increasing rate of partial protection (σ) of recovered individuals. Te region of backward bifurcation for te model increases wit decreasing average life span of mosquitoes. Acknowledgments Te autors are very grateful to te anonymous referees and te andling editor for teir careful reading, elpful comments, and suggestions, wic ave elped tem to improve te presentation of tis work significantly. I. H. Jung was supported by te National Fiseries Researc and Development Institute (RP-2012-FR-041). A. A. Lasari is very grateful to Mini Gos for er elp in numerical results. Partially, tis researc was supported by te National Natural Science Foundation of Cina (no ), University Key Teacer Foundation of He nan Province (2009GGJS-076). References [1] WHO, Malaria, ttp:// [2] NIAID, Malaria, ttp:// Pages/default.aspx. [3] ttp:// [4] J. L. Aron, Matematical modeling of immunity to malaria, Matematical Biosciences, vol. 90, no. 1-2, pp , [5] N. T. J. Bailey, Te Biomatematics of Malaria, Carles Griffin, London, UK, [6] G. A. Ngwa, Modelling te dynamics of endemic malaria in growing populations, Discrete and Continuous Dynamical Systems B,vol.4,no.4,pp ,2004. [7] P. 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