MATHEMATICAL ANALYSIS OF A MODEL FOR HIV-MALARIA CO-INFECTION. Zindoga Mukandavire. Abba B. Gumel. Winston Garira. Jean Michel Tchuenche

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1 MATHEMATICAL BIOSCIENCES doi:1.3934/mbe AND ENGINEERING Volume 6 Number 2 April 29 pp MATHEMATICAL ANALYSIS OF A MODEL FOR HIV-MALARIA CO-INFECTION Zindoga Mukandavire Department of Applied Mathematics National University of Science and Technology Box AC 939 Ascot Bulawayo Zimbabwe Abba B. Gumel Department of Mathematics University of Manitoba Winnipeg Manitoba R3T 2N2 Canada Winston Garira Department Of Mathematics And Applied Mathematics University of Venda Private Bag X55 Thohoyanndou 95 South Africa Jean Michel Tchuenche Mathematics Department University of Dar es Salaam P.O. Box 3562 Dar es Salaam Tanzania Abstract. A deterministic model for the co-interaction of HIV and malaria in a community is presented and rigorously analyzed. Two sub-models namely the HIV-only and malaria-only sub-models are considered first of all. Unlike the HIV-only sub-model which has a globally-asymptotically stable diseasefree equilibrium whenever the associated reproduction number is less than unity the malaria-only sub-model undergoes the phenomenon of backward bifurcation where a stable disease-free equilibrium co-exists with a stable endemic equilibrium for a certain range of the associated reproduction number less than unity. Thus for malaria the classical requirement of having the associated reproduction number to be less than unity although necessary is not sufficient for its elimination. It is also shown using centre manifold theory that the full HIV-malaria co-infection model undergoes backward bifurcation. Simulations of the full HIV-malaria model show that the two diseases co-exist whenever their reproduction numbers exceed unity (with no competitive exclusion occurring). Further the reduction in sexual activity of individuals with malaria symptoms decreases the number of new cases of HIV and the mixed HIV-malaria infection while increasing the number of malaria cases. Finally these simulations show that the HIV-induced increase in susceptibility to malaria infection has marginal effect on the new cases of HIV and malaria but increases the number of new cases of the dual HIV-malaria infection. 2 Mathematics Subject Classification. Primary: 92D3; Secondary: 92B5; 34D23. Key words and phrases. HIV-malaria model co-infection basic reproduction number equilibrium stability. Corresponding author: A.B. Gumel. gumelab@cc.umanitoba.ca. 333

2 334 Z. MUKANDAVIRE A. B. GUMEL W. GARIRA AND J. M. TCHUENCHE 1. Introduction. HIV/AIDS has killed more than 25 million people since it was first recognized in 1981 making it one of the most destructive epidemics in recorded history (UNAIDS/WHO [45]). It remains one of the leading causes of death in the world with its effect most devastating in sub-saharan Africa where HIV prevalence can range between 12% to 42% (Roseberry et al. [41]). One of the key factors that fuels the high incidence of HIV in sub-saharan Africa is the dual infection with malaria (Abu-Raddad et al. [1]). HIV has been shown to increase the risk of malaria infection and accelerate the development of clinical symptoms of malaria with the greatest impact in immune-suppressed persons. Conversely malaria has been shown to induce HIV-1 replication in vitro and in vivo. A biological explanation for these interactions lies in the cellular-based immune responses to HIV and malaria. Studies have shown that when HIV-infected individuals are attacked by malaria their body immune system weakens significantly creating a conducive environment for the HIV virus to replicate (virtually unchallenged) resulting in an increase in the viral load (the amount of HIV virus in the body). Hence since viral load is correlated with infectiousness [38] such a process (co-infection with malaria) leads to an increase in the number of new HIV cases in the population. Humans acquire malaria infection from infected female Anopheles mosquitoes (after taking blood meal). Of the four mosquito species that infect humans (P. falciparum P. vivax P. ovale and P. malarie) P. falciparum is the most virulent and potentially lethal to humans. Plasmodium falciparum stimulate release of interleukin-6 (IL-6) and tumor necrosis factor alpha (TNF-α) and the complications of severe falciparum malaria are mostly due to release of these cytokines. The increased levels of such cytokines stimulate the replication of HIV in vivo (thus increasing the HIV levels in patients [49]). HIV-1 proviral loads are significantly higher in patients with malaria than those without; and remain higher for at least 4 weeks after treatment [26]. Thus malaria infection could cause faster progression of HIV-1 disease [34]. A study in rural Tanzania shows a significantly higher prevalence of symptomless malarial parasitemia in HIV-infected adults and higher mortality due to malaria in these individuals [6]. In a large study carried out in Uganda HIV-1 infection has been found to increase the frequency of clinical malaria and parasite density with tendency to greater parasitemia with advancing immunosuppression [47]. Furthermore morbidity is higher in HIV-infected individuals [ ]. Recent studies of dual HIV-malaria infection confirm and extend earlier findings [ ] by showing that co-infection leads to a near one-log increase in viral load in chronic-stage HIV-infected patients during febrile malaria episodes and that HIV infection substantially increases susceptibility to malaria infection [1 39]. This symbiotic relationship between HIV and malaria is a double blow to sub- Saharan Africa region because of the high prevalence of HIV/AIDS and incidence of malaria [44]. This highlights the need for a robust qualitative assessment of the population-level implications of the immune-mediated interaction of the two diseases [1 48]. Abu-Raddad et al. [1] recently presented a mathematical model to study the transmission dynamics of HIV and malaria co-infection. It quantifies the size of the epidemic synergy between HIV-1 and malaria. In this study we formulate and analyse a realistic mathematical model for HIVmalaria co-infection which incorporates the key epidemiological and biological features of each of the two diseases. The main contribution of this study is in carrying out a detailed qualitative analysis of the resulting model; an activity not carried

3 DYNAMICS OF HIV-MALARIA CO-INFECTION 335 out in [1]. It is our view that this study represents the very first modelling work that provides an in-depth analysis of the qualitative dynamics of HIV-malaria coinfection. Additionally there are some important differences between the model in [1] and the one in this paper. For instance whilst we used an exponential distribution waiting time to model the exposed class a discrete time delay was used for the same purpose in [1]. Further seasonality variations were used in [1] to model the birth rate of mosquitoes whereas the current study uses a constant birth rate. Mathematically speaking while the model considered in [1] is non-autonomous the model considered in the current study is autonomous. Furthermore unlike in many other modelling studies of HIV transmission dynamics in a population this study assumes that individuals in the AIDS stage of HIV infection do transmit the disease to susceptible individuals. This is owing to the fact that epidemiologic evidence supports the hypothesis that AIDS patients are capable of and do engage in risky sexual behavior defined in terms of inconsistent condom use or having multiple sex partners [3]. It is worth stating that although the acquisition of immunity to malaria is a slow and complex process [4] the effect of partial host immunity on the transmission dynamics of malaria in areas where malaria is endemic can be significant. Such partial immunity which develops after several years of endemic exposure results from many factors such as antigenic polymorphism poor immunogenicity of individual antigens the ability of the parasite to interfere with the development of immune responses and the interaction of maternal and neonatal immunity [14]. Many infected individuals in endemic areas are asymptomatic (that is they may harbour large numbers of parasites without exhibiting signs and symptoms of the disease) [2 5 14]. In areas of low malaria transmission immunity develops slowly and malaria affects all age groups [22]. As noted in [33] since HIV infection interferes with cellular immune function HIV may interfere with the development of partial immunity to malaria particularly amongst children. This complicates the explicit modelling of the role of partial immunity to malaria in HIV/malaria transmission dynamics. The paper is organized as follows. The model is formulated in Section 2. The sub-models for HIV and malaria are presented and analyzed in Sections 3 and 4 respectively. The analysis of the full HIV-malaria co-infection model is carried out in Section 5. Numerical simulations and concluding remarks are presented in Section Model description. The model sub-divides the total sexually-active human population at time t denoted by N H (t) into the following sub-populations of susceptible individuals (S H (t)) individuals exposed to malaria parasite only (E M (t)) individuals with malaria symptoms only (I M (t)) individuals infected with HIV only but display no clinical symptoms of AIDS (I H (t)) HIV-infected individuals (with no symptoms of AIDS) exposed to malaria (E HM (t)) individuals dually-infected with HIV and malaria displaying clinical symptoms of malaria but no AIDS symptoms (I HM (t)) HIV-infected individuals displaying AIDS symptoms (A H (t)) AIDS individuals exposed to malaria (E AM (t)) and AIDS individuals dually-infected with malaria and displaying clinical symptoms (A HM (t)) so that N H (t) = S H (t) + E M (t) + I M (t) + I H (t) + E HM (t) + I HM (t) + A H (t) + E AM (t) + A HM (t).

4 336 Z. MUKANDAVIRE A. B. GUMEL W. GARIRA AND J. M. TCHUENCHE The total vector (mosquito) population at time t denoted by N V (t) is subdivided into susceptible mosquitoes (S V (t)) mosquitoes exposed to the malaria parasite (E V (t)) and infectious mosquitoes (I V (t)) so that N V (t) = S V (t) + E V (t) + I V (t). It is assumed that susceptible humans are recruited into the population at a constant rate Λ H. Susceptible individuals acquire HIV infection following effective contact with HIV-infected individuals (at a rate λ H ) and acquire infection with malaria following effective contact with infected mosquitoes (at a rate λ M ). It is assumed that individuals with malaria infection only may recover and return to the susceptible class (at a rate φ 1 ). Further natural death occurs in all human subpopulations (at a rate µ H ). The force of infection associated with HIV infection denoted by λ H is given by λ H = β H {I H + η HM (E HM + θ HM I HM ) + η A [A H + η HM (E AM + θ HM A HM )]}. N H (1) In (1) β H is the effective contact rate for HIV infection (contact sufficient to result in HIV infection) the modification parameter η HM 1 accounts for the relative infectiousness of individuals asymptomatically-infected with HIV exposed to malaria (E HM ) or displaying clinical symptoms of malaria (I HM ) in comparison to those with HIV infection alone but with no AIDS symptoms (I H ). In other words it is assumed that HIV-infected individuals (with no AIDS symptoms) who are also infected with malaria are more infectious than HIV-infected individuals with no AIDS symptoms and no malaria infection (similar comparisons are made for HIV-infected individuals with AIDS symptoms alone in relation to those with AIDS symptoms and malaria infection). Further the parameter θ HM 1 models the fact that dually-infected individuals with no symptoms of AIDS but displaying symptoms of malaria (I HM ) are more infectious than the corresponding duallyinfected individuals who are only exposed to malaria (E HM ). Finally the parameter η A > 1 captures the fact that individuals in the AIDS stage of HIV infection are more infectious than HIV-infected individuals displaying no clinical symptoms of AIDS. This is due to the fact that individuals in the AIDS stage have higher viral load compared to other HIV-infected individuals with no AIDS symptoms (this is owing to the aforementioned correlation between HIV viral load and infectiousness). Similarly humans acquire malaria infection following effective contact with infected mosquitoes at a rate λ M given by λ M = β M b M I V N H (2) where β M is the transmission probability per bite and b M is the per capita biting rate of mosquitoes (the form of the disease incidence function is obtained by taking into account conservation of bites ; that is the total number of bites made by mosquitoes equals the number of bites received by the human hosts [8] ). Susceptible individuals infected with malaria are moved to the exposed class (E M ) at the rate λ M and then progress to the infectious class (I M ) following the development of clinical symptoms (at a rate γ H ). Individuals exposed to malaria can be infected with HIV (at a rate λ H ). That is individuals in the E M class are moved into the E HM class upon acquiring HIV infection. Furthermore individuals

5 DYNAMICS OF HIV-MALARIA CO-INFECTION 337 with symptoms of malaria can acquire HIV infection at a rate σλ H where the parameter < σ 1 models the expected decrease in sexual activity (contact) by individuals with malaria symptoms (because of ill health). Individuals with malaria symptoms recover (at the rate φ 1 ) and suffer disease-induced death (at a rate δ M ). The population of individuals infected with HIV only (and displaying no symptoms of AIDS) is generated following infection (at the rate λ H ) and by the recovery of malaria infection by those dually-infected with HIV and malaria (at a rate φ 2 ). Individuals in this class acquire malaria infection (at a rate ϑλ M where ϑ > 1 accounts for the assumed increase in susceptibility to malaria infection as a result of HIV infection). This population is further decreased following progression to AIDS (at a rate κ). Individuals infected with HIV and exposed to malaria develop symptoms of malaria at a rate ǫγ H where ǫ 1 represents the assumption that HIV infected individuals exposed to malaria develop malaria at a faster rate compared to those not infected with HIV. Individuals in the I HM class die due to malaria (at the rate τδ M where τ 1 accounts for the increased mortality of the I HM individuals in comparison to individuals with malaria symptoms but not infected with HIV) recover (at a rate φ 2 ) and progress to AIDS (at a rate ξκ where ξ 1 represents the assumption that HIV infected individuals dually-infected with malaria progress to AIDS at a faster rate compared to those with HIV only). The population of individuals with AIDS symptoms only is generated following the progression to AIDS by individuals with HIV only (at the rate κ) as well as the recovery from malaria of individuals with AIDS symptoms and malaria (at a rate φ 3 ). Individuals in this class also acquire malaria infection (at the rate ϑλ M ) and die of AIDS-related illness (at a rate δ H ). Individuals in the class of people with AIDS symptoms exposed to malaria develop symptoms of malaria at the accelerated rate ǫγ H. The population of individuals with symptoms of both malaria and AIDS is generated by progression to AIDS by individuals dually-infected with HIV and malaria (at the rate ξκ) and the development of malaria symptoms by individuals with AIDS exposed to malaria (at the rate ǫγ H ). This population is diminished by natural death (at the rate µ H ) recovery from malaria infection (at a rate φ 3 ) death due to AIDS (at a rate ψδ H where ψ > 1 accounts for the assumed increase in HIV-related mortality due to the dual infection with malaria) and death due to malaria (at the rate τδ M ). Susceptible mosquitoes (S V ) are generated at a constant rate Λ V and acquire malaria infection (following effective contacts with humans infected with malaria) at a rate λ V where the force of infection λ V is given by λ V = β V b M I M + η V (I HM + θ V A HM ) N H (3) where β V is the transmission probability for mosquito infection b M is the biting rate of mosquitoes η V 1 is a modification parameter accounting for the increased likelihood of infection of vectors from humans with dual HIV-malaria infection in relation to acquiring infection from humans with malaria only. The parameter θ V 1 is similarly defined. Mosquitoes are assumed to suffer natural death at a rate µ V regardless of their infection status. Newly-infected mosquitoes are moved into the exposed class (E V ) and progress to the class of symptomatic mosquitoes (I V ) following the development of symptoms (at a rate γ V ).

6 338 Z. MUKANDAVIRE A. B. GUMEL W. GARIRA AND J. M. TCHUENCHE Putting the above formulations and assumptions together gives the following system of differential equations (where a prime represents differentiation with respect to time). S H = Λ H + φ 1 I M λ H S H λ M S H µ H S H E M = λ M S H λ H E M (γ H + µ H )E M I M = γ H E M σλ H I M (µ H + δ M + φ 1 )I M I H = λ HS H + φ 2 I HM ϑλ M I H (µ H + κ)i H E HM = λ HE M + ϑλ M I H (ǫγ H + µ H + κ)e HM I HM = σλ H I M + ǫγ H E HM (µ H + τδ M + φ 2 + ξκ)i HM A H = κi H + φ 3 A HM ϑλ M A H (µ H + δ H )A H (4) E AM = ϑλ MA H + κe HM (ǫγ H + µ H + δ H )E AM A HM = ξκi HM + ǫγ H E AM (µ H + φ 3 + τδ M + ψδ H )A HM S V = Λ V λ V S V µ V S V E V = λ V S V (γ V + µ V )E V I V = γ V E V µ V I V. The model flow diagram is depicted in Figure 1 and the associated parameters are described in Table 1. Since the model (4) monitors human populations all associated state variables and parameters are non-negative for all time t. Further before analyzing the dynamics of the full model (4) it is instructive to analyze the sub-models (HIV-only and malaria-only) first of all. This is done below. 3. HIV-only model. We begin by analysing the HIV-only model (obtained by setting E M = I M = E HM = E AM = A HM = S V = E V = I V = in (4)) given by S H = Λ H λ H S H µ H S H I H = λ HS H (µ H + κ)i H (5) A H = κi H (µ H + δ H )A H where now λ H = βh(ih+ηaah) N H and N H = S H + I H + A H. Consider the region } Ω H = {(S H I H A H ) R 3 + : N H Λ H /µ H. It can be shown (see for instance [42 43]) that all solutions of the system (5) starting in Ω H remain in Ω H for all t. Thus Ω H is positively-invariant (hence it is sufficient to consider the dynamics of (5) in Ω H ).

7 DYNAMICS OF HIV-MALARIA CO-INFECTION 339 The HIV-only model (5) has a DFE given by ) ( ΛH ) E h = (S H I H A H =. (6) µ H The stability of this equilibrium will be investigated using the next generation operator [17 46]. Using the notation in [46] on the system (5) the matrices F and V for the new infection terms and the remaining transfer terms are respectively given by F = ( βh β H η A ) and V = ( ) µh + κ. κ µ H + δ H It follows that the associated basic reproduction number [ ] denoted by R H is given by R H = ρ(fv 1 ) = β H(δ H + κη A + µ H ) (κ + µ H )(δ H + µ H ) (7) where ρ represents the spectral radius (the dominant eigenvalue in magnitude) of FV 1. Using Theorem 2 of [46] the following result is established. Lemma 1. The DFE of the HIV-only model (5) is locally-asymptotically stable (LAS) if R H < 1 and unstable if R H > 1. The basic reproduction number (R H ) measures the average number of new infections generated by a single infected individual in a completely susceptible population. Thus Lemma 1 implies that HIV can be eliminated from the community (when R H < 1) if the initial sizes of the sub-populations of the model are in the basin of attraction of the DFE E h. To ensure that elimination of the virus is independent of the initial sizes of the sub-populations it is necessary to show that the DFE is globally-asymptotically stable. The following results can be established (using for instance the techniques in [43] where a treatment model for HIV is considered): Theorem 2. The DFE of the model (5) given by E h is globally-asymptotically stable (GAS) whenever R H 1. Lemma 3. The HIV-only model has a unique endemic equilibrium if and only if R H > 1. The global stability property of the endemic equilibrium of the HIV-only model is now investigated for a special case Global stability of the endemic equilibrium for δ H =. Consider the HIV-only model (5) with δ H = given by S H = Λ H λ H S H µ H S H I H = λ HS H (µ H + κ)i H (8) A H = κi H µ H A H. The system (8) for the special case above has the same unique endemic equilibrium as the HIV-only model (5) but with δ H =. Let

8 34 Z. MUKANDAVIRE A. B. GUMEL W. GARIRA AND J. M. TCHUENCHE { } Ω H = (S H I H A H ) Ω h : I H = A H = We claim the following and R H1 = R H δh=. Theorem 4. The endemic equilibrium of the HIV-only model with δ H = is GAS in Ω H \ Ω H whenever R H1 > 1. Proof. It can be shown as for the case of Lemma 3 that the unique endemic equilibrium for this case exists only if R H1 > 1. Further N H = Λ H /µ H as t. Thus using S H = Λ H /µ H I H A H and substituting in (8) gives the following limiting system I H = λ H (Λ H /µ H I H A H ) (µ H + κ)i H A H = κi H µ H A H. (9) Using the Dulac s multiplier 1/I H A H it follows that [ βh (I H + η A A H ) (Λ H /µ H I H A H ) (κ + µ ] H) I H I H A H Λ H /µ H A H ( κ + µ ) H A H A H I H [ βh µ H = + β ( Hη A µ H Λ H Λ H IH 2 1 A ) H + κ ] Λ H /µ H A 2 H < since A H Λ H /µ H in Ω H. Thus by Dulac s criterion there are no periodic orbits in Ω H \Ω H. Since Ω H is positively invariant and the endemic equilibrium exists whenever R H1 > 1 then it follows from the Poincaré-Bendixson Theorem [4] that all solutions of the limiting system originating in Ω H remain in Ω H for all t. Further the absence of periodic orbits in Ω H implies that the unique endemic equilibrium of the special case of the HIV-only model is GAS whenever R H1 > 1. It may be possible to show using a regular perturbation argument (as in [8]) that the above proof holds for δ H > but small. In summary the model with HIV alone has a globally-asymptotically stable disease-free equilibrium whenever R H 1 and a unique endemic equilibrium whenever R H > 1. The unique endemic equilibrium is globally-asymptotically stable for the special case δ H = if R H1 > 1. The dynamics of the malaria-only model is now studied.

9 DYNAMICS OF HIV-MALARIA CO-INFECTION Malaria-only model. Consider the malaria-only model (obtained by setting I H = E HM = I HM = A H = E AM = A HM = in (4)) given by S H = Λ H + φ 1 I M λ M S H µ H S H E M = λ MS H (γ H + µ H )E M I M = γ H E M (µ H + δ M + φ 1 )I M S V = Λ V λ V S V µ V S V (1) E V = λ V S V (γ V + µ V )E V I V = γ V E V µ V I V I V I M where now λ M = β M b M λ V = β V b M and N H = S H + E M + I M. The N H N H model is a slight modification of the dengue transmission model in [21]. Consider the region } Ω M = {(S H E M I M S V E V I V ) R 6 + : N H Λ H /µ H N V Λ V /µ V. It can be shown that the region Ω M is positively-invariant (so that it is sufficient to consider the dynamics of the model (1) in Ω M ) Local stability of the disease-free equilibrium. The DFE of the malariaonly model (1) is given by ) ( ΛH E M = (S H E M I M S V E V I V = Λ ) V. (11) µ H µ V Here the associated next generation matrices are given by so that b M β M F = b Mβ V Λ V µ H Λ Hµ V γ H + µ H V = γ H δ M + µ H + φ 1 γ V + µ V γ V µ V R M = ρ(fv 1 b 2 M ) = β Mβ V γ H γ V µ H Λ V Λ H µ 2 V (γ H + µ H )(γ V + µ V )(δ M + µ H + φ 1 ). (12) Thus using Theorem 2 of [46] we have established the following result. Theorem 5. The DFE of the malaria-only model (1) is LAS if R M < 1 and unstable if R M > 1.

10 342 Z. MUKANDAVIRE A. B. GUMEL W. GARIRA AND J. M. TCHUENCHE 4.2. Existence of backward bifurcation. It is shown from Theorem 5 that the DFE of the malaria-only model is LAS if R M < 1. However this equilibrium may not be GAS in Ω M for R M < 1 owing to the possibility of backward bifurcation where the stable DFE co-exists with a stable endemic equilibrium when R M < 1 (see for instance [ ] and the references therein for further discussion on backward bifurcation). The public health implication of backward bifurcation is that the classical requirement of having the basic reproduction number less than unity although necessary is no longer sufficient for disease control. The possibility of the backward bifurcation phenomenon in the system (1) is investigated below. Solving the malaria-only model at an arbitrary equilibrium denoted by E M1 = (SH E M I M S V E V I V ) gives where S H = E M = I M = S V = (γ H+µ H)(δ M+µ H+φ 1)Λ H µ H(λ M +µh)(δm+µh+φ1)+γh(δm(λ M +µh)+µh(λ M +µh+φ1)) (δ M+µ H+φ 1)λ M ΛH µ H(λ M +µh)(δm+µh+φ1)+γh(δm(λ M +µh)+µh(λ M +µh+φ1)) γ Hλ M ΛH µ H(λ M +µh)(δm+µh+φ1)+γh(δm(λ M +µh)+µh(λ M +µh+φ1)) ΛV λ E V +µv V = λ V ΛV (γ V +µ V )(λ V +µv ) I V = γ V λ V ΛV µ V (γ V +µ V )(λ V +µv ) (13) and λ M = b M β M IV SH + E M + (14) I M λ b M β M IM V = SH + E M +. (15) I M Substituting (13) and (15) into (14) shows that the endemic equilibria of the malariaonly model (1) satisfy the following polynomial (in terms of λ M ) where λ M [ ] A(λ M) 2 + Bλ M + C = (16) A = Λ H µ V (γ V + µ V )(γ H + δ M + µ H + φ 1 )A 1 B = Λ H µ V (γ H + µ H )(γ V + µ V )(δ M + µ H + φ 1 )B 1 ] b 2 M β Mβ V γ H γ V Λ V [γ H (δ M + µ H ) + µ H (δ M + µ H + φ 1 ) with C = Λ H µ 2 V (γ V + µ V )(γ H + µ H ) 2 (δ M + µ H + φ 1 ) 2 (1 R 2 M ) A 1 = b M β V γ H + µ V (γ H + δ M + µ H + φ 1 ) B 1 = b M β V γ H + 2µ V (γ H + δ M + µ H + φ 1 ).

11 DYNAMICS OF HIV-MALARIA CO-INFECTION 343 The root λ M = of (16) corresponds to the DFE E M (whose stability has already been analyzed). For backward bifurcation to occur multiple non-zero (endemic) equilibria must exist. It follows from (16) that the non-zero equilibria of the model satisfy f(λ M ) = A(λ M )2 + Bλ M + C = (17) so that the quadratic (17) can be analyzed for the possibility of multiple equilibria. It is worth noting that the coefficient A is always positive and C is positive (negative) if R M is less than (greater than) unity respectively. Hence we have established the following result. Theorem 6. The malaria-only model (1) has (i) precisely one unique endemic equilibrium if C < (i.e. R M > 1) (ii) precisely one unique endemic equilibrium if B < and C = or B 2 4AC = (iii) precisely two endemic equilibria if C > (i.e. R M < 1) B < and B 2 4AC > (iv) no endemic equilibrium otherwise. The possible presence of two endemic equilibria (Case (iii)) above indicates the possibility of backward bifurcation in the model (1). This is explored further below using the Centre Manifold theory [ ]. To apply this theory the following simplification and change of variables are made first of all. Let S H = x 1 E M = x 2 I M = x 3 S V = x 4 E V = x 5 and I V = x 6 so that N H = x 1 + x 2 + x 3 and N V = x 4 + x 5 + x 6. Further by using vector notation x = (x 1 x 2 x 3 x 4 x 5 x 6 ) T the malaria-only model (1) can be written in the form dx = F(x) with F = (f 1 f 2 f 3 f 4 f 5 f 6 ) T as follows: dx 1 dx 2 dx 3 dx 4 dx 5 = f 1 = Λ H + φ 1 x 3 λ M x 1 µ H x 1 = f 2 = λ M x 1 (γ H + µ H )x 2 = f 3 = γ H x 2 (µ H + δ M + φ 1 )x 3 = f 4 = Λ V λ V x 4 µ V x 4 = f 5 = λ V x 4 (γ V + µ V )x 5 (18) with dx 6 = f 6 = γ V x 5 µ V x 6 λ M = β Mb M x 6 and λ V = β V b M x 3. x 1 + x 2 + x 3 x 1 + x 2 + x 3 The method entails evaluating the Jacobian of the system (18) at the DFE E M denoted by J(E M ). This gives:

12 344 Z. MUKANDAVIRE A. B. GUMEL W. GARIRA AND J. M. TCHUENCHE where J(E M ) = µ H φ 1 J 1 J 2 J 1 γ H J 3 J 4 µ V J 4 J 5 γ V µ V J 1 = β M b M J 2 = γ H + µ H J 3 = µ H + δ M + φ 1 J 4 = (β V b M Λ V µ H )/(µ V Λ H ) J 5 = γ V + µ V. Consider next the case when R M = 1. Suppose further that β M = β is chosen as a bifurcation parameter. Solving for β M from R M = 1 gives β M = β = Λ Hµ 2 V (γ H + µ H )(µ H + δ M + φ 1 )(µ V + γ V ) b 2 M β V γ H γ V µ H Λ V. It follows that the Jacobian (J(E M )) of (18) at the DFE with β M = β denoted by J β has a simple zero eigenvalue (with all other eigenvalues having negative real part). Hence the Centre Manifold theory [11] can be used to analyze the dynamics of the model (18). In particular the theorem in [13] (see also [ ]) reproduced below for convenience will be used to show that the model (18) (or equivalently (1)) undergoes backward bifurcation at R M = 1. Theorem 7. Castillo-Chavez and Song [13] Consider the following general system of ordinary differential equations with a parameter φ dx = f(x φ) f : Rn R R n and f C 2 (R n R) where is an equilibrium point of the system (that is f( φ) for all φ) and ( ) 1. A = D x f( ) = fi x j ( ) is the linearization matrix of the system around the equilibrium with φ evaluated at ; 2. Zero is a simple eigenvalue of A and all other eigenvalues of A have negative real parts; 3. Matrix A has a right eigenvector w and a left eigenvector v corresponding to the zero eigenvalue. Let f k be the kth component of f and a = b = n kij=1 v k w i w 2 f k j x i x j ( ) n v k w 2 f k i ( ) ki=1 x i φ then the local dynamics of the system around the equilibrium point is totally determined by the signs of a and b. Particularly if a > and b > then a backward bifurcation occurs at φ =.

13 DYNAMICS OF HIV-MALARIA CO-INFECTION 345 In order to apply the above theorem the following computations are necessary (it should be noted that we are using β as the bifurcation parameter in place of φ in Theorem 7). Eigenvectors of J β : For the case when R M = 1 it can be shown that the Jacobian of (18) at β M = β (denoted by J β ) has a right eigenvector given by w = [w 1 w 2 w 3 w 4 w 5 w 6 ] T where w 1 = φ 1w 3 β b M w 6 µ H w 2 = β b M w 6 γ H + µ H w 3 = γ H w 2 µ H + δ M + φ 1 w 4 = (β V b M µ H Λ V w 3 ) Λ H µ 2 V w 5 = µ V w 6 γ V w 6 = w 6. Further J β has a left eigenvector v = [v 1 v 2 v 3 v 4 v 5 v 6 ] where v 1 = v 2 = v 2 v 3 = γ H + µ H γ H v 4 = v 5 = γ V v 6 γ V + µ V v 6 = β b M v 2 µ V. Computations of a and b : It can be shown after some algebraic manipulations (involving computing the associated non-zero partial derivatives of F (at the DFE) to be used in the expression for a in Theorem 7) that where a = 2b Mµ H v 5 w 3 w 4 β V ( 1) Λ H

14 346 Z. MUKANDAVIRE A. B. GUMEL W. GARIRA AND J. M. TCHUENCHE and = v 2w 6 µ H Λ H β M (w 2 + w 3 ) + v 5 w 3 µ H Λ V β V (w 1 + w 2 + w 3 ) v 5 w 3 w 4 µ V Λ H β V b = v 2 w 6 b M >. Hence it follows (from Theorem 7 above) that the malaria-only model (1) undergoes backward bifurcation at R M = 1 whenever a = 2b Mµ H v 5 w 3 w 4 β V ( 1) Λ H >. (19) This result is summarized below. Theorem 8. The malaria-only model (1) undergoes a backward bifurcation at R M = 1 whenever inequality (19) holds. The backward bifurcation phenomenon is illustrated (Figure 2) by simulating the malaria-only model system (1) with the following set of parameter values (note that the parameters are chosen in order to illustrate the backward bifurcation and may not all be realistic epidemiologically (see [32] for some comments on whether or not backward bifurcation in the context of TB disease can occur with realistic parameter values)): Λ H =.99 Λ V =.89 β M =.7833 β V = b M =.58 γ H = 1 γ V =.981 µ H = µ V =.9 φ 1 =.656 δ M = Using the above set of parameter values it follows that R M = and a =.3354 with b =.58 (so that the inequality (19) is satisfied). Although the phenomenon of backward bifurcation has been observed in numerous epidemiological settings such as those for behavioural responses to perceived risk multi-groups vaccination TB dynamics with exogenous re-infection (see [ ] and the references therein) this is probably the first time such a phenomenon has been established in malaria transmission dynamics (Garba et al. [21] also established backward bifurcation in dengue disease another vector-borne disease). Finally it is worth stating that unlike in the HIV-only model the DFE of the malaria-only model (E M ) is not globally-asymptotically stable when the associated reproductive number (R M ) is less than unity owing to the phenomenon of backward bifurcation. Consequently this study shows that the control of malaria spread in a population when R M < 1 will depend on the initial sizes of the sub-populations of the malaria-only model (1). 5. Analysis of the HIV-malaria model Local stability of the disease-free equilibrium. Having analysed the dynamics of the two sub-models the full HIV-malaria model (4) is now considered. Its DFE is given by ( E = S H E M I M I H E HM I HM A H E AM A HM S V E V I V ) ( ΛH = Λ ) V. µ H µ V (2)

15 DYNAMICS OF HIV-MALARIA CO-INFECTION 347 It is easy to show using the next generation method (as in Sections 3 and 4) that the associated reproduction number for the full HIV-malaria model (4) (denoted by R HM ) is given by R HM = max{r H R M } (21) so that the following results follows from Theorem 2 of [46]. Theorem 9. The DFE of the HIV-malaria model (4) given by (2) is LAS if R HM < 1 and unstable if R HM > 1. Like in the case of the malaria-only model (1) the full HIV-malaria model (4) also undergoes backward bifurcation. We claim the following (see Appendix A for proof) Theorem 1. The full model (4) undergoes backward bifurcation at R HM = 1 whenever inequality (23) is satisfied. The backward bifurcation phenomenon of the full HIV-malaria model (4) is illustrated by depicting time series plots based on simulating the transformed model (22) with various initial conditions showing convergence to either the DFE (E ) or an endemic equilibrium (Figure 3). With the parameter values used in these simulations the coefficient a in the inequality (23) is given by a =.6 (and b is always positive). 6. Numerical simulations and concluding remarks. In order to illustrate some of the analytical results in this paper numerous numerical simulations of the full model (4) were carried out using a set of parameter values given in Table 1. Figure 4 illustrates the solution profiles of the populations of symptomatic individuals infected with malaria only (I M ) HIV only (I H ) both diseases (I HM ) and the symptomatic mosquito population (I V ) using various initial conditions. Simulating the model using the parameter values in Table 1 with β V =.2 and β H =.1 (so that R H =.376 R M =.465 and R HM =.465 < 1) shows convergence to the disease-free equilibrium (Fig. 4) in line with Theorem 7. Similarly choosing β H =.1 and β V =.9 (so that R H = R M =.8624 and R HM = 3.765) shows convergence to an endemic equilibrium (Fig. 5). Although the stability analysis of the endemic equilibrium of the HIV-malaria model (4) has not been carried out in this study this result is certainly expected (since the DFE is unstable in this case and typically the disease persists when the reproduction threshold (R HM ) exceeds unity; as is the case in these particular simulations). Simulations for the case where the two reproduction numbers exceed unity are carried out and depicted in Figure 6. These figures illustrate that for R H and R M greater than unity there is always co-existence of the two diseases no matter which of the reproduction numbers is greater. Further the simulations illustrate that the population of symptomatic individuals infected with malaria only (I M ) always has a higher steady-state value than that of the population of individuals in the HIV class I H for 1 < R M > R H 1 < R H < R M and R M = R H (it should be stated that in each of the pictures depicted in Figure 6 the steady-state value of the population of individuals in the I H class is non-zero). In other words these simulations suggest that for the set of parameter values used there would always be more cases of malaria at steady-state than cases of HIV infection in the community. The effect of reduction in sexual activity by individuals with malaria symptoms (I M ) exposed to HIV is monitored by varying the parameter σ. Figure 7B shows

16 348 Z. MUKANDAVIRE A. B. GUMEL W. GARIRA AND J. M. TCHUENCHE that while the cumulative number of new cases of malaria infection increases with decreasing σ the cumulative number of new cases of HIV (Fig. 7A) and that of the mixed (HIV-malaria) infection (Fig. 7C) decrease as σ decreases from 1 to. That is as individuals with symptoms of malaria decrease their risk of acquiring HIV infection (by decreasing their effective contact rate σλ H with < σ 1) the cumulative number of new cases of HIV and the mixed infection decrease; while the cumulative number of new cases of malaria rises. Simulations were carried out to monitor the effect of the assumed increase in susceptibility to malaria infection in individuals infected with HIV by varying the associated parameter ϑ. The simulations depicted in Figure 8 show that such an increase in susceptibility has marginal effect on the number of new cases of HIV (alone) and malaria infection (since the curves in Figure 8A and Figure 8B seem to generally coincide). However for the case of the mixed infection Figure 8C shows significant increase in the number of new cases as malaria susceptibility is increased by about 1-fold (the increase in the number of new cases remains constant as ϑ is increased further). The effect of increase in AIDS-related mortality in individuals dually-infected with HIV and malaria (with symptoms of malaria) is also monitored by varying ψ. Figure 9 shows an increase in HIV mortality as ψ increases and a decrease in mortality for individuals with the dual infection. This parameter seems to have no effect on the mortality of individuals infected with malaria only (albeit it shows a marginal decrease in mortality as ψ increases). In summary a deterministic compartmental model for the transmission dynamics of HIV and malaria in a given community is designed and rigorously analyzed. The model considered the epidemiologic synergy between sexually transmitted HIV and malaria in the context of Abu-Raddad et al. [1]. The HIV-only and malariaonly models were qualitatively examined first of all. The main theoretical results obtained are as follows: (i) The HIV-only model has a globally-asymptotically stable disease-free equilibrium whenever a certain epidemiological threshold (R H ) is less than unity (see also [42 43]); and unstable if this threshold exceeds unity; (ii) The HIV-only model has a unique endemic equilibrium whenever the aforementioned threshold exceeds unity. For the case where no AIDS-related mortality is considered this endemic equilibrium is globally-asymptotically stable whenever it exists; (iii) Unlike the HIV-only model the malaria-only model undergoes the phenomenon of backward bifurcation where the associated stable disease-free equilibrium co-exists with a stable endemic equilibrium when the corresponding reproduction number (R M ) is less than unity; (iv) The full HIV-malaria model is shown to have a locally-asymptotically stable disease-free equilibrium when its reproductive threshold is less than unity and unstable if the threshold exceeds unity. It also undergoes the phenomenon of backward bifurcation under certain conditions; Numerical simulations of the full HIV-malaria model show the following: (a) The two diseases co-exist whenever the reproduction number of each of the two diseases exceed unity (regardless of which number is larger); (b) The number of new cases of malaria at steady state seems to always exceeds that of HIV;

17 DYNAMICS OF HIV-MALARIA CO-INFECTION 349 (c) The assumed reduction in sexual activity of individuals with malaria symptoms results in decrease in the number of new cases of HIV and the mixed HIV-malaria infection while increasing the number of new cases of malaria; (d) The HIV-induced increase in susceptibility to malaria infection has marginal effect on the number of new cases of HIV but significantly increases the number of new cases of the dual HIV-malaria infection. This study provides the first in-depth mathematical analysis of a comprehensive model for the transmission dynamics of HIV and malaria in a population. There are a number of ways this study can be extended including incorporating preventive and therapeutic strategies for HIV (such as the use of anti-retroviral therapy condom use voluntary HIV testing and screening) and malaria (such as the use of treatment and prophylactic drugs vector-reduction strategies and personal protection against mosquito bites) and the acquisition of malaria immunity for adults in malariaendemic settings following repeated exposure (the latter would be somewhat of a daunting task since both diseases affect the immune system). It would also be interesting to consider the possible consequences of HIV-Malaria co-infection in mother-to-child transmission of HIV. Acknowledgements. This work is dedicated to Fred Brauer and Karl P. Hadeler on the occasion of their birthdays. The authors are grateful to African Millennium Mathematical Sciences Initiative (AMMSI) Center for Discrete Mathematics and Theoretical Computer Science (DIMACS) and the Mathematical Biosciences Institute (MBI) for the invitation to attend the Mathematical Biology Workshop in Nairobi Kenya December 26 where the work was initiated. The authors are grateful to F. Nyabadza (South African Centre for Epidemiological Modelling and Analysis) A. Niger O. Sharomi and C. Podder (of the University of Manitoba) for useful comments on the design of the model. One of the authors (ABG) acknowledges with thanks the support in part of the Natural Science and Engineering Research Council (NSERC) and Mathematics of Information Technology and Complex Systems (MITACS) of Canada. The authors are grateful to the anonymous reviewers for their very constructive comments which have enhanced the paper. Appendix A: Proof of Theorem 1. Proof. The proof is also based on using the Centre Manifold theory on the HIVmalaria model (4). As in Section 4.2 let S H = x 1 E M = x 2 I M = x 3 I H = x 4 E HM = x 5 I HM = x 6 A H = x 7 E AM = x 8 A HM = x 9 S V = x 1 E V = x 11 and I V = x 12 so that NH c = x 1 + x 2 + x 3 + x 4 + x 5 + x 6 + x 7 + x 8 + x 9 and NM c = x 1 +x 11 + x 12. Further by adopting the same vector notation as in Section 4.2 with x = (x 1 x 2 x 12 ) T the model (4) can be written in the form dx = F(x) where F = (f 1 f 2 f 12 ) T as follows: dx 1 = f 1 = Λ H + φ 1 x 3 λ c Hx 1 λ c Mx 1 µ H x 1 dx 2 = f 2 = λ c M x 1 λ c H x 2 K 1 x 2

18 35 Z. MUKANDAVIRE A. B. GUMEL W. GARIRA AND J. M. TCHUENCHE dx 3 dx 4 dx 5 dx 6 dx 7 dx 8 dx 9 dx 1 dx 11 = f 3 = γ H x 2 σλ c H x 3 K 2 x 3 = f 4 = λ c H x 1 + φ 2 x 6 ϑλ c M x 4 K 3 x 4 = f 5 = λ c H x 2 + ϑλ c M x 4 K 4 x 5 = f 6 = σλ c H x 3 + ǫγ H x 5 K 5 x 6 = f 7 = κx 4 + φ 3 x 9 ϑλ c M x 7 K 6 x 7 = f 8 = ϑλ c M x 7 K 7 x 8 + κx 5 = f 9 = ξκx 6 + ǫγ H x 8 K 8 x 9 = f 1 = Λ V λ c V x 1 µ V x 1 = f 11 = λ c V x 1 K 9 x 11 (22) where dx 12 = f 12 = γ V x 11 µ V x 12 K 1 = γ H + µ H K 2 = µ H + δ M + φ 1 K 3 = µ H + κ K 4 = ǫγ H + µ H + κ K 5 = µ H + τδ M + φ 2 + ξκ K 6 = µ H + δ H K 7 = ǫγ H + µ H + δ H K 8 = µ H + φ 3 + τδ M + ψδ H K 9 = µ V + γ V and λ c H = β H {x 4 + η HM (x 5 + θ HM x 6 ) + η A [x 7 + η HM (x 8 + θ HM x 9 )]} NH c λ c M = β Mb M NH c x 12 λ c V = β V b M N c H [x 3 + η V (x 6 + θ V x 9 )]. It can be shown by computing the eigenvalues of the associated Jacobian of the system (22) at the DFE (denoted by J(E )) that R HM = max{r H R M } as before. For convenience we re-write

19 DYNAMICS OF HIV-MALARIA CO-INFECTION 351 R H = β H(δ H + κη A + µ H ) b 2 M and R M = β Mβ V γ H γ V µ H Λ V K 3 K 6 Λ H µ 2 V K. 1K 2 K 9 Consider the case when R HM = 1 (that is R M < R H = 1). Suppose further that β H = β is chosen as a bifurcation parameter. Solving for β H from R H = 1 gives β H = β = K 3 K 6 (δ H + κη A + µ H ). Eigenvectors of J β : For the case when R HM = 1 it can be shown that the matrix J(E ) evaluated at β H = β denoted by J β has a right eigenvector given by where w = [w 1 w 2 w 3 w 4 w 5 w 6 w 7 w 8 w 9 w 1 w 11 w 12 ] T w 1 = φ 1w 3 β w 4 β η A w 7 β M b M w 12 µ H w 2 = w 2 w 3 = γ Hw 2 K 2 w 4 = w 4 w 5 = w 6 = w 7 = κw 4 K 6 w 8 = w 9 =

20 352 Z. MUKANDAVIRE A. B. GUMEL W. GARIRA AND J. M. TCHUENCHE w 1 = β V b M µ H Λ V w 3 Λ H µ 2 V w 11 = µ V w 12 γ V w 12 = K 1w 2 β M b M. Further the matrix J β has a left eigenvector v = [v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 v 9 v 1 v 11 v 12 ] where v 1 = v 2 = v 2 v 3 = K 1v 2 γ H v 4 = v 4 v 5 = β η HM v 4 + ǫγ H v 6 + κv 8 K 4 v 6 = (β η HM θ HM + φ 2 )v 4 K 5 + β V b M η V µ H Λ V v 11 Λ H µ V K 5 + ξκv 9 K 5 v 7 = β η A v 4 K 6 v 8 = β η A η HM v 4 + ǫγ H v 9 K 7 v 9 = β η A η HM θ HM Λ H µ V v 4 + Λ H µ V φ 3 v 7 + β V b M η V µ H Λ V θ V v 11 Λ H µ V K 8 v 1 = v 11 = K 2v 3 Λ H µ V β V b M µ H Λ V v 12 = β Mb M v 2 µ V. Computations of a and b : It can be shown after some tedious manipulations that

21 DYNAMICS OF HIV-MALARIA CO-INFECTION 353 a = 2µ H Λ 2 H µ (v 4 w4β 2 H Λ H µ V + v 11 w3β 2 V b M µ H Λ V V + v 11 w 1 w 3 β V b M µ H Λ V + v 11 w 2 w 3 β V b M µ H Λ V + v 11 w 3 w 4 β V b M µ H Λ V + v 11 w 3 w 7 β V b M µ H Λ V + v 2 w 2 w 4 β H Λ H µ V + v 2 w 2 w 7 β H η A Λ H µ V + v 2 w 2 w 12 β M b M Λ H µ V + v 2 w 3 w 12 β M b M Λ H µ V + v 2 w 4 w 12 β M b M Λ H µ V + v 2 w 7 w 12 β M b M Λ H µ V + v 3 w 3 w 4 σβ H Λ H µ V + v 3 w 3 w 7 σβ H η A Λ H µ V + v 4 w 2 w 4 β H Λ H µ V + v 4 w 2 w 7 β H η A Λ H µ V + v 4 w 3 w 4 β H Λ H µ V and + v 4 w 3 w 7 β H η A Λ H µ V + v 4 w 4 w 12 ϑβ M b M Λ H µ V + v 4 w7 2 β Hη A Λ H µ V v 5 w 2 w 4 β H Λ H µ V v 5 w 2 w 7 β H η A Λ H µ V v 5 w 4 w 12 ϑβ M b M Λ H µ V v 6 w 3 w 4 σβ H Λ H µ V v 6 w 3 w 7 σβ H η A Λ H µ V + v 7 w 7 w 12 ϑβ M b M Λ H µ V v 8 w 7 w 12 ϑβ M b M Λ H µ V v 11 w 3 w 1 β V b M Λ H µ V + v 4 w 4 w 7 β H Λ H µ V + v 4 w 4 w 7 β H Λ H µ V η A ) b = v 4 w 4 + v 4 w 7 η A >. Thus it follows from Theorem 7 that the full HIV-malaria model (4) undergoes backward bifurcation at R HM = 1 whenever a >. (23) Table 1. Model parameters and their interpretations Parameter Symbol Value Source Recruitment rate of humans Λ H day 1 Assumed Recruitment rate of mosquitoes Λ V 6 day 1 [15] Natural death rate of humans µ H day 1 [8] Natural death rate of mosquitoes µ V.1429 day 1 [16] HIV-induced death rate δ H day 1 [36] Malaria-induced death rate δ M day 1 [16] Effective contact rate for HIV infection β H Variable Variable Transmission probability for malaria in humans β M.8333 day 1 [16] Transmission probability for malaria in vectors β V (1) - Biting rate of mosquitoes b M (.25 1) day 1 Assumed Modification parameters η A η HM ξ Assumed Modification parameters θ HM σ τ Assumed Modification parameters ǫ ϑ ψ Assumed Recovery rate of humans from malaria φ 1 φ 2 φ Assumed Modification parameters η V θ V Assumed Rate of progression to AIDS stage κ.548 day 1 Assumed Rate at which humans exposed to malaria γ H.8333 day 1 [16] develop clinical symptoms Rate at which vectors exposed to malaria γ V.1 day 1 [16] develop symptoms REFERENCES [1] L. J. Abdu-Raddad P. Patnaik and J. G. Kublin Dual infection with HIV and Malaria fuels the spread of both diseases in sub-saharan Africa Science 314 (26) [2] T. N. Akenji and J. Deas Definition of populations at risk for plasmodium falciparum infection in three endemic areas of Cameroon J. Parisitol. 8 (1994)

22 354 Z. MUKANDAVIRE A. B. GUMEL W. GARIRA AND J. M. TCHUENCHE Figure 1. Flowchart of the HIV/malaria model (4) [3] R. M. Anderson and R. M. May Infectious Diseases of Humans Oxford Univesity Press London/New York [4] M. Arevalo-Herrara Y. Solarte F. Zamora F. Mendez F. M. Yasnot L. Rocha C. Long L. H. Miller and S. Herrara Plasmodium vivax: transmission-blocking immunity in a malariaendemic area of Colombia Am. J. Trop. Med. Hyg. (Suppl 5) 73 (25) [5] J. L. Aron Mathematical modelling of immunity to malaria Nonlinearity in biology and medicine (Los Alamos NM 1987) Math. Biosci. 9 (1988)

23 DYNAMICS OF HIV-MALARIA CO-INFECTION 355 Figure 2. Bifurcation diagram for the malaria-only model (1) using: Λ H =.99 Λ V =.89 β V = b M =.58 γ H = 1 γ V =.981 µ H = µ V =.9 φ 1 =.656 δ M = and various values of β M (with these parameter values and β M range a > and b > ). [6] C. Atzori A. Bruno and G. Chichino HIV1 and parasitic infections in rural Tanzania Ann. Trop. Med. Parasitol. 87 (1993) [7] G. Birkhoff and G. C. Rota Ordinary Differential Equations Ginn [8] C. Bowman A. B. Gumel P. van den Driessche J. Wu and H. Zhu A mathematical model for assessing control strategies against West Nile virus Bull. Math. Biol. 67 (25) [9] F. Brauer Backward bifurcation in simple vaccination models Journal of Mathematical Analysis and Application 298 (24) [1] F. Brauer and C. Castillo-Chavez Mathematical Models in Population Biology and Epidemiology In Texts in Applied Mathematics Series 4 Springer-Verlag New York 21. [11] J. Carr Applications Centre Manifold Theory Springer-Verlag New York [12] C. Castillo-Chavez Z. Feng and W. Huang On the computation of R and its role on global stability Mathematical Approaches for Emerging and Reemerging Infectious Diseases: An Introduction. The IMA Volumes in Mathematics and its Applications. Volume 125 pp Springer New York 22. Castillo-Chavez C. with S. Blower P. van den Driessche D. Kirschner and A.-A. Yakubu (Eds.). [13] C. Castillo-Chavez and B. Song Dynamical models of tuberculosis and their applications Mathematical Biosciences and Engineering 1 (24) [14] C. Chiyaka Development and Analysis of Mathematical Models of the Immunoepidemiology of Malaria Ph.D. Thesis National University of Science and Technology Zimbabwe 28.

24 356 Z. MUKANDAVIRE A. B. GUMEL W. GARIRA AND J. M. TCHUENCHE (A) Total HIV Cases Time (years) (B) 2 Total Malaria cases Time (years) Figure 3. Simulations of the HIV-malaria model (22) with different initial conditions illustrating the phenomenon of backward bifurcation. Parameter values used are: Λ H = 1 Λ V = 1 γ H = 1 γ V =.981 µ H = µ V =.9 φ 1 =.2 φ 2 =.4 φ 3 =.6 δ M =.1 δ H =.1 β V =.5723 b M =.58 β M = 3.3 κ = 1 ǫ = 1.2 η A = 1 η HM = 1 θ HM = 1 η V = 1 θ V = 1 ϑ = 1.2 τ = 1.1 ξ = 1.2 σ = 1 ψ = 1.2 β H =.15. (A) Total HIV cases (B) Total malaria cases. [15] C. Chiyaka W. Garira and S. Dube Mathematical modelling of the impact of vaccination on malaria epidemiology Qualitative Theory of Differential Equations and Applications To appear. [16] N. Chitnis J. M. Cushing and J. M. Hyman Bifurcation analysis of a mathematical model for malaria transmission SIAM J. Appl. Math. 67 (26) [17] O. Diekmann J. A. P. Heesterbeek and J. A. J. Metz On the definition and computation of the basic reproduction ratio R in models for infectious diseases in heterogeneous populations J. Math. Biol. 28 (199) [18] J. Dushoff W. Huang and C. Castillo-Chavez Backwards bifurcations and catastrophe in simple models of fatal diseases J. Math. Biol. 36 (1998)