This paper presents a deterministic model for assessing the role of repeated exposure on the transmission dynamics of malaria in a human population. Rigorous qualitative analysis of the model, which incorporates three immunity stages, reveals the presence of the phenomenon of backward bifurcation, where a stable diseasefree equilibrium co-exists with a stable endemic equilibrium when the associated reproduction threshold is less than unity. This phenomenon persists regardless of whether the standard or mass action incidence is used to model the transmission dynamics. It is further shown that the region for backward bifurcation increases with decreasing average life span of mosquitoes. Numerical simulations suggest that this region increases with increasing rate of re-infection of first-time infected individuals. In the absence of repeated exposure (re-infection) and loss of infection-acquired immunity, it is shown, using a non-linear Lyapunov function, that International Journal for Theory, Real World Modelling and Simulations 252 A.M. Niger and A.B. Gumel the resulting model with mass action incidence has a globallyasymptotically stable endemic equilibrium when the reproduction threshold exceeds unity.MSC: 92D30, 34D23.

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The immune response to malaria and the effects of an imperfect vaccine for this disease are modelled incorporating an n stage parasite life cycle, immune cells, and antibodies. A globally asymptotically stable parasite-free equilibrium occurs when the associated reproduction number is less than unity. An imperfect malaria vaccine that reduces the number of merozoites released per bursting infected red blood cell (IRBC) and that boosts immune response can reduce the concentration of IRBCs in vivo. Numerical simulations show that a vaccine efficacy of at least 87% is necessary to eliminate IRBC in vivo. The concentration of IRBCs varies with the capacity of the vaccine to modify the total number of merozoites released per bursting IRBC.immune response, malaria, ordinary differential equations, vaccine,

The distribution of HIV and malaria overlap globally. So there is always a chance of co-infection. In this paper the impact of medication on HIV-Malaria co-infection has been analyzed and we have developed a mathematical model using the idea of the models of Mukandavire, et al. [13] and Barley, et al. [3] where treatment classes are included. The disease-free equilibrium (DFE) of the HIV-only model is globally-asymptotically stable (GAS) when the reproduction number is less than one. But it is shown that in the malaria-only model, there is a coexistence of stable disease-free equilibrium and stable endemic equilibrium, for a certain interval of the reproduction number less than unity. This indicates the existence of backward bifurcation. Numerical simulations of the full model are performed to determine the impact of treatment strategies. It is shown that malaria-only treatment strategy reduces more new cases of the mixed infection than the HIV-only treatment strategy. Moreover, mixed treatment strategy reduces the least number of new cases compared to single treatment strategies. GANIT J. Bangladesh Math. Soc.Vol. 39 (2019) 45-62