The effect of incidence function in backward bifurcation for malaria model with temporary immunity

This paper addresses the effect of the choice of the incidence function for the occurrence of backward bifurcation in two malaria models, namely, malaria model with standard incidence rate and malaria model with nonlinear incidence rate. Rigorous qualitative analyzes of the models show that the malaria model with standard incidence rate exhibits the phenomenon of backward bifurcation whenever a certain epidemiological threshold, known as the basic reproduction number, is less than unity. The epidemiological consequence of this phenomenon is that the classical epidemiological requirement of making the reproductive number less than unity is no longer sufficient, although necessary, for effectively controlling the spread of malaria in a community. For the malaria model with nonlinear incidence rate, it is shown that this phenomenon does not occur and the disease-free equilibrium of the model is globally-asymptotically stable whenever the reproduction number is less than unity. When the associated basic reproduction number is greater than unity, the models have a unique endemic equilibrium which is globally asymptotically stable under certain conditions. The sensitivity analysis based on the mathematical technique has been performed to determine the importance of the epidemic model parameters in making strategies for controlling malaria.