Abstract
A deterministic model with variable human population for the transmission dynamics of malaria disease, which allows transmission by the recovered humans, is first developed and rigorously analyzed. The model reveals the presence of the phenomenon of backward bifurcation, where a stable disease-free equilibrium coexists with one or more stable endemic equilibria when the associated reproduction number is less than unity. This phenomenon may arise due to the reinfection of host individuals who recovered from the disease. The model in an asymptotical constant population is also investigated. This results in a model with mass action incidence. A complete global analysis of the model with mass action incidence is given, which reveals that the global dynamics of malaria disease with reinfection is completely determined by the associated reproduction number. Moreover, it is shown that the phenomenon of backward bifurcation can be removed by replacing the standard incidence function with a mass action incidence. Graphical representations are provided to study the effect of reinfection rate and to qualitatively support the analytical results on the transmission dynamics of malaria.
Highlights
Malaria is a mosquito-borne disease caused by a parasite
This paper presents a deterministic model for the transmission dynamics of malaria with partial immunity to reinfection
In comparison with the corresponding results of the model with mass action incidence, we can conclude that this phenomenon arises due to the use of standard incidence rate
Summary
Malaria is a mosquito-borne disease caused by a parasite. It is endemic and widespread in tropical and subtropical regions, including much of sub-Saharan Africa, Asia, and the Americas. In paper [13], Niger and Gumel constructed a mathematical model that includes multiple infected and recovered classes, to assess the role of the partial immunity on the transmission dynamics of malaria in a human population. Their results reveals the presence of the phenomenon of backward bifurcation in the standard incidence model with the disease-induced death in the human population. A transmission model of human malaria in a partially immune population is formulated in Wan and Cui’s paper [14] They established the basic reproduction number and explicit subthreshold conditions for the model, and showed that if the disease induced death rate is large enough, the model undergoes a backward bifurcation.
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