Mathematical analysis of the role of repeated exposure on malaria transmission dynamics

TLDR: In this paper, the authors present a deterministic model for assessing the role of repeated exposure on the transmission dynamics of malaria in a human population, which reveals the presence of the backward bifurcation, where a stable disease-free equilibrium co-exists with a stable endemic equilibrium when the associated reproduction threshold is less than unity.

Abstract: 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 disease-free 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 the resulting model with mass action incidence has a globally-asymptotically stable endemic equilibrium when the reproduction threshold exceeds unity.