Threshold Dynamics of a Time-Periodic Reaction-Diffusion Malaria Model with Distributed Latencies

Abstract

It is well-known that the transmission of malaria is caused by the bites of mosquitoes. Since the life habit of mosquitoes is influenced by seasonal factors such as temperature, humidity and rainfall, the transmission of malaria presents clear seasonable changes. In this paper, in order to take into account the incubation periods in humans and mosquitoes, we study the threshold dynamics of two periodic reaction-diffusion malaria models with distributed delay in terms of the basic reproduction number. Firstly, the basic reproduction number $R_0$ is introduced by virtue of the next generation operator method and the Poincaré mapping of a linear system. Secondly, the threshold dynamics is established in terms of $R_0$. It is proved that if $R_0 < 1$, then the disease-free periodic solution of the model is globally asymptotically stable; and if $R_0 > 1$, then the disease is persistent.