A Mathematical Model of Malaria Transmission with Structured Vector Population and Seasonality

Abstract

In this paper, we formulate a mathematical model of nonautonomous ordinary differential equations describing the dynamics of malaria transmission with age structure for the vector population. The biting rate of mosquitoes is considered as a positive periodic function which depends on climatic factors. The basic reproduction ratio of the model is obtained and we show that it is the threshold parameter between the extinction and the persistence of the disease. Thus, by applying the theorem of comparison and the theory of uniform persistence, we prove that if the basic reproduction ratio is less than $\mathrm{1}$, then the disease-free equilibrium is globally asymptotically stable and if it is greater than $\mathrm{1}$, then there exists at least one positive periodic solution. Finally, numerical simulations are carried out to illustrate our analytical results.