Global $L^{\infty }$-bounds and long-time behavior of a diffusive epidemic system in heterogeneous environment

Abstract

In this paper, we are concerned with an epidemic reaction-diffusion system with nonlinear incidence mechanism of the form $S^q{I}^p(p,q>0)$. The coefficients of the system are spatially heterogeneous and time dependent (particularly time periodic). We first establish the $L^{\infty }$-bounds of the solutions of a class of systems, which improve some previous results in [58]. Based on such estimates, we then study the long-time behavior of the solutions of the system. Our results reveal the delicate effect of the infection mechanism, transmission rate, recovery rate and disease-induced mortality rate on the infection dynamics. Our analysis can be adapted to some other types of infection incidence mechanisms.