Qualitative analysis of a diffusive SIR epidemic model with saturated incidence rate in a heterogeneous environment

In the present paper, we explore a diffusive SIR epidemic model with a saturated incidence rate, logistic growth rate for susceptibles, and Holling type II treatment, in a spatially heterogeneous environment. We establish some uniform bounds of solutions, the global stability of the disease-free equilibrium, uniform persistence, and the existence of endemic equilibria. When the spatial environment is heterogeneous, we determine the asymptotic profile of endemic equilibria if the diffusion rate of the susceptible or infected population is either small or large. Our numerical results show that heterogeneity in the transmission rate enhances the persistence of infection and that the disease may persist even when the basic reproduction number of our model is less than 1, provided that the initial number of infectives is high enough.