Threshold dynamics of a diffusive SIRI model with nonlinear incidence rate

Abstract

This work is concerned with the dynamics of a diffusive SIRI epidemic model with nonlinear incidence rate. We first establish the well-posedness of the model. Then we show the basic reproduction number R0 is a threshold parameter for the stability of the model. In fact, the disease-free equilibrium is globally asymptotically stable when R0<1, while the phenomena of uniform persistence occurs when R0>1. If R0=1, the disease-free equilibrium is globally asymptotically stable under some assumptions. We also clarify the relationship between R0 and the local basic reproduction number. Moreover, we establish the local and global stability of the endemic equilibrium provided that all parameters of this model are constants.