Threshold dynamics of a diffusive SIRI model with nonlinear incidence rate

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.