Dynamical analysis of a multi-group SIR epidemic model with nonlocal diffusion and nonlinear incidence rate

Abstract

In this paper, a multi-group SIR epidemic model with nonlocal diffusion and nonlinear incidence rate in spatially heterogeneous environment is proposed. The well-posedness, including the existence, positivity and boundedness of solutions, is achieved. The basic reproduction number R0 is defined, and the existence of principal eigenvalue is studied. Further, the threshold criteria on the global dynamics of the model are established in terms of R0. That is, when R0<1, the disease-free steady state is globally asymptotically stable, when R0>1, the disease is uniformly persistent. Particularly, a single-group degenerated SIR epidemic model is also studied. We give some properties on the principle eigenvalue and prove that there exists a compact global attractor for the solution semiflow. Furthermore, when the basic reproduction number R̄0>1, we obtain that the endemic steady state is unique to exist and globally asymptotically stable.