Global dynamics in a reaction–diffusion multi-group SIR epidemic model with nonlinear incidence

Abstract

In this paper, a reaction–diffusion multi-group SIR epidemic model with nonlinear incidence in spatially heterogeneous and homogeneous environment is investigated. In general spatially heterogeneous environments, the well-posedness of solutions, including the nonnegativity and ultimate boundedness of solutions, firstly is established. The basic reproduction number R0 is defined. The threshold criteria on the global dynamics of the model are established. That is, if R0<1, the disease-free steady state is globally asymptotically stable, while if R0>1, the model is uniformly persistent. Furthermore, for the homogeneous space and heterogeneous diffusion model, by using the Lyapunov functions method, the global asymptotic stability for the disease-free and endemic steady states is also established.