Time periodic traveling wave solutions for a Kermack–McKendrick epidemic model with diffusion and seasonality

Abstract

In this paper, we study the time periodic traveling wave solutions for a Kermack–McKendrick SIR epidemic model with individuals diffusion and environment heterogeneity. In terms of the basic reproduction number $$R_0$$ of the corresponding periodic ordinary differential model and the minimal wave speed $$c^*$$ , we establish the existence of periodic traveling wave solutions by the method of super- and sub-solutions, the fixed-point theorem, as applied to a truncated problem on a large but finite interval, and the limiting arguments. We further obtain the nonexistence of periodic traveling wave solutions for two cases involved with $$R_0$$ and $$c^*$$ .