Traveling waves for an epidemic patchy model with bilinear incidence

In this paper, we study epidemic models in a periodic patchy environment with bilinear incidence but without vital dynamics. We first show that there exists a minimal wave speed $c^{\ast }$ such that the system admits bounded traveling wave solutions if $c\ge c^{\ast }$ and $R_0>1$ , where $R_0$ is the basic reproduction number. Then we prove the non-existence of traveling wave solutions for the case where $R_0\le 1$ , or $R_0>1$ and $c\in (0,c_{}^{\ast })$ . Our analysis also indicates that the heterogeneity of transmission rates and removed rates can increase $c^{\ast }$ , meanwhile the heterogeneity of diffusion rate of the infected individuals decreases $c^{\ast }$ . Finally, we discuss how the parameters and their heterogeneity affect the final size of the epidemic via numerical simulations, and give some advice for disease control.