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 \(c^*\) such that the system admits bounded traveling wave solutions if \(c\ge c^*\) and \(\mathcal {R}_{0}>1\) , where \(\mathcal {R}_{0}\) is the basic reproduction number. Then we prove the non-existence of traveling wave solutions for the case where \(\mathcal {R}_{0}\le 1\) , or \(\mathcal {R}_{0}>1\) and \(c\in (0,c^{*})\) . Our analysis also indicates that the heterogeneity of transmission rates and removed rates can increase \(c^*\) , meanwhile the heterogeneity of diffusion rate of the infected individuals decreases \(c^*\) . 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.
