The Global Behavior of a Periodic Epidemic Model with Travel between Patches

Abstract

We establish an SIS (susceptible-infected-susceptible) epidemic model, in which the travel between patches and the periodic transmission rate are considered. As an example, the global behavior of the model with two patches is investigated. We present the expression of basic reproduction ratio $R_0$ and two theorems on the global behavior: if $R_0$ < 1 the disease-free periodic solution is globally asymptotically stable and if $R_0$ > 1, then it is unstable; if $R_0$ > 1, the disease is uniform persistence. Finally, two numerical examples are given to clarify the theoretical results.