The periodic traveling waves in a diffusive periodic SIR epidemic model with nonlinear incidence

In this paper, a reaction-diffusion SIR epidemic model is proposed. It takes into account the individuals mobility, the time periodicity of the infection rate and recovery rate, and the general nonlinear incidence function, which contains a number of classical incidence functions. In our model, due to the introduction of the general nonlinear incidence function, the boundedness of the infected individuals can not be obtained, so we study the existence and nonexistence of periodic traveling wave solutions of original system with the aid of auxiliary system. The basic reproduction number $R_0$ and the critical wave speed $c^{*}$ are given. We obtained the existence and uniqueness of periodic traveling waves for each wave speed $c>c^{*}$ using the Schauder's fixed points theorem when $R_0>1$. The nonexistence of periodic traveling waves for two cases (i) $R_0>1$ and $0<c<c^{*},$ (ii) $R_0\le 1$ and $c\ge 0$ are also obtained. These results generalize and improve the existing conclusions. Finally, the numerical experiments support the theoretical results. The differences of traveling wave solution between periodic system and general aperiodic coefficient system are analyzed by numerical simulations.