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 ${\mathcal{R}}_0$ 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 ${\mathcal{R}}_0>1$. The nonexistence of periodic traveling waves for two cases (i) ${\mathcal{R}}_0>1$ and $0<c<c^{*},$ (ii) ${\mathcal{R}}_0\le 1$ 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.

在本文中，提出了一种反应扩散SIR流行病模型。它考虑了个体的流动性，感染率和恢复率的时间周期性，以及一般的非线性关联函数，其中包含了一些经典的关联函数。在我们的模型中，由于引入了一般的非线性关联函数，无法获得感染个体的有界性，因此我们借助辅助系统研究了原系统周期行波解的存在与不存在。基本再生数${\mathcal{R}}_0$和临界波速$C^{*}$给出。我们得到了每个波速的周期行波的存在性和唯一性$C>C^{*}$使用 Schauder 的不动点定理时${\mathcal{R}}_0>\mathrm{1个}$. 两种情况 (i) 不存在周期性行波${\mathcal{R}}_0>\mathrm{1个}$和$0<C<C^{*},$(二)${\mathcal{R}}_0\le \mathrm{1个}$和$C\ge 0$也获得。这些结果推广并改进了现有结论。最后，数值实验支持了理论结果。通过数值模拟分析了周期系统与一般非周期系数系统行波解的差异。