This work is devoted to the time periodic traveling wave phenomena of a generalization of the classical Kermack–McKendrick model with seasonality and nonlocal interaction derived by mobility of individuals during latent period of disease. When the basic reproduction number $R_0$ is bigger than 1, we find a critical value $c^{\ast }$ and prove the existence of periodic traveling waves with the wave speed $c>c^{\ast }$. When $R_0$ is less than 1, we show that there is no periodic traveling wave with any wave speed $c\ge 0$. In addition, the influences of length of latency and seasonal factor on the critical value $c^{\ast }$ is explored by numerical simulations. Some novel epidemiological insights and biological interpretation are provided.

这项工作专门针对经典Kermack-McKendrick模型的时间周期行波现象，该模型具有季节性和由疾病潜伏期的个体活动性引起的非局部相互作用。基本再现次数${\mathrm{[R}}_0$ 大于1，我们发现一个临界值 $C^{\ast }$ 并以波速证明周期性行波的存在 $C>C^{\ast }$。什么时候${\mathrm{[R}}_0$ 小于1，我们表明没有任何波速的周期性行波 $C\ge 0$。此外，潜伏期的长度和季节因素对临界值的影响$C^{\ast }$通过数值模拟进行探索。提供了一些新颖的流行病学见解和生物学解释。