Time periodic traveling wave solutions of a time-periodic reaction–diffusion SEIR epidemic model with periodic recruitment

This paper focuses on the existence and nonexistence of a time periodic traveling wave solution of a time-periodic reaction–diffusion SEIR epidemic model. The main feature of the model is the possible deficiency of the classical comparison principle such that many known results do not directly work. If the basic reproduction number of the model, denoted by $R_0$, is larger than one, there exists a minimal wave speed $c^{\ast }>0$ satisfying for each $c>c^{\ast }$, the system admits a nontrivial time periodic traveling wave solution with wave speed $c$ and for $c<c^{\ast }$, there exists no nontrivial time periodic traveling waves such that the system; if $R_0<1$, the system admits no nontrivial time periodic traveling waves.