Time-periodic traveling wave solutions of a reaction–diffusion Zika epidemic model with seasonality

In this paper, the full information about the existence and nonexistence of a time-periodic traveling wave solution of a reaction–diffusion Zika epidemic model with seasonality, which is non-monotonic, is investigated. More precisely, if the basic reproduction number, denoted by \(R_{0}\) $R_0$ , is larger than one, there exists a minimal wave speed \(c^* > 0\) $c^{\ast }>0$ satisfying for each \(c > c^*\) $c>c^{\ast }$ , the system admits a nontrivial time-periodic traveling wave solution with wave speed c, and for \(c<c^*\) $c<c^{\ast }$ , there exist no nontrivial time-periodic traveling waves such that if \(R_0 \leqslant 1\) $R_0⩽1$ , the system admits no nontrivial time-periodic traveling waves.