Traveling waves in an SEIR epidemic model with the variable total population

In the present paper, we propose a simple diffusive SEIR epidemic modelwhere the total population is variable. We first give the explicit formula of thebasic reproduction number $\mathcal{R}_0$ for the model. And hence, we show that if$\mathcal{R}_0>1$, then there exists a constant $c^*>0$ such that for any $c>c^*$, themodel admits a nontrivial traveling wave solution, and if $\mathcal{R}_00$(or, $\mathcal{R}_0>1$ and $c\in(0,c^*)$), then the model has no nontrivial travelingwave solution. Consequently, we obtain the full information about the existence andnon-existence of traveling wave solutions of the model by determined by the constants$\mathcal{R}_0$ and $c^*$. The proof of the main results is mainly based on Schauderfixed point theorem and Laplace transform.