Traveling waves for a diffusive SEIR epidemic model

In this paper, we propose a diffusive SEIR epidemicmodel with saturating incidence rate. We first study the wellposedness of the model, and give the explicit formula of the basicreproduction number $\mathcal{R}_0$. And hence, we show that if$\mathcal{R}_0>1$, then there exists a positive constant $c^*>0$ such that for each$c>c^*$, the model admits a nontrivial traveling wave solution, andif $\mathcal{R}_0\leq1$ and $c\geq 0$ (or, $\mathcal{R}_0>1$ and$c\in[0,c^*)$), then the model has no nontrivial traveling wavesolutions. Consequently, we confirm that the constant $c^*$ is indeed theminimal wave speed. The proof of the main results is mainly based onSchauder fixed theorem and Laplace transform.