Traveling waves for a two-group epidemic model with latent period and bilinear incidence in a patchy environment

In this paper, we consider a two-group SIR epidemic model with bilinear incidence in a patchy environment. It is assumed that the infectious disease has a fixed latent period and spreads between two groups. Firstly, when the basic reproduction number \begin{document}$ \mathcal{R}_{0}>1 $\end{document} and speed \begin{document}$ c>c^{\ast} $\end{document} , we prove that the system admits a nontrivial traveling wave solution, where \begin{document}$ c^{\ast} $\end{document} is the minimal wave speed. Next, when \begin{document}$ \mathcal{R}_{0}\leq1 $\end{document} and \begin{document}$ c>0 $\end{document} , or \begin{document}$ \mathcal{R}_{0}>1 $\end{document} and \begin{document}$ c\in(0,c^{*}) $\end{document} , we also show that there is no positive traveling wave solution, where \begin{document}$ k = 1,2 $\end{document} . Finally, we discuss and simulate the dependence of the minimum speed \begin{document}$ c^{\ast} $\end{document} on the parameters.