A cross-infection model with diffusion and incubation period

In this paper, we study a cross-infection model with diffusion and incubation period. Firstly, we prove the global attractivity of the infection-free equilibrium and infected equilibrium for the spatially homogeneous system. Secondly, we establish the threshold dynamics for the spatially heterogeneous system in terms of the basic reproduction number \begin{document}$ \mathcal{R}_0 $\end{document} . It turns out that the infection-free steady state is globally attractive if \begin{document}$ \mathcal{R}_0<1 $\end{document} ; and the system is uniformly persistent if \begin{document}$ \mathcal{R}_0>1 $\end{document} . Finally, we explore the influence of different diffusion coefficients, spatial heterogeneity of the disease transmission rate and the incubation period on \begin{document}$ \mathcal{R}_0 $\end{document} . Our numerical results show that \begin{document}$ \mathcal{R}_0 $\end{document} are decreasing functions of the diffusion coefficients and the incubation period, respectively, while it is increasing with respect to the spatial heterogeneity.