A cross-infection model with diffusion and incubation period

<p style='text-indent:20px;'>In this paper, we study a cross-infection model with diffusion and incubation period. Firstly, prove the global attractivity of infection-free equilibrium infected for spatially homogeneous system. Secondly, establish threshold dynamics heterogeneous system in terms basic reproduction number <inline-formula><tex-math id="M1">\begin{document}$ \mathcal{R}_0 $\end{document}</tex-math></inline-formula>. It turns out that steady state is globally attractive if id="M2">\begin{document}$ \mathcal{R}_0<1 $\end{document}</tex-math></inline-formula>; uniformly persistent id="M3">\begin{document}$ \mathcal{R}_0>1 Finally, explore influence different coefficients, spatial heterogeneity disease transmission rate period on id="M4">\begin{document}$ Our numerical results show id="M5">\begin{document}$ $\end{document}</tex-math></inline-formula> are decreasing functions coefficients period, respectively, while it increasing respect to heterogeneity.</p>