Dynamics of an SIRS epidemic model with cross-diffusion

The dynamical behavior of an SIRS epidemic reaction-diffusion model with frequency-dependent mechanism in a spatially heterogeneous environment is studied, with a chemotaxis effect that susceptible individuals tend to move away from higher concentration of infected individuals. Regardless of the strength of the chemotactic coefficient and the spatial dimension \begin{document}$ n $\end{document} , it is established the unique global classical solution which is uniformly-in-time bounded. The model still recognizes the threshold dynamics in terms of the basic reproduction number \begin{document}$ \mathcal{R}_{0} $\end{document} even in the case of chemotaxis effects: if \begin{document}$ \mathcal{R}_{0} , the unique disease free equilibrium is globally stable; if \begin{document}$ \mathcal{R}_{0}>1 $\end{document} , the disease is uniformly persistent and there is at least one endemic equilibrium, which is globally stable in some special cases with weak chemotactic sensitivity. We also show the asymptotic profile of endemic equilibria (when exists) if the diffusion (migration) rate of the susceptible is small, which indicates that the disease always exists in the entire habitat in this case. Our results suggest that one cannot eradicate the SIRS disease model by only controlling the diffusion rate of susceptible individuals.