Threshold dynamics of a reaction-diffusion cholera model with seasonality and nonlocal delay

Abstract

<p style='text-indent:20px;'>In this paper, we investigate the threshold results for a nonlocal and time-delayed reaction-diffusion system involving the spatial heterogeneity and the seasonality. Due to the complexity of the model, we rigorously analyze the well-posedness of the model. The basic reproduction number <inline-formula><tex-math id="M1">\begin{document}$ \Re_0 $\end{document}</tex-math></inline-formula> is characterized with the next generation operator method. We show that the disease-free <inline-formula><tex-math id="M2">\begin{document}$ \omega $\end{document}</tex-math></inline-formula>-periodic solution is globally attractive when <inline-formula><tex-math id="M3">\begin{document}$ \Re_0 < 1 $\end{document}</tex-math></inline-formula>; while the system is uniformly persistent and a positive <inline-formula><tex-math id="M4">\begin{document}$ \omega $\end{document}</tex-math></inline-formula>-periodic solution exists when <inline-formula><tex-math id="M5">\begin{document}$ \Re_0 > 1 $\end{document}</tex-math></inline-formula>. In a special case that the parameters are all independent of the spatial heterogeneity and the seasonality, the global attractivity of the constant equilibria of the model is investigated by the technique of Lyapunov functionals.</p>