Global dynamics and bifurcations in a SIRS epidemic model with a nonmonotone incidence rate and a piecewise-smooth treatment rate

In this paper, we analyze a SIRS epidemic model with a nonmonotone incidence rate and a piecewise-smooth treatment rate. The nonmonotone incidence rate describes the "psychological effect": when the number of the infected individuals (denoted by \begin{document}$ I $\end{document} ) exceeds a certain level, the incidence rate is a decreasing function with respect to \begin{document}$ I $\end{document} . The piecewise-smooth treatment rate describes the situation where the community has limited medical resources, treatment rises linearly with \begin{document}$ I $\end{document} until the treatment capacity is reached, after which constant treatment (i.e., the maximum treatment) is taken.Our analysis indicates that there exists a critical value \begin{document}$ \widetilde{I_0} $\end{document} \begin{document}$ ( = \frac{b}{d}) $\end{document} for the infective level \begin{document}$ I_0 $\end{document} at which the health care system reaches its capacity such that:(i) When \begin{document}$ I_0 \geq \widetilde{I_0} $\end{document} , the transmission dynamics of the model is determined by the basic reproduction number \begin{document}$ R_0 $\end{document} : \begin{document}$ R_0 = 1 $\end{document} separates disease persistence from disease eradication. (ii) When \begin{document}$ I_0 , the model exhibits very rich dynamics and bifurcations, such as multiple endemic equilibria, periodic orbits, homoclinic orbits, Bogdanov-Takens bifurcations, and subcritical Hopf bifurcation.