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

<p style='text-indent:20px;'>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 <inline-formula><tex-math id="M1">\begin{document}$ I $\end{document}</tex-math></inline-formula>) exceeds a certain level, the incidence rate is a decreasing function with respect to <inline-formula><tex-math id="M2">\begin{document}$ I $\end{document}</tex-math></inline-formula>. The piecewise-smooth treatment rate describes the situation where the community has limited medical resources, treatment rises linearly with <inline-formula><tex-math id="M3">\begin{document}$ I $\end{document}</tex-math></inline-formula> until the treatment capacity is reached, after which constant treatment (i.e., the maximum treatment) is taken.Our analysis indicates that there exists a critical value <inline-formula><tex-math id="M4">\begin{document}$ \widetilde{I_0} $\end{document}</tex-math></inline-formula> <inline-formula><tex-math id="M5">\begin{document}$ ( = \frac{b}{d}) $\end{document}</tex-math></inline-formula> for the infective level <inline-formula><tex-math id="M6">\begin{document}$ I_0 $\end{document}</tex-math></inline-formula> at which the health care system reaches its capacity such that:<b>(i)</b> When <inline-formula><tex-math id="M7">\begin{document}$ I_0 \geq \widetilde{I_0} $\end{document}</tex-math></inline-formula>, the transmission dynamics of the model is determined by the basic reproduction number <inline-formula><tex-math id="M8">\begin{document}$ R_0 $\end{document}</tex-math></inline-formula>: <inline-formula><tex-math id="M9">\begin{document}$ R_0 = 1 $\end{document}</tex-math></inline-formula> separates disease persistence from disease eradication. <b>(ii)</b> When <inline-formula><tex-math id="M10">\begin{document}$ I_0 < \widetilde{I_0} $\end{document}</tex-math></inline-formula>, the model exhibits very rich dynamics and bifurcations, such as multiple endemic equilibria, periodic orbits, homoclinic orbits, Bogdanov-Takens bifurcations, and subcritical Hopf bifurcation.</p>