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

Abstract

<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>