Stability and Bogdanov-Takens Bifurcation of an SIS Epidemic Model with Saturated Treatment Function

Abstract

This paper introduces the global dynamics of an SIS model with bilinear incidence rate and saturated treatment function. The treatment function is a continuous and differential function which shows the effect of delayed treatment when the rate of treatment is lower and the number of infected individuals is getting larger. Sufficient conditions for the existence and global asymptotic stability of the disease-free and endemic equilibria are given in this paper. The first Lyapunov coefficient is computed to determine various types of Hopf bifurcation, such as subcritical or supercritical. By some complex algebra, the Bogdanov-Takens normal form and the three types of bifurcation curves are derived. Finally, mathematical analysis and numerical simulations are given to support our theoretical results.

Dynamics analysis and optimal control strategy for a SIRS epidemic model with two discrete time delays

“…There are many studies on incidence rate and treatment function in infectious disease models [1][2][3][4][5][6]. In [7], Wang et al investigated a SIR infectious disease model with constant treatment function and bilinear incidence rate, and presented the evidences of choosing a bilinear incidence rate to prove the existence of Hopf bifurcation.…”