Bifurcations of an SIRS epidemic model with nonlinear incidence rate. (English) Zbl 1207.92040

Hu, Zhixing; Bi, Ping; Ma, Wanbiao; Ruan, Shigui

Discrete Contin. Dyn. Syst., Ser. B 15, No. 1, 93-112 (2011).

Summary: The main purpose of this paper is to explore the dynamics of an epidemic model with a general nonlinear incidence \(\beta SI^p/(1 + \alpha I^q)\). The existence and stability of multiple endemic equilibria of the epidemic model are analyzed. Local bifurcation theory is applied to explore the rich dynamical behavior of the model. Normal forms of the model are derived for different types of bifurcations, including Hopf and Bogdanov-Takens bifurcations. Concretely speaking, the first Lyapunov coefficient is computed to determine various types of Hopf bifurcations. Next, with the help of the Bogdanov-Takens normal form, a family of homoclinic orbits is arising when a Hopf and a saddle-node bifurcation merge. Finally, some numerical results and simulations are presented to illustrate these theoretical results.

MSC:

92D30	Epidemiology
34D20	Stability of solutions to ordinary differential equations
34C60	Qualitative investigation and simulation of ordinary differential equation models
34K18	Bifurcation theory of functional-differential equations
34K20	Stability theory of functional-differential equations
34C23	Bifurcation theory for ordinary differential equations
37N25	Dynamical systems in biology

Keywords:

SIRS epidemic model; nonlinear incidence rate; stability; Hopf bifurcation, Bogdanov-Takes bifurcation