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Zbl 1207.92040
Hu, Zhixing; Bi, Ping; Ma, Wanbiao; Ruan, Shigui
Bifurcations of an SIRS epidemic model with nonlinear incidence rate. (English)
[J] Discrete Contin. Dyn. Syst., Ser. B 15, No. 1, 93-112 (2011). ISSN 1531-3492; ISSN 1553-524X/e

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 2000:: *92D30 Epidemiology
34D20 Lyapunov stability of ODE
34C60 Applications of qualitative theory of ODE
34K18 Bifurcation theory of functional differential equations
34K20 Stability theory of functional-differential equations
34C23 Bifurcation (periodic solutions)
37N25 Dynamical systems in biology

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

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