Bifurcations of an SIRS epidemic model with nonlinear
Zhixing Hu - Department of Applied Mathematics and Mechanics, University of Science and Technology Beijing, Beijing 100083, China (email)
Abstract: 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.
Keywords: SIRS epidemic model, nonlinear incidence rate, stability, Hopf
bifurcation, Bogdanov-Takens bifurcation.
Received: December 2009; Revised: July 2010; Available Online: October 2010.
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