# American Institute of Mathematical Sciences

January  2011, 15(1): 93-112. doi: 10.3934/dcdsb.2011.15.93

## Bifurcations of an SIRS epidemic model with nonlinear incidence rate

 1 Department of Applied Mathematics and Mechanics, University of Science and Technology Beijing, Beijing 100083, China 2 Department of Mathematics, East China Normal University, Shanghai 200062, China 3 Department of Applied Mathematics, School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083 4 Department of Mathematics, The University of Miami, P.O. Box 249085, Coral Gables, Florida 33124

Received  December 2009 Revised  July 2010 Published  October 2010

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.
Citation: Zhixing Hu, Ping Bi, Wanbiao Ma, Shigui Ruan. Bifurcations of an SIRS epidemic model with nonlinear incidence rate. Discrete & Continuous Dynamical Systems - B, 2011, 15 (1) : 93-112. doi: 10.3934/dcdsb.2011.15.93
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