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