Bifurcations of an SIRS epidemic model with nonlinearincidence rate

The main purpose of this paper is to explore the dynamics of anepidemic model with a general nonlinear incidenceβ SIᵖ/(1+α Iq). The existence and stabilityof multiple endemic equilibria of the epidemic model are analyzed.Local bifurcation theory is applied to explore the rich dynamicalbehavior of the model. Normal forms of the model are derived fordifferent types of bifurcations, including Hopf and Bogdanov-Takensbifurcations. Concretely speaking, the first Lyapunov coefficient iscomputed to determine various types of Hopf bifurcations. Next,with the help of the Bogdanov-Takens normal form, a family ofhomoclinic orbits is arising when a Hopf and a saddle-nodebifurcation merge. Finally, some numerical results and simulationsare presented to illustrate these theoretical results.