Bifurcation analysis for a SIRS model with a nonlinear incidence rate

In this paper, the main purpose is to explore an SIRS epidemic model with a general nonlinear incidence rate $f(I)S=\beta I(1+\upsilon I^{k-1})S$ ($k>0$). We analyzed the existence and stability of equilibria of the epidemic model. Local bifurcation theory is applied to explore the rich variety of dynamical behavior of the model. Normal forms of the epidemic model are derived for different types of bifurcation, including Bogdanov-Takens bifurcation, Nilpotent focus bifurcation and Hopf bifurcation. The first four focal values are computed to determine the codimension of the Hopf bifurcation, which can be undergo some limit cycles. Some numerical results and simulations are presented to illustrate these theoretical results.