Global dynamics of an epidemic model with an unspecified degree

In this paper, we give the global dynamical behaviors of a reduced SIRS epidemic model with a nonlinear incidence rate κIpSqκIpSq. We first discuss the qualitative properties of the equilibria in the interior of the first quadrant, and study the bifurcations including saddle–node bifurcation, transcritical bifurcation, Hopf bifurcation and Bogdanov–Takens bifurcation. Then we consider equilibria at infinity, determining the number of orbits in exceptional directions for the global tendency. In this discussion, the unspecified degree p,qp,q of polynomials and their high degeneracy prevent us from using the methods of blowing-up or normal sectors in some cases. We lastly discuss the existence and uniqueness of limit cycles.