In this paper we investigate the dynamics behavior of the discrete time SIR epidemic model with a nonlinear incidence rate λSpI. Firstly, we determine the topological type of the endemic fixed point, including the existence and stability of the fixed point. Furthermore, we analyze the bifurcation situations, and discuss the flip bifurcation on the center manifold and the Neimark Sacker bifurcation of this SIR system by center manifold theorem and normal form theory. Their bifurcation directions are given respectively. Finally, some biological explanations of our mathematical results are presented.
