Complex dynamics in an SIS epidemic model with nonlinear incidence

We study an epidemic model with nonlinear incidence rate, describing the saturated mass action and the psychological effect of certain serious diseases on the community. Firstly, the existence and local stability of disease-free and endemic equilibria are investigated. Then we prove the occurrence of backward bifurcations, saddle-node bifurcations, Hopf bifurcations and cusp type Bogdanov–Takens bifurcations of codimension 3. Finally, numerical simulations, including one limit cycle, two limit cycles, an unstable homoclinic loop and many other phase portraits are presented. These results show that the psychological effect of diseases and the behavior change of the susceptible individuals may affect the final spread level of an epidemic.