High codimension bifurcations of a simple epidemic model with a nonlinear incidence rate

In this paper, we study the high codimension bifurcation problems of an SIR epidemic model with a nonlinear incidence rate [Formula: see text]. Through bifurcation analysis of the system, it was found that the model can undergo high codimension bifurcation phenomena, including the codimension 3 cusp-type Bogdanov–Takens bifurcation, codimension 3 focus-type Bogdanov–Takens bifurcation, codimension 2 Bogdanov–Takens bifurcation, codimension 2 generalized Hopf bifurcation and codimension 2 cusp bifurcation of equilibrium points. In particular, considering the basic reproduction number [Formula: see text] as the bifurcation parameter, we found that the model exhibits complex dynamic behaviors during parameter variations, such as forward bifurcation with hysteresis, endemic bubble phenomena, mutual nesting of hysteresis and bubbles, multistability, and multi-periodicity. Our results show that high order nonlinear incidence rates may enable epidemic models to generate novel complex dynamic characteristics such as multiple steady states and large amplitude periodic oscillations. This study provides a new research perspective for analyzing the complex dynamic behaviors of epidemic systems with nonlinear incidence rates and their research achievements are of great significance for the prevention and control of infectious diseases.