Fast-slow dynamics of an SIS epidemic model with nonlinear incidence

This paper analyzes the dynamics of an SIS epidemic model with a nonlinear incidence rate. By employing geometric singular perturbation theory, we demonstrate that the system exhibits rich dynamical phenomena including supercritical Hopf bifurcation, canard explosion, relaxation oscillation, heteroclinic and homoclinic orbits. Our results show that higher-order nonlinear incidence rates can induce complex and diverse dynamic behaviors in simple SIS models. Moreover, the existence of canard phenomenon indicates that even when disease transmission is under control but not yet eliminated, there remains the possibility of sudden or even periodic outbreaks. At the same time, numerical simulations verify the theoretical results.