Slow–Fast Bifurcation of the SIRS Epidemic Model with a Generalized Nonmonotone Incidence Rate

In this paper, we propose an SIRS epidemic model with a generalized nonmonotone incidence rate, describing the psychological effects of communities on some infectious diseases as the numbers of infective individuals increase. Utilizing geometric singular perturbation theory together with blow-up techniques, we demonstrate that this model can exhibit richer dynamics, such as supercritical singular Hopf bifurcation, canard explosion and relaxation oscillations, and provide a complete classification of bifurcation phenomena near a transcritical singularity. Notably, we identify transitory canard cycles with beards, and the canard cycles with head pass nearby a canard point and a transcritical point during their transition from canard cycles without head to relaxation oscillations. This is a novel dynamical phenomenon. Numerical simulations are provided to illustrate and validate our theoretical results.