Canard Phenomenon in an SIRS Epidemic Model with Nonlinear Incidence Rate

In this paper, we propose an SIRS epidemic model with a new complex nonlinear incidence rate, which describes the psychological effect of some diseases on the community as the number of infective individuals increases, including linear and nonlinear hazards of infection. The canard phenomenon for the model is analyzed, and its epidemiological meaning is discussed. By using geometrical singular perturbation theory and blow up technique, we investigate the relaxation oscillation of the model with the special fold point B. The unique existence of the limit cycle is proved. We verify the existence of the canard cycle without head by using singular perturbation theory and analyze the cyclicity of the limit cycle. The detailed formula for slow divergence integral of the model is presented. We also discuss and prove the existence of the canard cycle with head. Numerical simulations are done to demonstrate our theoretical results.