Bifurcations of an SIRS model with generalized non-monotone incidence rate

Abstract

We consider an SIRS epidemic model with a more generalized non-monotone incidence: $\chi(I)=\frac{\kappa I^{p}}{1+I^{q}}$ with $0< p< q$ , describing the psychological effect of some serious diseases when the number of infective individuals is getting larger. By analyzing the existence and stability of disease-free and endemic equilibrium, we show that the dynamical behaviors of $p<1$ , $p=1$ and $p>1$ distinctly vary. On one hand, the number and stability of disease-free and endemic equilibrium are different. On the other hand, when $p\leq1$ , there do not exist any closed orbits and when $p>1$ , by qualitative and bifurcation analyses, we show that the model undergoes a saddle-node bifurcation, a Hopf bifurcation and a Bogdanov–Takens bifurcation of codimension 2. Besides, for $p=2$ , $q=3$ , we prove that the maximal multiplicity of weak focus is at least 2, which means at least 2 limit cycles can arise from this weak focus. And numerical examples of 1 limit cycle, 2 limit cycles and homoclinic loops are also given.