We consider an SIRS epidemic model with a more generalized non-monotone incidence: χ(I)=κIp1+Iq$\chi(I)=\frac{\kappa I^{p}}{1+I^{q}}$ with 01$, by qualitative and bifurcation analyses, we show that the model undergoes a saddle-node bifurcation,… Click to show full abstract
We consider an SIRS epidemic model with a more generalized non-monotone incidence: χ(I)=κIp1+Iq$\chi(I)=\frac{\kappa I^{p}}{1+I^{q}}$ with 01$p>1$ distinctly vary. On one hand, the number and stability of disease-free and endemic equilibrium are different. On the other hand, when p≤1$p\leq1$, there do not exist any closed orbits and when p>1$p>1$, 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$p=2$, q=3$q=3$, 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.
