Bifurcation analysis of an SIRS epidemic model with a generalized nonmonotone and saturated incidence rate

In this paper, we study a susceptible-infectious-recovered (SIRS) epidemic model with a generalized nonmonotone and saturated incidence rate $\frac{k{I}^{2}S}{1+\beta I+\alpha I^2}$ , in which the infection function first increases to a maximum when a new infectious disease emerges, then decreases due to psychological effect, and eventually tends to a saturation level due to crowding effect. It is shown that there are a weak focus of multiplicity at most two and a cusp of codimension at most two for various parameter values, and the model undergoes saddle-node bifurcation, Bogdanov-Takens bifurcation of codimension two, Hopf bifurcation, and degenerate Hopf bifurcation of codimension two as the parameters vary. It is shown that there exists a critical value $\alpha ={\alpha }_0$ for the psychological effect, and two critical values $k=k_0,k_1(k_0<k_1)$ for the infection rate such that: (i) when $\alpha >{\alpha }_0$ , or $\alpha \le {\alpha }_0$ and $k\le k_0$ , the disease will die out for all positive initial populations; (ii) when $\alpha ={\alpha }_0$ and $k_0<k\le k_1$ , the disease will die out for almost all positive initial populations; (iii) when $\alpha ={\alpha }_0$ and $k>k_1$ , the disease will persist in the form of a positive coexistent steady state for some positive initial populations; and (iv) when $\alpha <{\alpha }_0$ and $k>k_0$ , the disease will persist in the form of multiple positive periodic coexistent oscillations and coexistent steady states for some positive initial populations. Numerical simulations, including the existence of one or two limit cycles and data-fitting of the influenza data in Mainland China, are presented to illustrate the theoretical results.