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 [Formula: see text] , 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 [Formula: see text] for the psychological effect, and two critical values [Formula: see text] for the infection rate such that: (i) when [Formula: see text] , or [Formula: see text] and [Formula: see text] , the disease will die out for all positive initial populations; (ii) when [Formula: see text] and [Formula: see text] , the disease will die out for almost all positive initial populations; (iii) when [Formula: see text] and [Formula: see text] , the disease will persist in the form of a positive coexistent steady state for some positive initial populations; and (iv) when [Formula: see text] and [Formula: see text] , 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.