Bifurcation Analysis Of A SIRS Epidemic Model With A Generalized Nonmonotone And Saturated Incidence Rate

In this paper,we study a SIRS epidemic model with a generalized nonmonotone and saturated incidence rate kI²S/1+?I+?I²,in which the infection function firstincreases 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 also shown that,there exists a critical value ? = ?₀ for the psychological effect,and two critical values k = k₀,k?k₀<k₁?for the infection rate such that:???when ?>?₀,or ???₀and k?k₀,the disease will die out for all positive initial populations;???when ?= ?₀ and k₀<k?k₁,the disease will die out for almost,all positiveinitial populations:???when ? = ?₀ and k>k₁.the disease.will persist in the formof a positive coexistent steady state for some positive initial populations;and???when ?<?₀ and k>k₀,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.