Global Dynamics of a Susceptible-Infectious-Recovered Epidemic Model with a Generalized Nonmonotone Incidence Rate

A susceptible-infectious-recovered (SIRS) epidemic model with a generalized nonmonotone incidence rate $$ ( $$ such that $$ for all $$ ) is considered in this paper. It is shown that the basic reproduction number $$ does not act as a threshold value for the disease spread anymore, and there exists a sub-threshold value $$ such that: (i) if $$ , then the disease-free equilibrium is globally asymptotically stable; (ii) if $$ , then there is a unique endemic equilibrium which is a nilpotent cusp of codimension at most three; (iii) if $$ , then there are two endemic equilibria, one is a weak focus of multiplicity at least three, the other is a saddle; (iv) if $$ , then there is again a unique endemic equilibrium which is a weak focus of multiplicity at least three. As parameters vary, the model undergoes saddle-node bifurcation, backward bifurcation, Bogdanov-Takens bifurcation of codimension three, Hopf bifurcation, and degenerate Hopf bifurcation of codimension three. Moreover, it is shown that there exists a critical value $$ for the psychological effect $$ , a critical value $$ for the infection rate k, and two critical values $$ for $$ that will determine whether the disease dies out or persists in the form of positive periodic coexistent oscillations or coexistent steady states under different initial populations. Numerical simulations are given to demonstrate the existence of one, two or three limit cycles.