Bifurcations of an SIRS epidemic model with a general saturated incidence rate

This paper is concerned with the bifurcations of a susceptible-infectious-recovered-susceptible (SIRS) epidemic model with a general saturated incidence rate $ k I^p/(1+\alpha I^p) $. For general $ p > 1 $, it is shown that the model can undergo saddle-node bifurcation, Bogdanov-Takens bifurcation of codimension two, and degenerate Hopf bifurcation of codimension two with the change of parameters. Combining with the results in [1] for $ 0 < p\leq 1 $, this type of SIRS model has Hopf cyclicity $ 2 $ for any $ p > 0 $. These results also improve some previous ones in [2] and [3], which are dealt with the special case of $ p = 2 $.