Saddle-node bifurcation and Bogdanov-Takens bifurcation of a SIRS epidemic model with nonlinear incidence rate

The Bogdanov-Takens bifurcation of the SIRS epidemic model with nonlinear incidence rate was studied by Ruan and Wang (2003) [11], Tang et al. (2008) [13] and Lu et al. (2019) [9] in recent years. The results in the mentioned papers showed that the SIRS epidemic model with nonlinear incidence rate kI2/(1 + omega I2) can undergo a Bogdanov-Takens bifurcation of codimension two. In this paper we study the SIRS epidemic model with nonlinear incidence rate kIp/(1 + omega Iq) for general p and q. The bifurcation analysis indicates that there is a saddle-node or a cusp of codimension two for various parameter values and the model can undergo a saddle-node bifurcation or a Bogdanov-Takens bifurcation of codimension two if suitable bifurcation parameters are selected. It means that there are some SIRS epidemic models which have a limit cycle or a homoclinic loop. Moreover, it is also shown that the codimension of BogdanovTakens bifurcation is at most two. (c) 2023 Elsevier Inc. All rights reserved.