Stability And Bogdanov-Takens Bifurcation Of An SIRS Epidemic Model With Standard Incidence Rate And Saturated Treatment Function

An epidemic model with standard incidence rate and saturated treatment function of infective individuals is proposed to understand the e?ect of the capacity for treatment of infective individuals on the disease spread in this paper. The treatment function in this paper is a continuous and di?erential function which exhibits the e?ect of delayed treatment when the rate of treatment is lower and the number of infected individuals is getting larger. It is proved that the existence and stability of the disease-free and endemic equilibria for the model is not only related to the basic reproduction number but also the capacity for treatment of infective individuals. And a backward bifurcation is found when the capacity is not enough. By computing the first Lyapunov coe?cient, we can determine the type of Hopf bifurcation, i.e., subcritical Hopf bifurcation or supercritical Hopf bifurcation. We also show that the model undergoes Bogdanov-Takens bifurcation under some conditions, i.e., Hopf bifurcation, saddle-node bifurcation and homoclinic bifurcation.