Bifurcation Analysis Of An Sirs Epidemic Model With Saturated Incidence Rate And Saturated Treatment Function

In this paper,an SIRS epidemic model with saturated incidence rate and saturated treatment function of infective individuals is introduced to understand the effect of the treatment on the spread of the disease.The treatment function is a continuous and differential function which shows the effect of delayed treatment when the medical condition is limited and the number of infected individuals is getting larger.The existence and global asymptotical stability of the disease-free and endemic equilibria is proved for the model.The sufficient condition of nonexistence of limit cycles is also given.The results of the article show that the dynamic behavior of the disease-free and endemic equilibria is not only related to the basic reproduction number but also the capacity for treatment of infective individuals.When the capacity of the treatment is low,a backward bifurcation is found.It indicates that we should improve the capacity of the treatment to control the spread of diseases.By computing the first Lyapunov coefficient,we can determine the type of Hopf bifurcation.By mathematical analysis,under certain conditions,we also show that the model undergoes Bogdanov-Takens bifurcation,i.e.,there are saddle-node bifurcation,Hopf bifurcation and homoclinic bifurcation in the system.