Hopf Bifurcation of a Delayed Epidemic Model with Information Variable and Limited Medical Resources

Abstract

We consider SIR epidemic model in which population growth is subject to logistic growth in absence of disease. We get the condition for Hopf bifurcation of a delayed epidemic model with information variable and limited medical resources. By analyzing the corresponding characteristic equations, the local stability of an endemic equilibrium and a disease-free equilibrium is discussed. If the basic reproduction ratio , we discuss the global asymptotical stability of the disease-free equilibrium by constructing a Lyapunov functional. If ${ℛ}_0>1$, we obtain sufficient conditions under which the endemic equilibrium $E^{*}$ of system is locally asymptotically stable. And we also have discussed the stability and direction of Hopf bifurcations. Numerical simulations are carried out to explain the mathematical conclusions.