Stability analysis for a delayed SIR model with a nonlinear incidence rate

We develop an SIR vector-bone epidemic model incorporating incubation time delay and the nonlinear incidence rate, where the growth of susceptibles is governed by the logistic equation. The threshold parameter R0 is used to determine whether the disease persists in the population. The model always has the trivial equilibrium and the disease-free equilibrium whereas admits the endemic equilibrium if R0 exceeds one. The disease-free equilibrium is globally asymptotically stable if R0 is less than one, while the system is persistent if R0 is greater than one. Furthermore, by applying the time delay as a bifurcation parameter, the local stability of the endemic equilibrium is discussed and it loses stability and Hopf bifurcation occurs as the length of the time delay increases past τ 0 under certain conditions. An example is carried out to illustrate the main results.