Bifurcation analysis of a delayed SIR model

Hopf bifurcation of an SIR epidemic model with incubation time and saturated incidence rate is studied, where the susceptibles are assumed to satisfy the logistic equation and the incidence term is of saturated form with the infective. The threshold value R₀ determining whether the disease dies out is found. If R₀ >; 1, by using the time delay (i.e., incubation time) as a bifurcation parameter, the local stability of the endemic equilibrium is investigated, and the conditions for Hopf bifurcation to occur are derived. Numerical simulations are presented to illustrate our main results.