A delayed SIR model with general nonlinear incidence rate

An SIR epidemic model is investigated and analyzed based on incorporating an incubation time delay and a general nonlinear incidence rate, where the growth of susceptible individuals is governed by the logistic equation. The threshold parameter σ0[12pt]{minimal} {amsmath} {wasysym} {amsfonts} {amssymb} {amsbsy} {mathrsfs} {upgreek} {}{-69pt} {document}σ₀{document} is defined to determine whether the disease dies out in the population. The model always has the trivial equilibrium and the disease-free equilibrium whereas it admits the endemic equilibrium if σ0[12pt]{minimal} {amsmath} {wasysym} {amsfonts} {amssymb} {amsbsy} {mathrsfs} {upgreek} {}{-69pt} {document}σ₀{document} exceeds one. The disease-free equilibrium is globally asymptotically stable if σ0[12pt]{minimal} {amsmath} {wasysym} {amsfonts} {amssymb} {amsbsy} {mathrsfs} {upgreek} {}{-69pt} {document}σ₀{document} is less than one, while it is unstable if σ0[12pt]{minimal} {amsmath} {wasysym} {amsfonts} {amssymb} {amsbsy} {mathrsfs} {upgreek} {}{-69pt} {document}σ₀{document} is greater than one. By applying the time delay as a bifurcation parameter, the local stability of the endemic equilibrium is studied and the condition which is absolutely stable or conditionally stable is established. Furthermore, a Hopf bifurcation occurs under certain conditions. Numerical simulations are carried out to illustrate the main results.