endemic equilibrium

globally asymptotically

backward bifurcation

infection model

numerical simulations

basic reproduction number

asymptotically stable

normal form

global stability

Chaos, Solitons & Fractals

• A delayed epidemic model with logistic growth and saturated treatment is studied. • Backward bifurcation, Hopf bifurcation and Bogdanov-Takens bifurcation may occur. • Effects of varying parameters are studied in numerical simulations. • Presence of logistic growth and high treatment rate can decrease infection levels. In this paper, we incorporate a nonlinear incidence rate and a logistic growth rate into a SIR epidemic model for a vector-borne disease with incubation time delay and Holling type II saturated treatment. We compute the basic reproduction number and show that it completely determines the local stability of the disease-free equilibrium. Sufficient conditions for the existence of backward bifurcation and Hopf bifurcation are also established. Furthermore, we determine the direction and stability of the Hopf bifurcation around the endemic equilibrium by means of the center manifold theory. Our study reveals that the model admits a Bogdanov–Takens bifurcation when the time delay and the maximal disease transmission rate are varied. Numerical simulations are presented to illustrate the dynamics of the model and to study the effects caused by varying the treatment rate and delay parameters.

Dynamics of a time-delayed SIR epidemic model with logistic growth and saturated treatment