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

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. (C) 2019 Elsevier Ltd. All rights reserved.