Hopf bifurcation of a heroin model with time delay and saturated treatment function

In this paper, local stability and Hopf bifurcation of a delayed heroin model with saturated treatment function are discussed. First of all, sufficient conditions for local stability and existence of Hopf bifurcation are obtained by regarding the time delay as a bifurcation parameter and analyzing the distribution of the roots of the associated characteristic equation. Directly afterward, properties of the Hopf bifurcation, such as the direction and stability, are investigated with the aid of the normal form theory andthe manifold center theorem. Finally, numerical simulations are presented to justify the obtained theoretical results, and some suggestions are offered for controlling heroin abuse in populations.