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

Zizhen Zhang,Yougang Wang

doi:10.1186/s13662-019-2009-4

Abstract

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 and the 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.

Highlights

Heroin is an opiate drug that is synthesized from morphine [1, 2]
A delayed heroin model with saturated treatment function in the form of αU1 1+ηU1 is discussed, which is different from the model in [21]
We assume that the heroin users cannot be cured instantaneously and it needs a period to cure heroin users, which is more in line with truth

Summary

Introduction

Heroin is an opiate drug that is synthesized from morphine [1, 2]. It causes somatic and psychological effects for heroin users, and brings social panic and economic loss to the entire human society. Some mathematical models have been formulated to describe the epidemic dynamics of heroin users. Motivated by the work of Samanta et al [11], Abdurahman et al [15] derived a discretized heroin epidemic model with a distributed time delay and studied its stability and permanence. As stated in [16], all the heroin models above assume that all individuals have the same level of susceptibility.

Results

Conclusion