Hopf Bifurcation Analysis of a Synthetic Drug Transmission Model with Time Delays

Abstract

This paper is concerned with the Hopf bifurcation of a synthetic drug transmission model with two delays. Firstly, some sufficient conditions of delay-induced bifurcation for such a model are captured by using different combinations of the two delays as the bifurcation parameter. Secondly, based on the center manifold theorem and normal form theory, some sufficient conditions determining properties of the Hopf bifurcation such as the direction and the stability are established. Finally, to underline the effectiveness of the obtained results, some numerical simulations are ultimately addressed.

A Scoping Review of Drug Epidemic Models

“…The study obtained sufficient conditions for the threshold dynamics of the system and conducted sensitivity analysis as well as numerical simulations. This model was revisited by Zhang et al shortly after, who considered time delays in two rehabilitation processes [ 56 ]. The study investigated sufficient conditions for the existence of delay-induced Hopf bifurcation and determined the direction of the Hopf bifurcation as well as the stability of the bifurcating periodic solutions.…”

Modeling the Dynamics of Drug Spreading in China

“…In this article, we formulate a drug epidemic model with a complete addiction–rehabilitation–recovery process. Unlike many other studies, we do not allow self-abstinence without treatment or any other unrealistic assumptions incompatible with the social background in China [ 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 ]. After qualitative analyses of the theoretical behaviors of the model system, we fit the model to historical data of drug abuse in China and make projections of the future.…”