Modeling and stability analysis of substance abuse in women with control policies

Abstract

Substance abuse is considered to be predominantly a problem for all in society, and much substance abuse research is done with a male majority. Recent drug addiction studies, however, reveal that there are substantial gender variations in substance-related epidemiology. The epidemiology of women’s substance misuse raises issues distinct from those addressed by men’s substance abuse. Women with addictions are more likely than males to experience several impediments to accessing and entering substance misuse treatment. In this regard, we framed a mathematical model by considering the whole population as a women population. For this system, we found pair of equilibrium points namely the Addiction-free equilibrium point and the Non-trivial equilibrium point. Moreover, the addicts generation number (threshold parameter) RA has been found to investigate the spread of addiction. Also, the local and global stability analysis for both the equilibrium points have investigated. Further, to analyze the decrease and increase of the infected population and recovery population respectively, an optimum control analysis is done with two control parameters using Pontryagin’s maximum principle. The influence of addict generation numbers on significant parameters involved in RA has been shown. An optimum control study projected that social media awareness is substantially more efficient than the fixed control in optimizing drug addiction among women. Through numerical simulations we have exhibited, the effect of with and without control on each population. Finally, it produced positive efficacy by reducing the number of infected and increasing the rehabilitation population.

Keywords

• Lyapunov’s function
• Pontryagin’s maximum principle
• Sensitive analysis