Mathematical analysis for a dynamic model of illicit drug consumption

This paper presents a comprehensive mathematical model for analyzing the dynamics of illicit drug consumption within a population. Building upon and extending classical epidemiological models such as the susceptible, infected, and recovered (SIR) framework, the proposed model categorizes individuals into four compartments: Non-susceptible, susceptible, addicted, and rehabilitated. The model incorporates key social factors such as peer influence, intervention efforts, and the probability of relapse. A nonlinear system of differential equations was developed to describe the transitions between these states. The basic reproduction number $R_0$ was derived to assess the potential spread of addiction and stability analysis of the equilibrium points was carried out. Numerical simulations explore the impact of various intervention levels, highlighting how increased societal and family involvement can reduce the prevalence of addiction. Furthermore, an optimal control problem was formulated and solved using Pontryagin's maximum principle to determine the most cost-effective intervention strategy over time. The results demonstrate that adaptive, well-balanced policies can significantly reduce both susceptibility and addiction rates while minimizing social and economic costs.