Modeling the Spread and Control of Drug Addiction: A Mathematical and Sensitivity Analysis Approach

Abstract

This paper presents a comprehensive mathematical model for the dynamics of drug addiction, incorporating high-risk and low-risk susceptible populations (Si, Sd), prone individuals (P), untreated drug abusers (ID), treatment interventions (hospitalized (TH) and non-hospitalized (Tp)), and recovered individuals (R). The model captures key transitions between risk classes and treatment pathways, reflecting the effects of behavioral, social, and medical interventions. Analytical results include the derivation of the basic reproduction number R0 and a sensitivity analysis identifying the most influential parameters (β1,Δ, η, and the treatment rate σ) affecting the spread and control of drug abuse. Numerical simulations demonstrate that increasing the treatment rate σ, the central control parameter— leads to a marked reduction in both high-risk and low-risk susceptible populations, as well as a decline in the prevalence of drug abuse. Phase portraits further elucidate the dynamic interactions between key compartments, while parameter variation analyses highlight the critical role of transmission, risk transition, and especially treatment rates in shaping epidemic outcomes. The results underscore the importance of timely and effective treatment-based control strategies, providing valuable insights for designing public health policies aimed at reducing drug addiction and promoting sustained recovery.