Compartmental Model for Substance Abuse Dynamics: Threshold, Stability Analysis and Numerical Simulations

Abstract

 Substance abuse continues to pose a serious public health threat worldwide, fueling illness, premature death, crime, family breakdown, and heavy economic losses. To tackle this crisis effectively, we first need to understand how substance use takes root and persists across communities. With this in mind, we developed a deterministic compartmental model that tracks the movement of individuals through distinct stages from susceptible to initial use, dependence, treatment, recovery, and possible relapse. We ensured the model makes biological and mathematical sense by confirming that all solutions remain non-negative and bounded within a realistic region. We identified both drug-free and endemic steady states and computed the Threshold number, $R_0$, using the standard next-generation matrix approach. Our threshold analysis shows that when $R_0 < 1$, the drug-free equilibrium is locally stable, meaning the problem can die out over time. But once $R_0 > 1$, this stability breaks down, and the condition can persist. We also explored conditions under which the endemic equilibrium remains stable. To pinpoint which factors matter most, we carried out a sensitivity analysis, revealing the parameters that drive the spread and endurance of substance abuse. Finally, we ran numerical simulations to back up our theoretical results and to visualize how changing key parameters alters the course of the epidemic. Taken together, our findings suggest that curbing the rate at which people start using substances, reducing peer pressure, expanding access to treatment, improving recovery success, and preventing relapse could significantly lighten the burden of substance abuse. The framework we offer here is not only a tool for deeper scientific inquiry but also a practical guide for policymakers and health professionals designing long-term prevention and response strategies.