A Mathematical Model of Substance Abuse–Driven Criminality: Exploring the Dynamics of Addiction and Crime

Drug abuse and crime are deeply interconnected, forming a vicious cycle that exacerbates public health and criminal justice challenges. In South Africa’s Gauteng province, substance abuse remains a major socioeconomic burden with far-reaching consequences. This study develops a seven-compartment deterministic model using ordinary differential equations to analyze the dynamics between drug abuse and criminal activity. The model incorporates homogeneous population mixing and accounts for removal rates associated with drug-related crime and rehabilitation. Analytical results indicate two equilibrium states: a narcocriminality-free equilibrium and a persistent (endemic) equilibrium. This study establishes that the narcocriminality-free equilibrium is globally asymptotically stable when the drug abuse criminogenic growth number (DGN, $\mathcal{D}_0<1$), while the endemic equilibrium exists when $\mathcal{D}_0 > 1$. Sensitivity analysis identifies the initiation rate as the most influential parameter on $\mathcal{D}_0$, showing that $\mathcal{D}_0$ increases with the progression rates $\alpha$ (light drug users) and $\rho$ (heavy drug users). Conversely, $\mathcal{D}_0$ shows a decrease with higher incarceration rates ($\epsilon$, $\gamma_1$) and rehabilitation rates ($\gamma_2$). These findings have important policy implications related to early intervention strategies targeting the drug-crime cycle, and enhancing rehabilitation programs and incarceration efficacy to reduce drug-driven criminality.