Fractal–Fractional Modeling of Drug Addiction Dynamics: Capturing Memory-Driven Effects

This study presents a novel fractal–fractional model to investigate the dynamics of drug addiction within a specific region, capturing memory-driven behaviors and complex interactions among different subpopulations of drug users. The model employs fractal–fractional differential equations with a nonsingular and nonlocal kernel, enabling an accurate representation of long-term dependencies and the chronic nature of addiction. We conduct a theoretical analysis to establish the existence and stability of drug-free and persistent drug-use equilibria, highlighting the role of historical exposure and social influence in shaping addiction dynamics. Numerical solutions are obtained using the Adams–Bashforth method adapted for fractal–fractional derivatives, ensuring computational efficiency despite the memory effects introduced by the model. Simulations illustrate the progression of light, hidden, and treated drug users, as well as recovered individuals, providing insights into the effectiveness of early interventions, rehabilitation programs, and relapse prevention strategies. Overall, the proposed framework offers a robust tool for understanding addiction dynamics and informing public health policies aimed at mitigating the impact of substance abuse in communities.