This paper introduces a novel mathematical model for capturing the complex dynamics of drug abuse within populations. Departing from conventional methodologies, the model employs global derivatives to integrate non-local effects, thereby offering enhanced insight into the spread and evolution of drug abuse. The stability analysis and numerical simulations conducted in this study reveal critical thresholds and dynamic behaviors that are instrumental in understanding the persistence and potential escalation of abuse within communities. Numerical simulations also demonstrate the long-term behavior for different orders of $α,$ and the effects of the function $g(x)$ are presented, further elucidating the intricate interplay of factors that govern the system’s dynamics. These findings not only shed light on the underlying mechanisms driving the temporal and spatial patterns of drug abuse but also provide valuable guidance for designing effective intervention strategies aimed at mitigating its spread. By systematically manipulating key parameters, the model serves as a powerful tool for exploring the driving factors behind drug abuse diffusion and control. The insights gained from this research have significant implications for public health policy, offering a rigorous mathematical framework to inform targeted efforts in curbing the epidemic of drug abuse.