Mathematical Insights into Substance Addiction and Abuse Dynamics via Global Nonlocal Operators (African Journal for Industrial and Applied Mathematics, 2025)
This paper introduces a novel mathematical model for capturing thecomplex dynamics of drug abuse within populations. Departing from conventionalmethodologies, the model employs global derivatives to integrate non-local effects,thereby offering enhanced insight into the spread and evolution of drug abuse. Thestability analysis and numerical simulations conducted in this study reveal critical thresholds and dynamic behaviors that are instrumental in understanding thepersistence and potential escalation of abuse within communities. Numerical simulations also demonstrate the long-term behavior for different orders of $α,$ and theeffects of the function $g(x)$ are presented, further elucidating the intricate interplay of factors that govern the system’s dynamics. These findings not only shedlight on the underlying mechanisms driving the temporal and spatial patterns ofdrug abuse but also provide valuable guidance for designing effective interventionstrategies 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 havesignificant implications for public health policy, offering a rigorous mathematicalframework to inform targeted efforts in curbing the epidemic of drug abuse.