Stability analysis and optimal control of a mathematical model for heroin use dynamics

Abstract

The widespread abuse of heroin, a potent opioid derived from morphine, continues to pose serious challenges to public health due to its high addiction potential and neurotoxic effects. In response to the growing demand for evidence-based intervention strategies, this study develops a compartmental mathematical model to better understand the transmission dynamics of heroin dependence. The model incorporates a distinct compartment for recovering individuals, explicitly accounting for the risk of relapse. Analytical investigation yields the equilibrium states of the system, and the basic reproduction number R0 is derived to characterize the threshold behavior. Stability analyses -both local and global- are conducted for the heroin-free and endemic equilibria using the Routh-Hurwitz criterion and Lyapunov functionals. A sensitivity analysis is performed to identify key parameters that influence the persistence of addiction. To evaluate possible interventions, we introduce optimal control strategies that emphasize awareness programs and nonpharmaceutical approaches. Numerical simulations carried out using MATLAB support the theoretical results and highlight the potential effectiveness of combined strategies in curbing the spread of heroin use. These findings provide a theoretical foundation for informed public health decision-making in the context of opioid addiction.