Modeling and Analyzing the Addiction of Gambling Through Fractional Calculus With Optimal Control Strategies

This study develops and analyzes a fractional-order compartmental model to describe the spread of gambling addiction within a community, incorporating Caputo derivatives to account for memory effects that influence individuals’ susceptibility over time. The model’s mathematical properties were rigorously investigated, confirming the boundedness and non-negativity of solutions, identifying addiction-free and endemic equilibrium points, and analyzing conditions for backward bifurcation when the effective reproduction number Reff κ is below unity. Local and global stability of these equilibrium points were examined, and an optimal control problem was formulated to determine strategies that effectively reduce gambling prevalence while considering the costs and intensities of interventions. Using Pontryagin’s maximum principle, qualitative analyses were performed, and numerical simulations based on the fourth-order Runge–Kutta method in MATLAB evaluated the dynamic responses of the system under different scenarios. Preventive and treatment strategies were assessed not only for their impact on addiction reduction but also for their cost-effectiveness. The influence of two fractional orders κ=1 (without memory) and κ=0.7 (with memory) on system dynamics was also explored. Results indicate that a dual strategy—combining public awareness campaigns with targeted counseling and treatment—is the most effective and cost-efficient approach for controlling gambling addiction in a community. This study provides novel insights into addiction dynamics and offers a practical framework for designing adaptive, economically viable intervention strategies.