Fractional order modelling of alcoholism: optimal control analysis and simulations with deep-neural network approach

Classical integer-order models provide useful insights into alcoholism dynamics but cannot adequately capture the memory and hereditary effects associated with addiction, relapse, and recovery. In this work, the Caputo fractional derivative is used to study an alcoholism model describing susceptible to alcohol consumption ([Formula: see text]), moderate drinkers ([Formula: see text]), heavy drinkers ([Formula: see text]), affluent heavy drinkers receiving care in private treatment facilities ([Formula: see text]), disadvantaged heavy drinkers attending public treatment centers ([Formula: see text]), and those who have stopped drinking ([Formula: see text]). Using the concepts of the fixed point theory and nonlinear analysis, the existence, uniqueness, and stability of solutions of the considered fractional alcoholism model are rigorously proved. Using the Newton interpolation polynomial, the numerical analysis and simulations of the fractional alcoholism model are carried out to visualize the behavior of the considered system and to examine the effects of various crucial parameters on its dynamics. Optimal control analysis for the fractional order model is demonstrated to show its dynamics with and without control. A deep neural network surrogate is also developed to approximate the numerical trajectories generated by the fractional-order model.