Fractional-order modeling of social media addiction dynamics with initial control and deep learning prediction

Social media addiction has emerged as a major public health risk due to its strong association with anxiety, depression, sleep disorders, and reduced quality of life. To better understand these interconnected psychological conditions, we propose a novel fractional order model that incorporates susceptible individuals, exposed users, mild addiction, severe addiction, anxiety, depression, treatment, recovery, digital detox, and family support compartments. The model further integrates awareness campaigns, psychological counseling, digital detox interventions, and AI-based screen-time management as optimal control strategies. The theoretical analysis verifies the boundedness, positivity, existence, uniqueness, equilibrium points, and stability of the proposed system. The basic reproduction number is calculated through the next generation matrix approach, and statistical sensitivity identifies the highest influential variables governing addiction dynamics. The fractional model is solved numerically using the Liouville-Caputo Fractional Taylor Method (LCFTM), and the numerical solutions are validated through a Deep Neural Network model.