A fractional social media addiction model: graphical interpretation and numerical analysis

Abstract

This paper investigates the complex dynamics of Social Media Addiction (SMA) through the angle of fractional calculus. We employ a five-compartment deterministic model as Susceptible (S), Exposed (E), Addicted (A), Recovered (R), and Quit (Q) which map the transmission pathways of this behavioral addiction. The model is formulated using the Caputo fractional derivative to effectively capture the memory and hereditary properties inherent in human behavioral patterns, which integer-order models often overlook. We conduct a comprehensive qualitative analysis, establishing the existence, uniqueness, non-negativity, and boundedness of the solutions. The stability of the model’s equilibria as addiction-free and endemic states is rigorously examined through local and global stability analyses based on the basic reproduction number (R0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$R_{0}$\end{document}). To approximate the solution of the fractional system, we implement the robust Adams-Bashforth-Moulton predictor-corrector algorithm. Numerical simulations and a detailed sensitivity analysis are presented to illustrate the theoretical findings, demonstrating the influence of key parameters and the fractional order α on the system’s dynamics. The results confirm that fractional-order models offer a more nuanced and realistic framework for understanding the propagation of social media addiction. The main findings show that the fractional-order model provides slower convergence and reduced addiction peaks compared to the classical integer-order case, highlighting the significant role of memory effects. The analysis confirms that the basic reproduction number governs the persistence of addiction, while sensitivity results identify recovery and quitting rates as the most effective control parameters. These results demonstrate the importance of incorporating memory in accurately describing and controlling social media addiction dynamics.