Mathematical Model of Social Media Addiction: An Optimal Control Approach

Abstract

In this study, we developed a deterministic mathematical model to analyze social media addiction, incorporating an optimal control strategy. The basic model captures the dynamics through which individuals become exposed to and eventually addicted to social media platforms. To enhance the model, we introduced two time-dependent control variables: one representing awareness campaigns through advertising and education, and the other representing treatment interventions for individuals suffering from addiction. An optimal control framework was then formulated based on these interventions. By applying Pontryagin’s Minimum Principle, we derived the necessary conditions for optimality and constructed the corresponding optimality system. Numerical simulations of the optimal control problem were conducted using the forward-backward sweep method to assess the effectiveness of the proposed strategies. The results demonstrate that the integrated control strategy—combining public awareness efforts with treatment interventions— substantially reduces the number of individuals exposed to and addicted to social media. Compared to scenarios without intervention, the number of affected individuals was significantly lower. These findings underscore the importance of implementing combined strategies rather than isolated measures. Therefore, this integrated approach is strongly recommended for policymakers and stakeholders as a practical and effective means to mitigate the adverse effects of social media addiction on public health and societal well-being..

Keywords: Social Media Addiction, Mathematical Model, Pontryagin Minimum Principle, Optimal Control.