Mathematical modeling and optimal control of social media addiction with stability and sensitivity analysis

This research formulates a precise mathematical model for social media addiction and integrates an optimal control mechanism. We analyzed key equilibrium points, including the equilibrium of dependence-free (E0) and dependence-endemic (E∗), and computed the basic reproduction number (R0).The addiction-free equilibrium is locally and globally asymptotically stable for R0<1, as shown by stability analysis and the Castillo-Chavez theorem. The endemic equilibrium retains local asymptotic stability when R0>1. The application of the Center Manifold theorem confirms the occurrence of a forward bifurcation in the model at R0=1. The normalized forward sensitivity index was used to do sensitivity analysis. We also evaluated the model’s Lyapunov exponents, which confirm global asymptotic stability, and highlighted the critical role of R0 in determining the addiction dynamics. In addition, we presented awareness and punishment as controls that change throughout time to enhance the model and developed a problem for optimum control.The optimal system was determined by using the maximum principle of Pontryagin. Numerical simulations using the finite difference method evaluated the efficiency of the suggested strategies. The results indicate that effectively controlling Social Media Addiction requires an integrated approach by stakeholders and policymakers.