Fractional Derivative Technique for Modeling the Dynamics of Social Media Impacts

The advent of social media (SM) platforms has transformed communications, information dissemination, and interpersonal relationships on a global scale. As SM continues to evolve and proliferate, its impact on various aspects of society has become increasingly complex and multifaceted. For this reason and over the past decades, several controversies have been held about whether SM is good or bad. However, the mathematical modeling technique inculcating SM impacts (positive and negative) has not been studied in the existing works. This article considers a mathematical model approach using the ABC-fractional derivative technique to study the dynamics of SM impacts. We provide the various definitions and the properties needed to study the model. Also, we use the fixed point theorem and a nonlinear analytic approach to demonstrate the theoretical solutions of the existence of solutions for the proposed model. Furthermore, the fundamental reproduction number is computed, and the stability of the model is achieved using the Ulam–Hyers (HU) criteria. We again perform a sensitivity study for the SM impact model and the effects of the sensitive parameters are presented in 3D and contour plots. In addition, a numerical algorithm of the predictor–corrector type of the Adams–Bashforth method for determining the approximate solution of the model is developed and the results are discussed. The effects of the most sensitive parameters on affected individuals in the model with a constant fractional order are shown and discussed. The simulation results indicate that as individuals become aware of the negative impacts of SM, the number of positively impacted individuals rises.