Abstract
A mathematical model for the dynamic systems of SMA involving the ABC-fractional derivative is considered in this manuscript. We examine the basic reproduction number and analyze the stability of the equilibrium points. We prove the theoretical results of the existence and Ulam's stability of the solutions for the proposed model using fixed point theory and nonlinear analytic techniques. Using the Adams type predictor-corrector rule for the ABC-fractional integral operator, a numerical scheme is devised for obtaining the approximate solution of the proposed model. Different numerical plots corresponding to various fractional orders are presented. In addition, we demonstrate a numerical simulation for the transmission of social media addiction in two cases with the basic reproduction numbers greater than and less than one.