On analysis of a nonlinear fractional system for social media addiction involving Atangana–Baleanu–Caputo derivative

Abstract

A mathematical model for the dynamic systems of $\mathbb{SMA}$ SMA involving the $\mathbb{ABC}$ 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 $\mathbb{ABC}$ 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.

A mathematical study on a fractional COVID-19 transmission model within the framework of nonsingular and nonlocal kernel

“…Following similar lines of argument as in [31] , [18] , we sum up all equations of the fractional model (3.4) to obtain An application of the Laplace transform yields where , denotes the total initial population and is the two-parameter Mittag-Leffler function [15] defined by By invoking the following property for the two-parameter Mittag-Leffler function [15] the inequality in (3.7) simplifies to Clearly, as due to the asymptotic behaviour of the Mittag-Leffler function [15] . Thus, all solutions of the fractional model (3.4) with the non-negative initial conditions in will remain in .…”

“…The idea of incorporating fractional order derivatives in the mathematical modeling of infectious diseases is not anything new (see, for instance [17] , [18] , [19] , [20] , [21] and the references therein). Within the past nineteen months, there have been extensive studies on COVID-19 from different mathematical perspectives.…”

Mathematical modelling, analysis and numerical simulation of social media addiction and depression

“…where λ Individuals that leave recovered class 0.4 [42] τ Natural death rate 0.05-0.25 [42,45] α The rate at which depression is brought on by media impact 0.3-0.5 Approximate β Susceptibles who avoid and/or stop using social media 0.01 Assume χ Contact rate of susceptibles with addicted population 0.25 [42] υ Depressed individuals that move to recovered individuals through the treatment 0.7 [40] B Individuals that leave exposed class 0.25 [40] ω Probability that treatment is successful 0.8 [43] ψ…”

A simple method for studying asymptotic stability of discrete dynamical systems and its applications

“…Many important processes and phenomena in real-world situations can be mathematically modeled by autonomous dynamical systems described by differential equations associated with the classical and fractional derivative operators [1][2][3][4][5][6][7][8]. While differential equation models with the classical derivatives have been formed and studied for a long time [1,3,5,6,8], mathematical models based on fractional differential equations have been strongly developed in recent years (see, for example, [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28]). The stability analysis of differential equation models has been a central and prominent problem with many useful applications.…”