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<p>Social media addiction driven by social influence has emerged as a pervasive issue across all demographics, mimicking the characteristics of contagious behavior. Understanding and mitigating its spread requires advanced mathematical tools. While several models based on ordinary and fractional differential equations with bilinear transmission rates exist, they often fail to account for the non-linear responses of individuals to intervention strategies. Notably, the application of a Beddington-DeAngelis (non-linear) transmission rate in modeling social media addiction remains unexplored. This study presents a fractional-order mathematical model for social media addiction, leveraging the Beddington-DeAngelis transmission rate to capture the complex dynamics of addiction spread and recovery. The population is divided into three groups: non-addicted individuals (<i>N</i>), addicted individuals (<i>A</i>), and rehabilitated individuals (<i>R</i>). The model incorporates Caputo and Caputo-Fabrizio derivatives to account for memory effects and behavioral shifts in social media usage. Competitive and cooperative interactions are represented using Beddington-DeAngelis and Holling II functional response rates. The model identifies two equilibrium states: an addiction-free equilibrium, which is stable when the addiction generation number <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\((A_g)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msub> <mi>A</mi> <mi>g</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is below one, and an addiction-persistence equilibrium, which arises when this number exceeds one. Using real-world data on social media usage from 2015 to 2024, model parameters are calibrated. Sensitivity analysis and transcritical bifurcation studies highlight critical factors influencing addiction spread. Numerical simulations performed using the Adams-Bashforth method provide valuable insights, supporting the development of effective strategies to mitigate social media addiction.</p>

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Social media addiction using non-linear dynamics: insights from Caputo and Caputo-Fabrizio derivatives

  • 期刊论文

  • 发表日期：

    2026年2月25日

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  • Komal Bansal

摘要

Social media addiction driven by social influence has emerged as a pervasive issue across all demographics, mimicking the characteristics of contagious behavior. Understanding and mitigating its spread requires advanced mathematical tools. While several models based on ordinary and fractional differential equations with bilinear transmission rates exist, they often fail to account for the non-linear responses of individuals to intervention strategies. Notably, the application of a Beddington-DeAngelis (non-linear) transmission rate in modeling social media addiction remains unexplored. This study presents a fractional-order mathematical model for social media addiction, leveraging the Beddington-DeAngelis transmission rate to capture the complex dynamics of addiction spread and recovery. The population is divided into three groups: non-addicted individuals (N), addicted individuals (A), and rehabilitated individuals (R). The model incorporates Caputo and Caputo-Fabrizio derivatives to account for memory effects and behavioral shifts in social media usage. Competitive and cooperative interactions are represented using Beddington-DeAngelis and Holling II functional response rates. The model identifies two equilibrium states: an addiction-free equilibrium, which is stable when the addiction generation number \((A_g)\) ( A g ) is below one, and an addiction-persistence equilibrium, which arises when this number exceeds one. Using real-world data on social media usage from 2015 to 2024, model parameters are calibrated. Sensitivity analysis and transcritical bifurcation studies highlight critical factors influencing addiction spread. Numerical simulations performed using the Adams-Bashforth method provide valuable insights, supporting the development of effective strategies to mitigate social media addiction.