Mathematical Model Dynamics of Cyber Accounts for Vices, Recovery and Relapse

Abstract

In this study, we develop a mathematical model through a system of first-order nonlinear ordinary differential equations. This model covers the dynamics between vulnerable cyber accounts and those implicated in cyber vices such as bullying, scams, spreading of misinformation, and the creation of harmful digital footprints. It further explores the mechanisms of recovery and relapse among these accounts. Through some mathematical analysis, we apply relevant theorems to affirm the model’s fundamental properties, which includes its existence, uniqueness, positivity, and boundedness. We also determine the model’s cyber vice-free and endemic equilibrium states, analyzing their local and global asymptotic stability based on when the basic reproduction number $R_{cb}$ is greater or less than one. Simulation exercises are conducted to substantiate our theoretical findings and demonstrate the model’s behavior in relation to $R_{cb}.$ The simulation outcomes reveal an escalating trend in cyber vices, showing the necessity for targeted interventions that promote a more secure online environment for users and the broader cyber space.