AMathematical Model Analysis on the Dynamics of Online GameAddiction with Optimal Control

Abstract

Excessive playing of online games leads to addiction, which causes academic under achievement and prevents the daily achievement of goals. Symptoms of addiction are spend ing more time gaming online and irritability, while risk factors are low self-worth, anxiety and depression. This thesis study focuses on the mathematical model analysis of the dynam ics of online game addiction with optimal control. We demonstrated the existence, positivity and boundedness of the solution of the dynamical system. The system has two equilibrium points, namely the online game addiction free and endemic equilibrium points, and their stability was determined using the linearization technique, the Castillo-Chavrz theorem and LaSalles’s invariant principle. The basic reproduction number (R0) of the model was cal culated using the principle of the next generation matrix and the sensitivity indices of R0 to the model parameters were investigated. Additionally, bifurcation analysis was performed to verify the backward and forward bifurcations, and from the analysis, we observed that the model system exhibits forward bifurcation at R0 = 1. The optimal control problem of the model was analyzed using Pontryagin’s Maximum Principle and the characterization of the optimal control was constructed. Numerical simulations were performed to verify the accuracy of the analytical results using MATLAB software. From the numerical simula tion of sensitivity analysis, decreasing the contact rate with addicted(β1) and incompletely recovered(β2) individuals, the addiction rate(δ), the re-addiction rate(ω) and the incomplete recovery rate of treated individuals(τ) decreases the reproduction number. Using optimal control, the combined strategy of minimizing the contact rate with addicted individuals and the re-addiction rate of incompletely recovered individuals minimizes the exposed and ad dicted individuals as well as the associated cost