Optimal Control Strategy of an Online Game Addiction Model with Incomplete Recovery

Since the global COVID-19 pandemic in 2020, some people who have dropped out of online game have become re-addicted to it due to the order of stay-at-home, making the phenomenon of online game addiction even worse. Controlling the prevalence of online game addiction is of great significance to protect people’s healthy life. For this purpose, a mathematical model of online game addiction with incomplete recovery and relapse is established. First, we analyze the basic properties of the model and obtain the expression of the basic reproduction number and all equilibria. By constructing suitable Lyapunov functions, the global asymptotical stability of the equilibria are proved. Then in the numerical simulation, we use the least squares estimation method to fit the real data of e-sports users in China from 2010 to 2020, and obtain the estimated value of all parameters. The approximation value of the basic reproduction number is obtained as [Formula: see text] . The result reflects that the spread of game addiction in China is very serious. The stability of the equilibria are proved by using the estimated parameter values. Finally, the simulation results between with control and without control during 2020 to 2050 are compared, and the optimal control strategy is found by comparing the total infectious people. The results of optimal control suggest that if we increase our continuous attention to incompletely recovered people, we can prevent more people from becoming addicted to games. The findings in this paper reveal new mechanisms of game addiction transmission and demonstrate a more detailed and reliable control strategy.