In this paper, we establish a new online game addiction model with low and high risk exposure. With the help of the next generation matrix, the basic reproduction number $ R_{0} $ is obtained. By constructing a suitable Lyapunov function, the equilibria of the model are Globally Asymptotically Stable. We use the optimal control theory to study the optimal solution problem with three kinds of control measures (isolation, education and treatment) and get the expression of optimal control. In the simulation, we first verify the Globally Asymptotical Stability of Disease-Free Equilibrium and Endemic Equilibrium, and obtain that the different trajectories with different initial values converges to the equilibria. Then the simulations of nine control strategies are obtained by forward-backward sweep method, and they are compared with the situation of without control respectively. The results show that we should implement the three kinds of control measures according to the optimal control strategy at the same time, which can effectively reduce the situation of game addiction.

中文翻译：

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具有低风险和高风险暴露的网络游戏成瘾模型的最优控制策略

在本文中，我们建立了一种新的具有低风险和高风险暴露的网络游戏成瘾模型。借助下一代矩阵，得到基本再生数$R_{0}$。通过构造合适的李雅普诺夫函数，该模型的平衡是全局渐近稳定的。我们用最优控制理论研究了三种控制措施（隔离、教育和治疗）的最优解问题，得到了最优控制的表达式。在模拟中，我们首先验证了无病平衡和地方病平衡的全局渐近稳定性，得到不同初始值的不同轨迹收敛到平衡。然后通过前向后向扫描法得到九种控制策略的仿真结果，并分别与无控制情况进行比较。结果表明，应同时按照最优控制策略实施三种控制措施，可有效减少游戏成瘾的情况。