# American Institute of Mathematical Sciences

October  2021, 26(10): 5355-5382. doi: 10.3934/dcdsb.2020347

## Optimal control strategies for an online game addiction model with low and high risk exposure

 School of Science, Guilin University of Technology, Guilin, Guangxi 541004, China

* Corresponding author: Tingting Li

Received  May 2020 Revised  August 2020 Published  October 2021 Early access  November 2020

Fund Project: The second author is supported by the Basic Competence Promotion Project for Young and Middle-aged Teachers in Guangxi, China (2019KY0269)

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.

Citation: Youming Guo, Tingting Li. Optimal control strategies for an online game addiction model with low and high risk exposure. Discrete and Continuous Dynamical Systems - B, 2021, 26 (10) : 5355-5382. doi: 10.3934/dcdsb.2020347
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##### References:
Transfer diagram of model
DFE $D_{0} = (829,0,0,0,0,0)$ is Globally Asymptotically Stable when $R_{0} = 0.5778 < 1$ and $\beta = 0.2$
EE $D^{*} = (358.829, 20.903, 31.354, 67.916, 25.648,324.35)$ is Globally Asymptotically Stable when $R_{0} = 2.3111 > 1$ and $\beta = 0.8$
Dynamical behavior of infected when $R_{0} = 0.5778$ and $\beta = 0.2$
Dynamical behavior of infected when $R_{0} = 2.3111$ and $\beta = 0.8$
Graphical results for strategy A
Graphical results for strategy B
Graphical results for strategy C
Graphical results for strategy D
Graphical results for strategy E
Graphical results for strategy F
Graphical results for strategy G
Graphical results for strategy H
Graphical results for strategy I
Estimation of parameters
 Parameters Descriptions Values $\mu$ Natural supplementary and death rate 0.05 per week $\theta$ Proportion of individuals who became low risk exposed 0.4 per week $\beta$ Contact transmission rate 0.1$\sim$ 0.8 per week $v_{1}$ Proportion of $E_{1}$ who become infected 0.2 per week $v_{2}$ Proportion of $E_{1}$ who become professional 0.2 per week $w_{1}$ Proportion of $E_{2}$ who become infected 0.3 per week $w_{2}$ Proportion of $E_{1}$ who become professional 0.1 per week $k_{1}$ Proportion of $I$ who become quitting 0.05 per week $k_{2}$ Proportion of $I$ who become professional 0.1 per week $\delta$ Proportion of $P$ who become quitting 0.5 per week $u_{1}$ The decreased proportion by isolation Variable $u_{2}$ The decreased proportion in $E_{1}$ by prevention Variable $u_{3}$ The decreased proportion in $E_{2}$ by prevention Variable $u_{4}$ The decreased proportion in $I$ by treatment Variable
 Parameters Descriptions Values $\mu$ Natural supplementary and death rate 0.05 per week $\theta$ Proportion of individuals who became low risk exposed 0.4 per week $\beta$ Contact transmission rate 0.1$\sim$ 0.8 per week $v_{1}$ Proportion of $E_{1}$ who become infected 0.2 per week $v_{2}$ Proportion of $E_{1}$ who become professional 0.2 per week $w_{1}$ Proportion of $E_{2}$ who become infected 0.3 per week $w_{2}$ Proportion of $E_{1}$ who become professional 0.1 per week $k_{1}$ Proportion of $I$ who become quitting 0.05 per week $k_{2}$ Proportion of $I$ who become professional 0.1 per week $\delta$ Proportion of $P$ who become quitting 0.5 per week $u_{1}$ The decreased proportion by isolation Variable $u_{2}$ The decreased proportion in $E_{1}$ by prevention Variable $u_{3}$ The decreased proportion in $E_{2}$ by prevention Variable $u_{4}$ The decreased proportion in $I$ by treatment Variable
Results of different control strategies
 Strategy Total infectious individuals ($\int_{0}^{t_f}(E_{1}+E_{2}+I)dt)$ Averted infectious individuals Objective function $J$ Without control 7461.1302 $-$ $8.5947\times 10^{6}$ Strategy A 526.3468 6934.7835 $1.3646\times 10^{6}$ Strategy B 1426.9073 6034.2229 $2.5242\times 10^{6}$ Strategy C 701.3874 6759.7428 $1.7413\times 10^{6}$ Strategy D 524.2143 6936.9159 $1.3592\times 10^{6}$ Strategy E 525.4126 6935.7176 $1.3619\times 10^{6}$ Strategy F 525.0718 6936.0585 $1.3618\times 10^{6}$ Strategy G 579.8124 6881.3178 $4.784\times 10^{6}$ Strategy H 1626.7971 5834.3331 $2.7511\times 10^{6}$ Strategy I 658.0017 6803.1286 $2.6232\times 10^{6}$
 Strategy Total infectious individuals ($\int_{0}^{t_f}(E_{1}+E_{2}+I)dt)$ Averted infectious individuals Objective function $J$ Without control 7461.1302 $-$ $8.5947\times 10^{6}$ Strategy A 526.3468 6934.7835 $1.3646\times 10^{6}$ Strategy B 1426.9073 6034.2229 $2.5242\times 10^{6}$ Strategy C 701.3874 6759.7428 $1.7413\times 10^{6}$ Strategy D 524.2143 6936.9159 $1.3592\times 10^{6}$ Strategy E 525.4126 6935.7176 $1.3619\times 10^{6}$ Strategy F 525.0718 6936.0585 $1.3618\times 10^{6}$ Strategy G 579.8124 6881.3178 $4.784\times 10^{6}$ Strategy H 1626.7971 5834.3331 $2.7511\times 10^{6}$ Strategy I 658.0017 6803.1286 $2.6232\times 10^{6}$
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