# 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 & Continuous Dynamical Systems - B, 2021, 26 (10) : 5355-5382. doi: 10.3934/dcdsb.2020347
##### References:
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He, 29 (2017), 673-682.  doi: 10.1177/1010539517739558.  Google Scholar [31] The 43rd Statistical Report on Internet Development in China, 2019. Available from: http://www.cac.gov.cn. Google Scholar [32] X. Tian, R. Xu and J. Lin, Mathematical analysis of a cholera infection model with vaccination strategy, Appl. Math. Comput., 361 (2019), 517-535.  doi: 10.1016/j.amc.2019.05.055.  Google Scholar [33] S. Ullah, M. A. Khan and J. F. Gómez-Aguilar, Mathematical formulation of hepatitis B virus with optimal control analysis, Optim. Contr. Appl. Met., 40 (2019), 529-544.  doi: 10.1002/oca.2493.  Google Scholar [34] R. Viriyapong and M. Sookpiam, Education campaign and family understanding affect stability and qualitative behavior ofan online game addiction model for children and youth in Thailand, Math. Method App. Sci., 42 (2019), 6906-6916.  doi: 10.1002/mma.5796.  Google Scholar [35] X. Wang, M. Shen, Y. Xiao and L. Rong, Optimal control and cost-effectiveness analysis of a Zika virus infection model with comprehensive interventions, Appl. Math. Comput., 359 (2019), 165-185.  doi: 10.1016/j.amc.2019.04.026.  Google Scholar [36] X. Wang, Y. Shi, D. Wang and C. Xu, Dynamic Analysis on a Kind of Mathematical Model Incorporating Online Game Addiction Model and Age-Structure, Journal of Beijing University of Civil Engineering and Architecture, 2 (2017), 54-58.   Google Scholar [37] World Health Statistics 2019, 2019. Available from: https://www.who.int/data/gho/publications/world-health-statistics. Google Scholar [38] T. A. Yıldız and E. Karaoǧlu, Optimal control strategies for tuberculosis dynamics with exogenous reinfections in case of treatment at home and treatment in hospital, Nonlinear Dynam., 97 (2019), 2643-2659.   Google Scholar [39] Z.-K. Zhang, C. Liu, X.-X. Zhan, X. Lu, C.-X. Zhang and Y.-C. Zhang, Dynamics of information diffusion and its applications on complex networks, Phys. Rep., 651 (2016), 1-34.  doi: 10.1016/j.physrep.2016.07.002.  Google Scholar [40] W. Zhou, Y. Xiao and J. M. Heffernan, Optimal media reporting intensity on mitigating spread of an emerging infectious disease, Plos. One, 3 (2019), E0213898. doi: 10.1371/journal.pone.0213898.  Google Scholar

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##### References:
 [1] F. B. Agusto and M. A. Khan, Optimal control strategies for dengue transmission in pakistan, Math. Biosci., 305 (2018), 102-121.  doi: 10.1016/j.mbs.2018.09.007.  Google Scholar [2] J. O. Akanni, F. O. Akinpelu, S. Olaniyi, A. T. Oladipo and A. W. Ogunsola, Modelling financial crime population dynamics: Optimal control and cost-effectiveness analysis, Int. J. Dyn. Control, 8 (2020), 531-544.  doi: 10.1007/s40435-019-00572-3.  Google Scholar [3] A. Barrea and M. E. Hernández, Optimal control of a delayed breast cancer stem cells nonlinear model, Optimal Control Appl. Methods, 37 (2016), 248-258.  doi: 10.1002/oca.2164.  Google Scholar [4] E. Bonyah, M. A. Khan, K. O. Okosun and J. F. Gómez-Aguilar, Modelling the effects of heavy alcohol consumption on the transmission dynamics of gonorrhea with optimal control, Math. Biosci., 309 (2019), 1-11.  doi: 10.1016/j.mbs.2018.12.015.  Google Scholar [5] D. K. Das, S. Khajanchi and T. K. Kar, The impact of the media awareness and optimal strategy on the prevalence of tuberculosis, Appl. Math. Comput., 366 (2020), 124732, 23 pp. doi: 10.1016/j.amc.2019.124732.  Google Scholar [6] C. Ding, Y. Sun and Y. Zhu, A schistosomiasis compartment model with incubation and its optimal control, Math. Methods Appl. Sci., 40 (2017), 5079-5094.  doi: 10.1002/mma.4372.  Google Scholar [7] P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48.  doi: 10.1016/S0025-5564(02)00108-6.  Google Scholar [8] G. Fan, H. R. Thieme and H. Zhu, Delay differential systems for tick population dynamics, J. Math. Biol., 71 (2015), 1017-1048.  doi: 10.1007/s00285-014-0845-0.  Google Scholar [9] W. H. Fleming and R. W. Rishel, Deterministic and Stochastic Optimal Control, Springer-Verlag, New York, 1975.  Google Scholar [10] D. Gao and N. Huang, Optimal control analysis of a tuberculosis model, Appl. Math. Model., 58 (2018), 47-64.  doi: 10.1016/j.apm.2017.12.027.  Google Scholar [11] Y. Guo and T. Li, Optimal control and stability analysis of an online game addiction model with two stages, Math. Method App. Sci., 43 (2020), 4391-4408.   Google Scholar [12] K. Hattaf, Optimal control of a delayed HIV infection model with immune response using an efficient numerical method, ISRN Biomathematics, (2012), Article ID215124. Google Scholar [13] J. M. Heffernan, R. J. Smith and L. M. Wahl, Perspectives on the basic reproductive ratio, J. R. Soc. Interface, 2 (2005), 281-293.  doi: 10.1098/rsif.2005.0042.  Google Scholar [14] H.-F. Huo, F.-F. Cui and H. Xiang, Dynamics of an SAITS alcoholism model on unweighted and weighted networks, Physica A, 496 (2018), 249-262.  doi: 10.1016/j.physa.2018.01.003.  Google Scholar [15] H.-F. Huo and X.-M. Zhang, Complex dynamics in an alcoholism model with the impact of Twitter, Math. Biosci., 281 (2016), 24-35.  doi: 10.1016/j.mbs.2016.08.009.  Google Scholar [16] M. A. Khan, S. W. Shah, S. Ullah and J. F. Gómez-Aguilar, A dynamical model of asymptomatic carrier zika virus with optimal control strategies, Nonlinear Anal. Real World Appl., 50 (2019), 144-170.  doi: 10.1016/j.nonrwa.2019.04.006.  Google Scholar [17] Y. Kuang, Delay Differential Equations with Application in Population Dynamics, Academic Press, Inc., Boston, MA, 1993.   Google Scholar [18] V. Lakshmikantham, S. Leela and A. A. Martynyuk, Stability Analysis of Nonlinear Systems, Marcel Dekker, Inc., New York, 1989.  Google Scholar [19] T. Li and Y. Guo, Stability and optimal control in a mathematical model of online game addiction, Filomat, 33 (2019), 5691-5711.   Google Scholar [20] Z. Lin and H. Zhu, Spatial spreading model and dynamics of West Nile virus in birds and mosquitoes with free boundary, J. Math. Biol., 75 (2017), 1381-1409.  doi: 10.1007/s00285-017-1124-7.  Google Scholar [21] Z. Lu, From E-Heroin to E-sports: The development of competitive gaming in China, The International Journal of the History of Sport, 33 (2017), 2186-2206.  doi: 10.1080/09523367.2017.1358167.  Google Scholar [22] D. L. Lukes, Differential Equations: Classical to Controlled, Matheatics in Science and Engineering, Academia Press, New York, 1982.   Google Scholar [23] M. McAsey, L. Mou and W. Han, Convergence of the forward-backward sweep method in optimal control, Comput. Optim. Appl., 53 (2012), 207-226.  doi: 10.1007/s10589-011-9454-7.  Google Scholar [24] K. O. Okosun, M. A. Khan, E. Bonyah and O. O. Okosun, Cholera-schistosomiasis coinfection dynamics, Optim. Contr. Appl. Met., 40 (2019), 703-727.  doi: 10.1002/oca.2507.  Google Scholar [25] K. A. Pawelek, A. Oeldorf-Hirsch and L. Rong, Modeling the impact of Twitter on influenza epidemics, Math. Biosci. Eng., 11 (2014), 1337-1356.  doi: 10.3934/mbe.2014.11.1337.  Google Scholar [26] M. Sana, R. Saleem, A. Manaf and M. Habib, Varying forward backward sweep method using Runge-Kutta, Euler and Trapezoidal scheme as applied to optimal control problems, Sci.Int.(Labore), 27 (2015), 839-843.   Google Scholar [27] O. Sharomi and A. B. Gumel, Curtailing smoking dynamics: A mathematical modeling approach, Appl. Math. Comput., 195 (2008), 475-499.  doi: 10.1016/j.amc.2007.05.012.  Google Scholar [28] Statistical Classification of Sports Industry, 2019. Available from: http://www.stats.gov.cn/tjgz/tzgb/201904/t20190409_1658556.html. Google Scholar [29] X. Sun, H. Nishiura and Y. Xiao, Modeling methods for estimating HIV incidence: A mathematical review, Theor. Biol. Med. Model, 17 (2020), 1-14.  doi: 10.1186/s12976-019-0118-0.  Google Scholar [30] C. S. Tang, Y. W. Koh and Y. Q. Gan, Addiction to internet use, online gaming, and online social networking among young adults in China, Singapore, and the United States, Asia Pac. J. Public. He, 29 (2017), 673-682.  doi: 10.1177/1010539517739558.  Google Scholar [31] The 43rd Statistical Report on Internet Development in China, 2019. Available from: http://www.cac.gov.cn. Google Scholar [32] X. Tian, R. Xu and J. Lin, Mathematical analysis of a cholera infection model with vaccination strategy, Appl. Math. Comput., 361 (2019), 517-535.  doi: 10.1016/j.amc.2019.05.055.  Google Scholar [33] S. Ullah, M. A. Khan and J. F. Gómez-Aguilar, Mathematical formulation of hepatitis B virus with optimal control analysis, Optim. Contr. Appl. Met., 40 (2019), 529-544.  doi: 10.1002/oca.2493.  Google Scholar [34] R. Viriyapong and M. Sookpiam, Education campaign and family understanding affect stability and qualitative behavior ofan online game addiction model for children and youth in Thailand, Math. Method App. Sci., 42 (2019), 6906-6916.  doi: 10.1002/mma.5796.  Google Scholar [35] X. Wang, M. Shen, Y. Xiao and L. Rong, Optimal control and cost-effectiveness analysis of a Zika virus infection model with comprehensive interventions, Appl. Math. Comput., 359 (2019), 165-185.  doi: 10.1016/j.amc.2019.04.026.  Google Scholar [36] X. Wang, Y. Shi, D. Wang and C. Xu, Dynamic Analysis on a Kind of Mathematical Model Incorporating Online Game Addiction Model and Age-Structure, Journal of Beijing University of Civil Engineering and Architecture, 2 (2017), 54-58.   Google Scholar [37] World Health Statistics 2019, 2019. Available from: https://www.who.int/data/gho/publications/world-health-statistics. Google Scholar [38] T. A. Yıldız and E. Karaoǧlu, Optimal control strategies for tuberculosis dynamics with exogenous reinfections in case of treatment at home and treatment in hospital, Nonlinear Dynam., 97 (2019), 2643-2659.   Google Scholar [39] Z.-K. Zhang, C. Liu, X.-X. Zhan, X. Lu, C.-X. Zhang and Y.-C. Zhang, Dynamics of information diffusion and its applications on complex networks, Phys. Rep., 651 (2016), 1-34.  doi: 10.1016/j.physrep.2016.07.002.  Google Scholar [40] W. Zhou, Y. Xiao and J. M. Heffernan, Optimal media reporting intensity on mitigating spread of an emerging infectious disease, Plos. One, 3 (2019), E0213898. doi: 10.1371/journal.pone.0213898.  Google Scholar
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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