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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 |
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.
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Optimal control strategies for dengue transmission in pakistan, Math. Biosci., 305 (2018), 102-121.
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Optimal control of a delayed breast cancer stem cells nonlinear model, Optimal Control Appl. Methods, 37 (2016), 248-258.
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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.
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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.
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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. |
[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. |
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G. Fan, H. R. Thieme and H. Zhu,
Delay differential systems for tick population dynamics, J. Math. Biol., 71 (2015), 1017-1048.
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W. H. Fleming and R. W. Rishel, Deterministic and Stochastic Optimal Control, Springer-Verlag, New York, 1975. |
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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. |
[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. |
[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. |
[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. |
[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.
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Stability and optimal control in a mathematical model of online game addiction, Filomat, 33 (2019), 5691-5711.
|
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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. |
[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. |
[22] |
D. L. Lukes, Differential Equations: Classical to Controlled, Matheatics in Science and Engineering, Academia Press, New York, 1982.
![]() |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
show all references
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[9] |
W. H. Fleming and R. W. Rishel, Deterministic and Stochastic Optimal Control, Springer-Verlag, New York, 1975. |
[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. |
[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. |
[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. |
[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. |
[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. |
[17] |
Y. Kuang, Delay Differential Equations with Application in Population Dynamics, Academic Press, Inc., Boston, MA, 1993.
![]() |
[18] |
V. Lakshmikantham, S. Leela and A. A. Martynyuk, Stability Analysis of Nonlinear Systems, Marcel Dekker, Inc., New York, 1989. |
[19] |
T. Li and Y. Guo,
Stability and optimal control in a mathematical model of online game addiction, Filomat, 33 (2019), 5691-5711.
|
[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. |
[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. |
[22] |
D. L. Lukes, Differential Equations: Classical to Controlled, Matheatics in Science and Engineering, Academia Press, New York, 1982.
![]() |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |












Parameters | Descriptions | Values |
Natural supplementary and death rate | 0.05 per week | |
Proportion of individuals who became low risk exposed | 0.4 per week | |
Contact transmission rate | 0.1 |
|
|
Proportion of |
0.2 per week |
|
Proportion of |
0.2 per week |
|
Proportion of |
0.3 per week |
|
Proportion of |
0.1 per week |
|
Proportion of |
0.05 per week |
|
Proportion of |
0.1 per week |
|
Proportion of |
0.5 per week |
The decreased proportion by isolation | Variable | |
|
The decreased proportion in |
Variable |
|
The decreased proportion in |
Variable |
|
The decreased proportion in |
Variable |
Parameters | Descriptions | Values |
Natural supplementary and death rate | 0.05 per week | |
Proportion of individuals who became low risk exposed | 0.4 per week | |
Contact transmission rate | 0.1 |
|
|
Proportion of |
0.2 per week |
|
Proportion of |
0.2 per week |
|
Proportion of |
0.3 per week |
|
Proportion of |
0.1 per week |
|
Proportion of |
0.05 per week |
|
Proportion of |
0.1 per week |
|
Proportion of |
0.5 per week |
The decreased proportion by isolation | Variable | |
|
The decreased proportion in |
Variable |
|
The decreased proportion in |
Variable |
|
The decreased proportion in |
Variable |
Strategy | Total infectious individuals ( |
Averted infectious individuals | Objective function |
Without control | 7461.1302 | ||
Strategy A | 526.3468 | 6934.7835 | |
Strategy B | 1426.9073 | 6034.2229 | |
Strategy C | 701.3874 | 6759.7428 | |
Strategy D | 524.2143 | 6936.9159 | |
Strategy E | 525.4126 | 6935.7176 | |
Strategy F | 525.0718 | 6936.0585 | |
Strategy G | 579.8124 | 6881.3178 | |
Strategy H | 1626.7971 | 5834.3331 | |
Strategy I | 658.0017 | 6803.1286 |
Strategy | Total infectious individuals ( |
Averted infectious individuals | Objective function |
Without control | 7461.1302 | ||
Strategy A | 526.3468 | 6934.7835 | |
Strategy B | 1426.9073 | 6034.2229 | |
Strategy C | 701.3874 | 6759.7428 | |
Strategy D | 524.2143 | 6936.9159 | |
Strategy E | 525.4126 | 6935.7176 | |
Strategy F | 525.0718 | 6936.0585 | |
Strategy G | 579.8124 | 6881.3178 | |
Strategy H | 1626.7971 | 5834.3331 | |
Strategy I | 658.0017 | 6803.1286 |
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