Research article

Global stability of a novel nonlinear diffusion online game addiction model with unsustainable control

  • Received: 19 June 2022 Revised: 31 August 2022 Accepted: 12 September 2022 Published: 26 September 2022
  • MSC : 35K57, 35B40, 34K09, 92D25

  • In this paper, we build a novel nonlinear diffusion online game addiction model with unsustainable control. The existence and boundedness of a solution are investigated by a $ C_0 $-semigroup and differential inclusion. Simultaneously, we study the global asymptotic stability of steady states of the model. Finally, a concrete example is theoretically analyzed and numerically simulated.

    Citation: Kaihong Zhao. Global stability of a novel nonlinear diffusion online game addiction model with unsustainable control[J]. AIMS Mathematics, 2022, 7(12): 20752-20766. doi: 10.3934/math.20221137

    Related Papers:

  • In this paper, we build a novel nonlinear diffusion online game addiction model with unsustainable control. The existence and boundedness of a solution are investigated by a $ C_0 $-semigroup and differential inclusion. Simultaneously, we study the global asymptotic stability of steady states of the model. Finally, a concrete example is theoretically analyzed and numerically simulated.



    加载中


    [1] World Health Organization, The 11th revision of the International Classification of Diseases (ICD-11), Geneva, 2018.
    [2] W. Feng, D. E. Ramo, S. R. Chan, J. A. Bourgeois, Internet gaming disorder: Trends in prevalence 1998–2016, Addict. Behav., 75 (2017), 17–24. https://doi.org/10.1016/j.addbeh.2017.06.010 doi: 10.1016/j.addbeh.2017.06.010
    [3] Diagnostic and statistical manual of mental disorders (DSM-5), 5 Eds., Washington: American Psychiatric Association, 2013. https://doi.org/10.1176/appi.books.9780890425596
    [4] F. W. Paulus, S. Ohmann, A. von Gontard, C. Popow, Internet gaming disorder in children and adolescents: a systematic review, Dev. Med. Child Neurol., 60 (2018), 645–659. https://doi.org/10.1111/dmcn.13754 doi: 10.1111/dmcn.13754
    [5] Y. M. Guo, T. T. Li, Optimal control and stability analysis of an online game addiction model with two stages, Math. Methods Appl. Sci., 43 (2020), 4391–4408. https://doi.org/10.1002/mma.6200 doi: 10.1002/mma.6200
    [6] T. T. Li, Y. M. Guo, Stability and optimal control in a mathematical model of online game addiction, Filomat, 33 (2019), 5691–5711. https://doi.org/10.2298/FIL1917691L doi: 10.2298/FIL1917691L
    [7] R. Viriyapong, M. Sookpiam, Education campaign and family understanding affect stability and qualitative behavior of an online game addiction model for children and youth in Thailand, Math. Methods Appl. Sci., 42 (2019), 6906–6916. https://doi.org/10.1002/mma.5796 doi: 10.1002/mma.5796
    [8] H. Seno, A mathematical model of population dynamics for the internet gaming addiction, Nonlinear Anal. Model. Control, 26 (2021), 861–883. https://doi.org/10.15388/namc.2021.26.24177 doi: 10.15388/namc.2021.26.24177
    [9] S. Djilali, S. Bentout, Pattern formations of a delayed diffusive predator-prey model with predator harvesting and prey social behavior, Math. Methods Appl. Sci., 44 (2021), 9128–9142. https://doi.org/10.1002/mma.7340 doi: 10.1002/mma.7340
    [10] S. Djilali, S. Bentout, Global dynamics of SVIR epidemic model with distributed delay and imperfect vaccine, Results Phys., 25 (2021), 104245. https://doi.org/10.1016/j.rinp.2021.104245 doi: 10.1016/j.rinp.2021.104245
    [11] Z. A. Khan, A. L. Alaoui, A. Zeb, M. Tilioua, S. Djilali, Global dynamics of a SEI epidemic model with immigration and generalized nonlinear incidence functional, Results Phys., 27 (2021), 104477. https://doi.org/10.1016/j.rinp.2021.104477 doi: 10.1016/j.rinp.2021.104477
    [12] K. H. Zhao, Local exponential stability of four almost-periodic positive solutions for a classic Ayala-Gilpin competitive ecosystem provided with varying-lags and control terms, Int. J. Control, 2022. https://doi.org/10.1080/00207179.2022.2078425 doi: 10.1080/00207179.2022.2078425
    [13] K. H. Zhao, Local exponential stability of several almost periodic positive solutions for a classical controlled GA-predation ecosystem possessed distributed delays, Appl. Math. Comput., 437 (2023), 127540. https://doi.org/10.1016/j.amc.2022.127540 doi: 10.1016/j.amc.2022.127540
    [14] W. J. Li, L. H. Huang, J. C. Ji, Globally exponentially stable periodic solution in a general delayed predator-prey model under discontinuous prey control strategy, Discrete Contin. Dyn. Syst. B, 25 (2020), 2639–2664. https://doi.org/10.3934/dcdsb.2020026 doi: 10.3934/dcdsb.2020026
    [15] Z. Y. Guo, L. H. Huang, X. F. Zou, Impact of discontinuous treatments on disease dynamics in an SIR epidemic model, Math. Biosci. Eng., 9 (2012), 97–110. https://doi.org/10.3934/mbe.2012.9.97 doi: 10.3934/mbe.2012.9.97
    [16] X. B. Zhang, H. Y. Zhao, Global stability of a diffusive predator-prey model with discontinuous harvesting policy, Appl. Math. Lett., 109 (2020), 106539. https://doi.org/10.1016/j.aml.2020.106539 doi: 10.1016/j.aml.2020.106539
    [17] W. M. Ni, M. X. Tang, Turing patterns in the Lengyel-Epstein system for the CIMA reaction, Trans. Amer. Math. Soc., 357 (2005), 3953–3969. https://doi.org/10.1090/S0002-9947-05-04010-9 doi: 10.1090/S0002-9947-05-04010-9
    [18] O. Arino, S. Gauthier, J. P. Penot, A fixed point theorem for sequentially continuous mapping with applications to ordinary differential equations, Funkc. Ekvacioj, 27 (1984), 273–279.
    [19] I. I. Vrabie, Compactness methods for nonlinear evolutions, 1 Ed., London: Longman Scientific and Technical, 1987.
    [20] J. Simsen, C. B. Gentile, On $p$-Laplacian differential inclusions–Global existence, compactness properties and asymptotic behavior, Nonlinear Anal., 71 (2009), 3488–3500. https://doi.org/10.1016/j.na.2009.02.044 doi: 10.1016/j.na.2009.02.044
    [21] J. I. Díaz, I. I. Vrabie, Existence for reaction diffusion systems: a compactness method approach, J. Math. Anal. Appl., 88 (1994), 521–540. https://doi.org/10.1006/jmaa.1994.1443 doi: 10.1006/jmaa.1994.1443
    [22] H. Brezis, Operateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, Amsterdam: North-Holland Publishing Company, 1973.
    [23] I. I. Vrabie, Compactness methods for nonlinear evolutions, 2 Eds., London: Longman Scientific and Technical, 1995.
    [24] S. L. Hollis, R. H. Martin, M. Pierre, Global existence and boundedness in reaction-diffusion systems, SIAM J. Math. Anal., 18 (1987), 744–761. https://doi.org/10.1137/0518057 doi: 10.1137/0518057
    [25] K. J. Brown, P. C. Dunne, R. A. Gardner, A semilinear parabolic system arising in the theory of superconductivity, J. Differ. Equ., 40 (1981), 232–252.
    [26] K. H. Zhao, Stability of a nonlinear ML-nonsingular kernel fractional Langevin system with distributed lags and integral control, Axioms, 11 (2022), 350. https://doi.org/10.3390/axioms11070350 doi: 10.3390/axioms11070350
    [27] K. H. Zhao, Existence, stability and simulation of a class of nonlinear fractional Langevin equations involving nonsingular Mittag-Leffler kernel, Fractal Fract., 6 (2022), 469. https://doi.org/10.3390/fractalfract6090469 doi: 10.3390/fractalfract6090469
    [28] H. Huang, K. H. Zhao, X. D. Liu, On solvability of BVP for a coupled Hadamard fractional systems involving fractional derivative impulses, AIMS Math., 7 (2022), 19221–19236. https://doi.org/10.3934/math.20221055 doi: 10.3934/math.20221055
  • Reader Comments
  • © 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(933) PDF downloads(51) Cited by(8)

Article outline

Figures and Tables

Figures(12)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog