Research article

A delayed synthetic drug transmission model with two stages of addiction and Holling Type-II functional response

  • Received: 24 July 2020 Accepted: 22 September 2020 Published: 28 September 2020
  • MSC : 34C23

  • This paper gropes the stability and Hopf bifurcation of a delayed synthetic drug transmission model with two stages of addiction and Holling Type-II functional response. The critical point at which a Hopf bifurcation occurs can be figured out by using the escalating time delay of psychologically addicts as a bifurcation parameter. Directly afterwards, properties of the Hopf bifurcation are explored with aid of the central manifold theorem and normal form theory. Specially, global stability of the model is proved by constructing a suitable Lyapunov function. To underline effectiveness of the obtained results and analyze influence of some influential parameters on dynamics of the model, some numerical simulations are ultimately addressed.

    Citation: Yougang Wang, Anwar Zeb, Ranjit Kumar Upadhyay, A Pratap. A delayed synthetic drug transmission model with two stages of addiction and Holling Type-II functional response[J]. AIMS Mathematics, 2021, 6(1): 1-22. doi: 10.3934/math.2021001

    Related Papers:

  • This paper gropes the stability and Hopf bifurcation of a delayed synthetic drug transmission model with two stages of addiction and Holling Type-II functional response. The critical point at which a Hopf bifurcation occurs can be figured out by using the escalating time delay of psychologically addicts as a bifurcation parameter. Directly afterwards, properties of the Hopf bifurcation are explored with aid of the central manifold theorem and normal form theory. Specially, global stability of the model is proved by constructing a suitable Lyapunov function. To underline effectiveness of the obtained results and analyze influence of some influential parameters on dynamics of the model, some numerical simulations are ultimately addressed.


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    [1] 2019 World Drug Report, UNODC. Available from: http://www.hqrw.com.cn/2019/0627/87860.shtml.
    [2] 2017 China Drug Report, 2018. Available from: http://www.nncc626.com/2018-06/25/c_129900461.htm.
    [3] 2018 China Drug Report, 2019. Available from: http://www.nncc626.com/2019-06/17/c_1210161797.htm.
    [4] J. L. Liu, T. L. Zhang, Global behaviour of a heroin epidemic model with distributed delays, Appl. Math. Lett., 24 (2011), 1685-1692. doi: 10.1016/j.aml.2011.04.019
    [5] M. J. Ma, S. Y. Liu, J. Li, Does media coverage influence the spread of drug addiction? Commun. Nonlinear Sci., 50 (2017), 169-179.
    [6] J. L. Manthey, A. Y. Aidoo, K. Y. Ward, Campus drinking: an epidemiological model, J. Biol. Dyn., 2 (2008), 346-356. doi: 10.1080/17513750801911169
    [7] H. F. Huo, N. N. Song, Global stability for a binge drinking model with two stages, Discrete Dyn. Nat. Soc., 2012 (2012), 1-15.
    [8] H. Xiang, N. N. Song, H. F. Huo, Modelling effects of public health educational campaigns on drinking dynamics, J. Bio. Dyn., 10 (2016), 164-178. doi: 10.1080/17513758.2015.1115562
    [9] B. Khajji, A. Labzai, O. Balatif, M. Rachik, Mathematical modeling and analysis of an alcohol drinking model with the influence of alcohol treatment centers, Int. J. Math. Math. Sci., 2020 (2020), 1-12.
    [10] C. C. Garsow, G. J. Salivia, A. R. Herrera, Mathematical models for dynamics of tobacco use, recovery and relapse, Technical 390 Report Series BU-1505-M, Cornell University, Ithaca, NY, 2000.
    [11] G. Rahman, R. P. Agarwal, L. L. Liu, A. Khan, Threshold dynamics and optimal control of an age-structured giving up smoking model, Nonlinear Anal-Real, 43 (2018), 96-120. doi: 10.1016/j.nonrwa.2018.02.006
    [12] X. K. Zhang, Z. Z. Zhang, J. Y. Tong, M. Dong, Ergodicity of stochastic smoking model and parameter estimation, Adv. Differ. Equ., 274 (2016), 1-20.
    [13] G. U. Rahman, R. P. Agarwal, Q. Din, Mathematical analysis of giving up smoking model via harmonic mean type incidence rate, Appl. Math. Comput., 354 (2019), 128-148.
    [14] S. Ucar, E. Ucar, N. Ozdemir, Z. Hammouch, Mathematical analysis and numerical simulation for a smoking model with Atangana-Baleanu derivative, Chaos, Solitons Fractals, 118 (2019), 300-306. doi: 10.1016/j.chaos.2018.12.003
    [15] Z. Z. Zhang, R. B. Wei, W. J. Xia, Dynamical analysis of a giving up smoking model with time delay, Adv. Differ. Equ., 88 (2017), 1-16.
    [16] J. L. Wang, J. Wang, T. Kuniya, Analysis of an age-structured multi-group heroin epidemic model, Appl. Math. Comput., 347 (2019), 78-100.
    [17] I. M. Wangari, L. Stone, Analysis of a heroin epidemic model with saturated treatment function, J. Appl. Math., 2017 (2017), 1-21.
    [18] S. T. Liu, L. Zhang, X. B. Zhang, A. B. Li, Dynamics of a stochastic heroin epidemic model with bilinear incidence and varying population size, Int. J. Biomath., 12 (2019), 1-21.
    [19] Y. C. Wei, Q. G. Yang, G. J. Li, Dynamics of the stochastically perturbed Heroin epidemic model under non-degenerate noises, Physica A, 526 (2019), 120914. doi: 10.1016/j.physa.2019.04.150
    [20] F. Nyabadza, J. B. H. Njagarah, R. J. Smith, Modelling the dynamics of crystal meth(tik) abuse in the presence of drug-supply chain in South Africa, B. Math. Biol., 75 (2013), 24-28. doi: 10.1007/s11538-012-9790-5
    [21] P. Y. Liu, L. Zhang, Y. F. Xing, Modeling and stability of a synthetic drugs transimission model with relapse and treatment, J. Appl. Math. Comput., 60 (2019), 465-484. doi: 10.1007/s12190-018-01223-0
    [22] M. J. Ma, S. Y. Liu, H. Xiang, J. Liu, Dynamics of synthetic drugs transmission model with psychological addicts and general incidence rate, Physica A, 491 (2018), 641-649. doi: 10.1016/j.physa.2017.08.128
    [23] S. Saha, G. P. Samanta, Synthetic drugs transmission:stability analysis and optimal control, Letters in Biomath, 6 (2019), 1-31. doi: 10.30707/LiB6.1Banuelos
    [24] Y. X. Guo, N. N. Ji, B. Niu, Hopf bifurcation analysis in a predator-prey model with time delay and food subsidies, Adv. Differ. Equ., 99 (2019), 1-22.
    [25] S. Kundu, S. Maitra, Dynamics of a delayed predator-prey system with stage structure and cooperation for preys, Chaos, Solitons Fractals, 114 (2018), 453-460. doi: 10.1016/j.chaos.2018.07.013
    [26] Y. Z. Bai, Y. Y. Li, Stability and Hopf bifurcation for a stage-structured predator-prey model incorporating refuge for prey and additional food for predator, Adv. Differ. Equ., 42 (2019), 1-20.
    [27] C. D. Huang, H. L, J. D. Cao, A novel strategy of bifurcation control for a delayed fractional predator-prey model, Appl. Math. Comput., 347 (2019), 808-838.
    [28] K. Zheng, X. L. Zhou, Z. H. Wu, Z. M. Wang, T. J. Zhou, Hopf bifurcation controlling for a fractional order delayed paddy ecosystem in the fallow season, Adv. Differ. Equ., 309 (2019), 1-14.
    [29] H. Miao, C. J. Kang, Stability and Hopf bifurcation analysis for an HIV infection model with Beddington-DeAngelis incidence and two delays, J. Appl. Math. Comput., 60 (2019), 265-290. doi: 10.1007/s12190-018-1213-9
    [30] Z. Z. Zhang, S. Kundu, J. P. Tripathi, S. Bugalia, Stability and Hopf bifurcation analysis of an SVEIR epidemic model with vaccination and multiple time delays, Chaos Solitons Fractals, 131 (2020), 1-17.
    [31] Z. Z. Zhang, S. Kumari, R. K. Upadhyay, A delayed e-epidemic SLBS model for computer virus, Adv. Differ. Equ., 414 (2019), 1-24.
    [32] Z. Z. Zhang, T. Zhao, Bifurcation analysis of an e-SEIARS model with multiple delays for pointto-group worm propagation, Adv. Differ. Equ., 228 (2019), 1-26.
    [33] X. L. Li, J. J. Wei, On the zeros of a fourth degree exponential polynomial with applications to a neural network model with delays, Chaos, Solitons Fractals, 26 (2005), 519-526. doi: 10.1016/j.chaos.2005.01.019
    [34] B. D. Hassard, N. D. Kazarinoff, Y. H. Wan, Theory and Applications of Hopf Bifurcation, Cambridge: Cambridge University Press, 1981.
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