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Stochastic Stability and Optimal Control Analysis for a Tobacco Smoking Model

Received: 26 November 2021     Accepted: 20 December 2021     Published: 29 December 2021
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Abstract

In this paper, a smoking model, which takes snuffing class and Brownian motion into consideration and is thus an extension of previously studied deterministic smoking models. We analytically show that this extended model system has one and only one positively bounded solution for any nonnegative initial values for the state variables. Interestingly, we find that the model system can exhibit sharp threshold characteristics whatever values of the basic reproductive number. By analyzing persistence, extinction and stationary distribution, we also find that the stochastic system is ergodic only when the coefficients of the noise terms are small. To eliminate gradually the infection out of the community, we introduce a stochastic system of two control variables and perform analysis, with results that can provide guidelines for tobacco control department. Results obtained by theoretical analysis are verified by numerical simulations.

Published in Applied and Computational Mathematics (Volume 10, Issue 6)
DOI 10.11648/j.acm.20211006.15
Page(s) 163-185
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2021. Published by Science Publishing Group

Keywords

Stochastic Tobacco Smoking Model, Itô Formula, Extinction, Persistence, Stationary Distribution, Stochastic Optimal Control, Numerical Simulation

References
[1] Brownlee, J.: Certain considerations on the causation and course of epidemics. Proc. R. Soc. Med. 2, 243-258 (1909).
[2] Brownlee, J.: The mathematical theory of random migrationandepidemicdistribution. Proc. R. Soc. Edinb. 31, 262-289 (1912).
[3] Kermack, W. O., McKendrick, A. G.: Contributions to the mathematical theory of epidemics, part 1. Proc. R. Soc. Edinb., Sect. A, Math., 115, 700-721 (1927).
[4] Chong, J.-R.: Analysis clarifies route of AlDS. Los Angeles Times (2007).
[5] Wang, K., Wang, W., Song, S.: Dynamics of an HBV model with difusion and deley. J. Theor. Biol. 253 (1), 36-44 (2008).
[6] Huo, H. F., Ma, Z. P.: Dynamics of a delayed epidemic model with non-monotonic incidence rate. Commun. Nonlinear Sci. Numer. Simul. 15 (2), 459-468 (2010).
[7] Din, Anwarud, and Yongjin Li. “Lévy noise impact on a stochastic hepatitis B epidemic model under real statistical data and its fractal¨Cfractional Atangana¨CBaleanu order model.” Physica Scripta 96, no. 12 (2021): 124008.
[8] Xu, R., Ma, Z.: Global stability of a SIR epidemic model with nonlinear incidence rate and time delay. Nonlinear Anal., Real World Appl. 10 (5), 3175-3189 (2009).
[9] Xu, R., Ma, Z.: Stability of a delayed SIRS epidemic model with a nonlinear incidence rate. Chaos Solitons Fractals 41 (5), 2319-2325 (2009).
[10] Song, X., Cheng, S.: A delay-differential equation model of HIV infection of CD4+ T-cells. J. Korean Math. Soc. 42 (5), 1071-1086 (2005).
[11] Takeuchi, Y., Ma, W., Beretta, E.: Global asymptotic properties of a delay SIR epidemic model with finite incubation times. Nonlinear Anal., Theory Methods Appl. 42 (6), 931-947 (2010).
[12] Van den Driessche, P., Watmough, J.: Further notes on the basic reproduction number. In: Mathematical Epidemiology. Lecture Notes in Mathematics, vol. 1945, pp. 159-178. Springer, Berlin (2008).
[13] d’Onofrio, A., Manfredi, P., Salinelli, E.: Bifurcation thresholds in an SIR model with information-dependent vaccination. Math. Model. Nat. Phenom. 2 (1), 26-43 (2007).
[14] Yi, N., Zhao, Z., Zhang, Q.: Bifurcations of an SEIQS epidemic model. Int. J. Inf. Syst. Sci. 5 (3 − 4), 296-310 (2009).
[15] Smith, Elizabeth A., and Ruth E. Malone. “‘Everywhere the soldier will be’: wartime tobacco promotion in the US military.” American journal of public health 99, no. 9 (2009): 1595-1602.
[16] Alkhudhari, Zainab, Sarah Al-Sheikh, and Salma Al- Tuwairqi. “Global dynamics of a mathematical model on smoking.” International Scholarly Research Notices 2014 (2014).
[17] Erturk, Vedat Suat, Gul Zaman, and Shaher Momani. “A numeric¨Canalytic method for approximating a giving up smoking model containing fractional derivatives.” Computers and Mathematics with Applications 64, no. 10 (2012): 3065-3074.
[18] Zeb, Anwar, Gul Zaman, and Shaher Momani. “Square- root dynamics of a giving up smoking model.” Applied Mathematical Modelling 37, no. 7 (2013): 5326-5334.
[19] Ain QT, Anjum N, Din A, Zeb A, Djilali S, Khan ZA. On the Analysis of Caputo Fractional Order Dynamics of Middle East Lungs Coronavirus (MERS-CoV) Model. Alexandria Engineering Journal. 2021 Oct 16.
[20] Din, Anwarud, Yongjin Li, Tahir Khan, and Gul Zaman. “Mathematical analysis of spread and control of the novel corona virus (COVID-19) in China.” Chaos, Solitons and Fractals 141 (2020): 110286.
[21] Atangana, Abdon. “Mathematical model of survival of fractional calculus, critics and their impact: How singular is our world?” (2021).
[22] Din, Anwarud, and Yongjin Li. “Controlling heroin addiction via age-structured modeling.” Advances in Difference Equations 2020, no. 1 (2020): 1-17.
[23] Wang, Xun-Yang, Khalid Hattaf, Hai-Feng Huo, and Hong Xiang. “Stability analysis of a delayed social epidemics model with general contact rate and its optimal control.” Journal of Industrial & Management Optimization 12, no. 4 (2016): 1267.
[24] Nana-Kyere, Sacrifice, et al. “Hepatitis B optimal control model with vertical transmission.” Appl. Math 7.1 (2017): 5-13.
[25] Din, Anwarud, Yongjin Li, and Qi Liu. “Viral dynamics and control of hepatitis B virus (HBV) using an epidemic model.” Alexandria Engineering Journal 59, no. 2 (2020): 667-679.
[26] Jianjun, J. I. A. O., C. H. E. N. Lansun, and C. A. I. Shaohong. “An SEIRS epidemic model with two delays and pulse vaccination.” Journal of Systems Science and Complexity 21, no. 2 (2008): 217-225.
[27] Din, Anwarud, Yongjin Li “The Complex Dynamics of Hepatitis B Infected Individuals with Optimal Control.” J Syst Sci Complex (2020) 33: 1¨C23.
[28] Osemwinyen, Amenaghawon C., and Aboubakary Diakhaby. “Mathematical modelling of the transmission dynamics of ebola virus.” Applied and Computational Mathematics 4, no. 4 (2015): 313-320.
[29] Din, Anwarud, Yongjin Li, and Abdullahi Yusuf. “Delayed hepatitis B epidemic model with stochastic analysis.” Chaos, Solitons & Fractals 146 (2021): 110839.
[30] Zhao, Yanan, and Daqing Jiang. “The threshold of a stochastic SIS epidemic model with vaccination.” Applied Mathematics and Computation 243 (2014): 718- 727.
[31] Din, Anwarud, Tahir Khan, Yongjin Li, Hassan Tahir, Asaf Khan, and Wajahat Ali Khan. “Mathematical analysis of dengue stochastic epidemic model.” Results in Physics 20 (2021): 103719.
[32] Yongjin Li. “Stochastic optimal control for norovirus transmission dynamics by contaminated food and water.” Chinese Physics B (2021).
[33] Zhang, X. B., Wang, X. D., & Huo, H. F. (2019). ExtinctionandstationarydistributionofastochasticSIRS epidemic model with standard incidence rate and partial immunity. Physica A: Statistical Mechanics and its Applications, 531, 121548.
[34] Agarwal, Ravi P., Qaisar Badshah, Ghaus ur Rahman, and Saeed Islam. “Optimal control and dynamical aspects of a stochastic pine wilt disease model.” Journal of the Franklin Institute 356, no. 7 (2019): 3991-4025.
[35] El Fatini, M., I. Sekkak, R. Taki, and T. El Guendouz. “A control treatment for a stochastic epidemic model with relapse and Crowly¨CMartin incidence.” The Journal of Analysis (2020): 1-17.
[36] Witbooi, Peter J., Grant E. Muller, and Garth J. Van Schalkwyk. “Vaccination control in a stochastic SVIR epidemic model.” Computational and mathematical methods in medicine 2015 (2015).
[37] Din, Anwarud, and Yongjin Li. “Stationary distribution extinction and optimal control for the stochastic hepatitis B epidemic model with partial immunity.” Physica Scripta 96, no. 7 (2021): 074005.
[38] Dalal, Nirav, David Greenhalgh, and Xuerong Mao. “A stochastic model for internal HIV dynamics.” Journal of Mathematical Analysis and Applications 341, no. 2 (2008): 1084-1101.
[39] Din, Anwarud, et al. “Stochastic dynamics of hepatitis B epidemics.” Results in Physics 19 (2020).
[40] Lei, Q., and Z. Yang. 2016. Dynamical behaviours of a stochastic SIRI epidemic model. Applicable Analysis 96: 1¨C13.
[41] Alzahrani, Ebraheem, and Anwar Zeb. “Stability analysis and prevention strategies of tobacco smoking model.” Boundary Value Problems 2020, no. 1 (2020): 1-13.
[42] Sitas, Freddy, Ben Harris-Roxas, Debbie Bradshaw, and Alan D. Lopez. “Smoking and epidemics of respiratory infections.” Bulletin of the World Health Organization 99, no. 2 (2021): 164.
[43] Khasminskii, Rafail. Stochastic stability of differential equations. Vol. 66. Springer Science & Business Media, 2011.
[44] Bao, Kangbo, and Qimin Zhang. “Stationary distribution and extinction of a stochastic SIRS epidemic model with information intervention.” Advances in Difference Equations 2017, no. 1 (2017): 1-19.
[45] Liu, Meng, and Chuanzhi Bai. “Optimal harvesting of a stochastic delay competitive model.” Discrete and Continuous Dynamical Systems-B 22.4 (2017): 1493.
[46] Liu, Lidan, and Xinzhu Meng. “Optimal harvesting control and dynamics of two-species stochastic model with delays.” Advances in Difference Equations 2017, no. 1 (2017): 1-17.
[47] W. H. Fleming, R. W. Rishel, Deterministic and Stochastic Optimal Control, Springer Verlag, New York, 1975.
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    Anwarud Din, Peijiang Liu, Ting Cui. (2021). Stochastic Stability and Optimal Control Analysis for a Tobacco Smoking Model. Applied and Computational Mathematics, 10(6), 163-185. https://doi.org/10.11648/j.acm.20211006.15

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    ACS Style

    Anwarud Din; Peijiang Liu; Ting Cui. Stochastic Stability and Optimal Control Analysis for a Tobacco Smoking Model. Appl. Comput. Math. 2021, 10(6), 163-185. doi: 10.11648/j.acm.20211006.15

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    AMA Style

    Anwarud Din, Peijiang Liu, Ting Cui. Stochastic Stability and Optimal Control Analysis for a Tobacco Smoking Model. Appl Comput Math. 2021;10(6):163-185. doi: 10.11648/j.acm.20211006.15

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  • @article{10.11648/j.acm.20211006.15,
      author = {Anwarud Din and Peijiang Liu and Ting Cui},
      title = {Stochastic Stability and Optimal Control Analysis for a Tobacco Smoking Model},
      journal = {Applied and Computational Mathematics},
      volume = {10},
      number = {6},
      pages = {163-185},
      doi = {10.11648/j.acm.20211006.15},
      url = {https://doi.org/10.11648/j.acm.20211006.15},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20211006.15},
      abstract = {In this paper, a smoking model, which takes snuffing class and Brownian motion into consideration and is thus an extension of previously studied deterministic smoking models. We analytically show that this extended model system has one and only one positively bounded solution for any nonnegative initial values for the state variables. Interestingly, we find that the model system can exhibit sharp threshold characteristics whatever values of the basic reproductive number. By analyzing persistence, extinction and stationary distribution, we also find that the stochastic system is ergodic only when the coefficients of the noise terms are small. To eliminate gradually the infection out of the community, we introduce a stochastic system of two control variables and perform analysis, with results that can provide guidelines for tobacco control department. Results obtained by theoretical analysis are verified by numerical simulations.},
     year = {2021}
    }
    

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    T1  - Stochastic Stability and Optimal Control Analysis for a Tobacco Smoking Model
    AU  - Anwarud Din
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    T2  - Applied and Computational Mathematics
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    UR  - https://doi.org/10.11648/j.acm.20211006.15
    AB  - In this paper, a smoking model, which takes snuffing class and Brownian motion into consideration and is thus an extension of previously studied deterministic smoking models. We analytically show that this extended model system has one and only one positively bounded solution for any nonnegative initial values for the state variables. Interestingly, we find that the model system can exhibit sharp threshold characteristics whatever values of the basic reproductive number. By analyzing persistence, extinction and stationary distribution, we also find that the stochastic system is ergodic only when the coefficients of the noise terms are small. To eliminate gradually the infection out of the community, we introduce a stochastic system of two control variables and perform analysis, with results that can provide guidelines for tobacco control department. Results obtained by theoretical analysis are verified by numerical simulations.
    VL  - 10
    IS  - 6
    ER  - 

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Author Information
  • Department of Mathematics Sun Yat-sen University, Guangzhou, P. R. China

  • School of Statistics and Mathematics, Guangdong University of Finance and Economics, Big Data and Educational Statistics Application Laboratory Guangzhou, P. R. China

  • School of Economics, Guangdong University of Finance and Economics, Guangzhou, People’s Republic of China

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