Stochastic Stability and Optimal Control Analysis for a Tobacco Smoking Model

Anwarud Din; Peijiang Liu; Ting Cui

doi:doi:10.11648/j.acm.20211006.15

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Stochastic Stability and Optimal Control Analysis for a Tobacco Smoking Model

Anwarud Din, Peijiang Liu, Ting Cui

Published in Applied and Computational Mathematics (Volume 10, Issue 6)

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

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APA Style

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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TY  - JOUR
T1  - Stochastic Stability and Optimal Control Analysis for a Tobacco Smoking Model
AU  - Anwarud Din
AU  - Peijiang Liu
AU  - Ting Cui
Y1  - 2021/12/29
PY  - 2021
N1  - https://doi.org/10.11648/j.acm.20211006.15
DO  - 10.11648/j.acm.20211006.15
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 163
EP  - 185
PB  - Science Publishing Group
SN  - 2328-5613
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

Anwarud Din

Department of Mathematics Sun Yat-sen University, Guangzhou, P. R. China
Peijiang Liu

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

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

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APA Style

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

Copy | Download

@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}
}

Copy | Download

TY  - JOUR
T1  - Stochastic Stability and Optimal Control Analysis for a Tobacco Smoking Model
AU  - Anwarud Din
AU  - Peijiang Liu
AU  - Ting Cui
Y1  - 2021/12/29
PY  - 2021
N1  - https://doi.org/10.11648/j.acm.20211006.15
DO  - 10.11648/j.acm.20211006.15
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 163
EP  - 185
PB  - Science Publishing Group
SN  - 2328-5613
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  -

Copy | Download