Journal of East China Normal University(Natural Science) >

2017 >Issue 4: 71 - 88

DOI: https://doi.org/10.3969/j.issn.1000-5641.2017.04.007

Mathematics

Permanence and extinction of stochastic smoking model

  • ZHANG Xue-kang ,
  • ZHANG Zhen-zhong
Expand
  • Department of Mathematics, Donghua University, Shanghai 201620, China

Received date: 2016-05-18

  Online published: 2017-07-20

Fold

Abstract

To characterize the effects of stochastic noises on smokers, a stochastic smoking model driven by Brownian motion has been proposed. First, a unique global positive solution is proved according to Lyapunov function method. Then, some sufficient and necessary conditions for permanence and extinction are presented. Finally, the parameters of stochastic smoking model are estimated by the pseudo-maximum likelihood estimation.

Key words： Brownian motion; extinction; permanence

Cite this article

ZHANG Xue-kang , ZHANG Zhen-zhong . Permanence and extinction of stochastic smoking model[J]. Journal of East China Normal University(Natural Science), 2017 , (4) : 71 -88 . DOI: 10.3969/j.issn.1000-5641.2017.04.007

References

[1] World Health Organization. Tobacco fact sheet [EB/OL]. [2016-03-01]. http://www.who.int/mediacentre/fact-sheets/fs339/en/.
[2] CASTILLO-GARSOW C, JORDAN-SALIVIA G, RODRIGUEZ-HERRERA A. Mathematical models for the dynamics of tobacco use, recovery and relapse [EB/OL]. [2016-03-01]. https://ecommons.cornell.edu/handle/1813/32095.
[3] SHAROMI O, GUMEL A. Curtailing smoking dyanamics: A mathematical modeling approach [J]. Applied Mathematics and Computation, 2008, 195(2): 475-499.
[4] ZAMAN G. Qualitative behavior of giving up smoking models [J]. Bulletin of the Malaysian Mathematical Sciences Sciences Society, 2011, 34(2): 403-415.
[5] ALKHUDHARI Z, SHEIKH S, TUWAIRQI S. Global dynamics of a mathematical model on smoking [J]. ISRN Applied Mathematics, 2014, Article ID 847075.
[6] GARD T C. Persistence in stochastic food web models [J]. Bulletin of Mathematical Biology, 1984, 46(3): 357-370.
[7] GRAY A, GREENHALGH D, HU L, et al. A stochastic differential equations SIS epidemic model [J]. Siam Journal on Applied Mathematics, 2011, 71(3): 876-902.
[8] BAO J, YUAN C. Stochastic population dynamics driven by Lévy noise [J]. Journal of Mathematical Analysis and Applications, 2012, 391(2): 363-375.
[9] LAHROUZ A, OMARI L, KIOUACH D, et al. Deterministic and stochastic stability of a mathematical model of smoking [J]. Statistics and Probability Letters, 2011, 81(8): 1276-1284.
[10] MAO X. Exponential Stability of Stochastic Differential Equations [M]. New York: Marcel Dekker, 1994.
[11] MAO X. Stochastic Differential Equations and Applications [M]. Chichest: Horwood Publishing Limited, 1999.
[12] LIU H, MA Z. The threshold of survival for system of two species in a polluted environment [J]. Journal of Mathematical Biology, 1991, 30(1): 49-61.
[13] MAO X, YUAN C. Stochastic Differential Equations with Markovian Switching [M]. London: Imperial College Press, 2006: 74.
[14] JIANG D, SHI N, LI X. Global stability and stochastic permanence of a non-autonomous logistic equation with random perturbation [J]. Journal of Mathematical Analysis and Applications, 2008, 340(1): 588-597.
[15] LI X, GRAY A, JING D, et al. Sufficient and necessary conditions of stochastic permanence and exctinction for stochastic logistic populations under regime switching [J]. Journal of Mathematical Analysis and Applications, 2011, 376(1): 11-28.
[16] KARATZAS I, SHREVE S. Brownian Motion and Stochastic Calculus [M]. Second edition. Berlin: Springer, 2007: 293.
[17] ZHANG X, ZHANG Z, TONG J, et al. Ergodicity of stochastic smoking model and parameter estimation [J]. Advances in Difference Equations, 2016, 274: 1-20.
[18] KLEBANER F C. Introduction to Stochastic Calculus with Applications [M]. Second edition. London: Imperial College Press, 2005: 219.
Options
Outlines

/

〈 〉