Modeling and analysis of a stochastic giving-up-smoking model with quit smoking duration

Yajuan Guo; Zijian Liu; Yuanshun Tan; Yawei Liu; Yajuan Guo; Zijian Liu; Yuanshun Tan; Yawei Liu

doi:10.3934/mbe.2023910

Mathematical Biosciences and Engineering

2023, Volume 20, Issue 12: 20576-20598. doi: 10.3934/mbe.2023910

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Modeling and analysis of a stochastic giving-up-smoking model with quit smoking duration

College of Mathematics and Statistics, Chongqing Jiaotong University, Chongqing 400074, China

Academic Editor: Carlo Cattani

Received: 31 August 2023 Revised: 03 November 2023 Accepted: 08 November 2023 Published: 14 November 2023

Smoking has gradually become a very common behavior, and the related situation in different groups also presents different forms. Due to the differences of individual smoking cessation time and the interference of environmental factors in the spread of smoking behavior, we establish a stochastic giving up smoking model with quit-smoking duration. We also consider the saturated incidence rate. The total population is composed of potential smokers, smokers, quitters and removed. By using Itô's formula and constructing appropriate Lyapunov functions, we first ensure the existence of a unique global positive solution of the stochastic model. In addition, a threshold condition for extinction and permanence of smoking behavior is deduced. If the intensity of white noise is small, and $ \widetilde{\mathcal{R}}_0 < 1 $, smokers will eventually become extinct. If $ \widetilde{\mathcal{R}}_0 > 1 $, smoking will last. Then, the sufficient condition for the existence of a unique stationary distribution of the smoking phenomenon is studied as $ R_0^s > 1 $. Finally, conclusions are explained by numerical simulations.
- stochastic giving-up-smoking model,
- saturated incidence rate,
- extinction,
- persistence,
- stationary distribution
Citation: Yajuan Guo, Zijian Liu, Yuanshun Tan, Yawei Liu. Modeling and analysis of a stochastic giving-up-smoking model with quit smoking duration[J]. Mathematical Biosciences and Engineering, 2023, 20(12): 20576-20598. doi: 10.3934/mbe.2023910

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Abstract

Smoking has gradually become a very common behavior, and the related situation in different groups also presents different forms. Due to the differences of individual smoking cessation time and the interference of environmental factors in the spread of smoking behavior, we establish a stochastic giving up smoking model with quit-smoking duration. We also consider the saturated incidence rate. The total population is composed of potential smokers, smokers, quitters and removed. By using Itô's formula and constructing appropriate Lyapunov functions, we first ensure the existence of a unique global positive solution of the stochastic model. In addition, a threshold condition for extinction and permanence of smoking behavior is deduced. If the intensity of white noise is small, and $ \widetilde{\mathcal{R}}_0 < 1 $, smokers will eventually become extinct. If $ \widetilde{\mathcal{R}}_0 > 1 $, smoking will last. Then, the sufficient condition for the existence of a unique stationary distribution of the smoking phenomenon is studied as $ R_0^s > 1 $. Finally, conclusions are explained by numerical simulations.

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