Dynamics and control for a stochastic giving up smoking model

Xin Yi; Rong Liu; Yanmei Wang; Xin Yi; Rong Liu; Yanmei Wang

doi:10.3934/math.20251164

AIMS Mathematics

2025, Volume 10, Issue 11: 26484-26510. doi: 10.3934/math.20251164

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Dynamics and control for a stochastic giving up smoking model

1.
School of Mathematics and Statistics, Taiyuan Normal University, Jinzhong 030619, China
2.
School of Applied Mathematics, Shanxi University of Finance and Economics, Taiyuan 030006, China

Received: 16 August 2025 Revised: 29 October 2025 Accepted: 10 November 2025 Published: 17 November 2025
MSC : 60H10, 92D25, 92D40, 93E15

This paper focused on a stochastic giving-up-smoking model with harmonic mean-type incidence rate, in which the population was divided into four types. Firstly, we showed that the model has a unique global positive solution. Then, stochastic permanence of the model was discussed, which means that the population described by the model will not grow wildly or disappear. Next, sufficient conditions for the elimination of smokers (including occasional smokers, chain smokers, and quit smokers) were established. Additionally, sufficient conditions for the existence of an ergodic stationary distribution were derived, meaning that all types of smokers can be persistent. Moreover, we discussed how to control the size of the smoker population from the perspective of economics. Finally, some numerical simulations were introduced.
- stochastic smoking model,
- extinction and permanence,
- ergodic stationary distribution,
- smoking control
Citation: Xin Yi, Rong Liu, Yanmei Wang. Dynamics and control for a stochastic giving up smoking model[J]. AIMS Mathematics, 2025, 10(11): 26484-26510. doi: 10.3934/math.20251164

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Abstract

This paper focused on a stochastic giving-up-smoking model with harmonic mean-type incidence rate, in which the population was divided into four types. Firstly, we showed that the model has a unique global positive solution. Then, stochastic permanence of the model was discussed, which means that the population described by the model will not grow wildly or disappear. Next, sufficient conditions for the elimination of smokers (including occasional smokers, chain smokers, and quit smokers) were established. Additionally, sufficient conditions for the existence of an ergodic stationary distribution were derived, meaning that all types of smokers can be persistent. Moreover, we discussed how to control the size of the smoker population from the perspective of economics. Finally, some numerical simulations were introduced.

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