Survival analysis and probability density function of switching heroin model

Hui Jiang; Ling Chen; Fengying Wei; Quanxin Zhu; Hui Jiang; Ling Chen; Fengying Wei; Quanxin Zhu

doi:10.3934/mbe.2023590

Mathematical Biosciences and Engineering

2023, Volume 20, Issue 7: 13222-13249. doi: 10.3934/mbe.2023590

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Research article

Survival analysis and probability density function of switching heroin model

1.
School of Mathematics and Statistics, Fuzhou University, Fuzhou 350116, China
2.
Fujian Key Laboratory of Financial Information Processing, Putian University, Putian 351100, China
3.
Center for Applied Mathematics of Fujian Province, Fuzhou University, Fuzhou 350116, China
4.
School of Mathematics and Statistics, Hunan Normal University, Changsha 410081, China

Academic Editor: Yang Kuang

Received: 14 January 2023 Revised: 15 March 2023 Accepted: 03 April 2023 Published: 08 June 2023

We study a switching heroin epidemic model in this paper, in which the switching of supply of heroin occurs due to the flowering period and fruiting period of opium poppy plants. Precisely, we give three equations to represent the dynamics of the susceptible, the dynamics of the untreated drug addicts and the dynamics of the drug addicts under treatment, respectively, within a local population, and the coefficients of each equation are functions of Markov chains taking values in a finite state space. The first concern is to prove the existence and uniqueness of a global positive solution to the switching model. Then, the survival dynamics including the extinction and persistence of the untreated drug addicts under some moderate conditions are derived. The corresponding numerical simulations reveal that the densities of sample paths depend on regime switching, and larger intensities of the white noises yield earlier times for extinction of the untreated drug addicts. Especially, when the switching model degenerates to the constant model, we show the existence of the positive equilibrium point under moderate conditions, and we give the expression of the probability density function around the positive equilibrium point.
- heroin model,
- stationary distribution,
- extinction,
- regime switching,
- Fokker-Planck equation,
- probability density function
Citation: Hui Jiang, Ling Chen, Fengying Wei, Quanxin Zhu. Survival analysis and probability density function of switching heroin model[J]. Mathematical Biosciences and Engineering, 2023, 20(7): 13222-13249. doi: 10.3934/mbe.2023590

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Abstract

We study a switching heroin epidemic model in this paper, in which the switching of supply of heroin occurs due to the flowering period and fruiting period of opium poppy plants. Precisely, we give three equations to represent the dynamics of the susceptible, the dynamics of the untreated drug addicts and the dynamics of the drug addicts under treatment, respectively, within a local population, and the coefficients of each equation are functions of Markov chains taking values in a finite state space. The first concern is to prove the existence and uniqueness of a global positive solution to the switching model. Then, the survival dynamics including the extinction and persistence of the untreated drug addicts under some moderate conditions are derived. The corresponding numerical simulations reveal that the densities of sample paths depend on regime switching, and larger intensities of the white noises yield earlier times for extinction of the untreated drug addicts. Especially, when the switching model degenerates to the constant model, we show the existence of the positive equilibrium point under moderate conditions, and we give the expression of the probability density function around the positive equilibrium point.

References

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