Stationary distribution, extinction and probability density function of a stochastic tuberculosis model with Ornstein-Uhlenbeck process

Huimei Liu; Wencai Zhao; Huimei Liu; Wencai Zhao

doi:10.3934/math.2025876

AIMS Mathematics

2025, Volume 10, Issue 8: 19642-19674. doi: 10.3934/math.2025876

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Research article

Stationary distribution, extinction and probability density function of a stochastic tuberculosis model with Ornstein-Uhlenbeck process

Huimei Liu ,
Wencai Zhao ^,

College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China

Received: 25 June 2025 Revised: 14 August 2025 Accepted: 19 August 2025 Published: 27 August 2025
MSC : 34D05, 60H10, 92D30

In order to investigate the impact of environmental disturbances on disease transmission, this paper established a stochastic tuberculosis model, where the infection rate satisfied the Ornstein-Uhlenbeck (OU) process. For the corresponding deterministic model, the endemic equilibrium and its stability were investigated. For the stochastic system under OU noise perturbation, we first established the existence and uniqueness of the global positive solution. Subsequently, properly constructed Lyapunov functions were used to deduce sufficient criteria to ensure the existence of stationary distribution and the eradication of the disease. When $ R_{0}^{s} > 1 $, the system has a stationary distribution, which means that the disease will persist. When $ R_{0}^{E} < 1 $, the disease becomes extinct. Furthermore, an analytical expression for the probability density in the vicinity of the endemic equilibrium was obtained by solving the five-dimensional Fokker-Planck equation. Finally, the accuracy of these theoretical conclusions was corroborated through numerical simulations.
- stochastic tuberculosis model,
- Ornstein-Uhlenbeck process,
- stationary distribution,
- extinction,
- probability density function
Citation: Huimei Liu, Wencai Zhao. Stationary distribution, extinction and probability density function of a stochastic tuberculosis model with Ornstein-Uhlenbeck process[J]. AIMS Mathematics, 2025, 10(8): 19642-19674. doi: 10.3934/math.2025876

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Abstract

In order to investigate the impact of environmental disturbances on disease transmission, this paper established a stochastic tuberculosis model, where the infection rate satisfied the Ornstein-Uhlenbeck (OU) process. For the corresponding deterministic model, the endemic equilibrium and its stability were investigated. For the stochastic system under OU noise perturbation, we first established the existence and uniqueness of the global positive solution. Subsequently, properly constructed Lyapunov functions were used to deduce sufficient criteria to ensure the existence of stationary distribution and the eradication of the disease. When $ R_{0}^{s} > 1 $, the system has a stationary distribution, which means that the disease will persist. When $ R_{0}^{E} < 1 $, the disease becomes extinct. Furthermore, an analytical expression for the probability density in the vicinity of the endemic equilibrium was obtained by solving the five-dimensional Fokker-Planck equation. Finally, the accuracy of these theoretical conclusions was corroborated through numerical simulations.

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