Conditions for extinction and ergodicity of a stochastic Mycobacterium tuberculosis model with Markov switching

Ying He; Bo Bi; Ying He; Bo Bi

doi:10.3934/math.20241482

AIMS Mathematics

2024, Volume 9, Issue 11: 30686-30709. doi: 10.3934/math.20241482

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Research article

Conditions for extinction and ergodicity of a stochastic Mycobacterium tuberculosis model with Markov switching

Ying He ¹,
Bo Bi ^{2
,
,}

1.
School of Mathematics and Statistics, Northeast Petroleum University, Daqing163318, China
2.
School of Public Health, Hainan Medical University and Hainan Academy of Medical Sciences, Haikou 571199, China

Received: 16 July 2024 Revised: 07 October 2024 Accepted: 21 October 2024 Published: 29 October 2024
MSC : 37H05, 37H30, 60H10

This paper is concerned with a stochastic Mycobacterium tuberculosis model, which is perturbed by both white noise and colored noise. First, we prove that the stochastic model has a unique global positive solution. Second, we derive an important condition $ R_0^* $ depending on environmental noise for this stochastic model. We construct an appropriate Lyapunov function, and show that the model possesses a unique ergodic stationary distribution when $ R_0^* < 0 $, in other words, it indicates the long-term persistence of the disease. Finally, we investigate the related conditions of extinction.
- Mycobacterium tuberculosis,
- Markov switching,
- stationary distribution,
- extinction
Citation: Ying He, Bo Bi. Conditions for extinction and ergodicity of a stochastic Mycobacterium tuberculosis model with Markov switching[J]. AIMS Mathematics, 2024, 9(11): 30686-30709. doi: 10.3934/math.20241482

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Abstract

This paper is concerned with a stochastic Mycobacterium tuberculosis model, which is perturbed by both white noise and colored noise. First, we prove that the stochastic model has a unique global positive solution. Second, we derive an important condition $ R_0^* $ depending on environmental noise for this stochastic model. We construct an appropriate Lyapunov function, and show that the model possesses a unique ergodic stationary distribution when $ R_0^* < 0 $, in other words, it indicates the long-term persistence of the disease. Finally, we investigate the related conditions of extinction.

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