Ergodic stationary distribution and extinction of stochastic pertussis model with immune and Markov switching

Jia Chen; Yuming Chen; Qimin Zhang; Jia Chen; Yuming Chen; Qimin Zhang

doi:10.3934/era.2025121

Electronic Research Archive

2025, Volume 33, Issue 5: 2736-2761. doi: 10.3934/era.2025121

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Ergodic stationary distribution and extinction of stochastic pertussis model with immune and Markov switching

1.
School of Mathematics and Statistics, Ningxia University, Yinchuan 750021, China
2.
Department of Mathematics, Wilfrid Laurier University, Waterloo, ON N2L 3C5, Canada

Received: 05 March 2025 Revised: 22 April 2025 Accepted: 25 April 2025 Published: 06 May 2025

Temperature, humidity, and other environmental factors can influence the spread of diseases. To investigate the impact of environmental perturbations and state changes on pertussis, this study established a random pertussis model with immunity and Markov switching. This stochastic model presented a global positive solution. Subsequently, using Itô's lemma and Lyapunov function, we concluded that the disease will become extinct. Then, a critical value $ \mathcal R_{0}^e $ was introduced. It was established that the stochastic model with Markov switching has an ergodic stationary distribution when $ \mathcal R_{0}^e > 1 $, which implies that this infectious disease will persist and remain prevalent. Some examples are presented to further substantiate our theoretical conclusions.
- stochastic pertussis model,
- Markov switching,
- ergodic stationary distribution
Citation: Jia Chen, Yuming Chen, Qimin Zhang. Ergodic stationary distribution and extinction of stochastic pertussis model with immune and Markov switching[J]. Electronic Research Archive, 2025, 33(5): 2736-2761. doi: 10.3934/era.2025121

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Abstract

Temperature, humidity, and other environmental factors can influence the spread of diseases. To investigate the impact of environmental perturbations and state changes on pertussis, this study established a random pertussis model with immunity and Markov switching. This stochastic model presented a global positive solution. Subsequently, using Itô's lemma and Lyapunov function, we concluded that the disease will become extinct. Then, a critical value $ \mathcal R_{0}^e $ was introduced. It was established that the stochastic model with Markov switching has an ergodic stationary distribution when $ \mathcal R_{0}^e > 1 $, which implies that this infectious disease will persist and remain prevalent. Some examples are presented to further substantiate our theoretical conclusions.

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