Threshold behaviour of a stochastic SIRS $ \mathrm {L\acute{e}vy} $ jump model with saturated incidence and vaccination

Yu Zhu; Liang Wang; Zhipeng Qiu; Yu Zhu; Liang Wang; Zhipeng Qiu

doi:10.3934/mbe.2023063

Mathematical Biosciences and Engineering

2023, Volume 20, Issue 1: 1402-1419. doi: 10.3934/mbe.2023063

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Threshold behaviour of a stochastic SIRS $ \mathrm {L\acute{e}vy} $ jump model with saturated incidence and vaccination

1.
School of Mathematics and Statistics, Nanjing University of Science and Technology, Nanjing 210094, China
2.
Interdisciplinary Center for Fundamental and Frontier Sciences, Nanjing University of Science and Technology, Jiangyin 214443, China

Academic Editor: Sanling Yuan

Received: 25 August 2022 Revised: 14 October 2022 Accepted: 20 October 2022 Published: 28 October 2022

A stochastic SIRS system with $ \mathrm {L\acute{e}vy} $ process is formulated in this paper, and the model incorporates the saturated incidence and vaccination strategies. Due to the introduction of $ \mathrm {L\acute{e}vy} $ jump, the jump stochastic integral process is a discontinuous martingale. Then the Kunita's inequality is used to estimate the asymptotic pathwise of the solution for the proposed model, instead of Burkholder-Davis-Gundy inequality which is suitable for continuous martingales. The basic reproduction number $ R_{0}^{s} $ of the system is also derived, and the sufficient conditions are provided for the persistence and extinction of SIRS disease. In addition, the numerical simulations are carried out to illustrate the theoretical results. Theoretical and numerical results both show that $ \mathrm {L\acute{e}vy} $ process can suppress the outbreak of the disease.
- stochastic SIRS model,
- $ \mathrm {L\acute{e}vy} $ process,
- Kunita's inequality,
- extinction
Citation: Yu Zhu, Liang Wang, Zhipeng Qiu. Threshold behaviour of a stochastic SIRS $ \mathrm {L\acute{e}vy} $ jump model with saturated incidence and vaccination[J]. Mathematical Biosciences and Engineering, 2023, 20(1): 1402-1419. doi: 10.3934/mbe.2023063

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Abstract

A stochastic SIRS system with $ \mathrm {L\acute{e}vy} $ process is formulated in this paper, and the model incorporates the saturated incidence and vaccination strategies. Due to the introduction of $ \mathrm {L\acute{e}vy} $ jump, the jump stochastic integral process is a discontinuous martingale. Then the Kunita's inequality is used to estimate the asymptotic pathwise of the solution for the proposed model, instead of Burkholder-Davis-Gundy inequality which is suitable for continuous martingales. The basic reproduction number $ R_{0}^{s} $ of the system is also derived, and the sufficient conditions are provided for the persistence and extinction of SIRS disease. In addition, the numerical simulations are carried out to illustrate the theoretical results. Theoretical and numerical results both show that $ \mathrm {L\acute{e}vy} $ process can suppress the outbreak of the disease.

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