Threshold dynamics and global stability of a SIRS epidemic model with information intervention and vaccination

Xiaoni Li; Qiang Li; Junxian Yang; Dongmei Fan; Hao Zhang; Xiaoni Li; Qiang Li; Junxian Yang; Dongmei Fan; Hao Zhang

doi:10.3934/mmc.2026021

Mathematical Modelling and Control

2026, Volume 6, Issue 3: 278-289. doi: 10.3934/mmc.2026021

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Theory article

Threshold dynamics and global stability of a SIRS epidemic model with information intervention and vaccination

1.
Basic Courses Department, Shaanxi University of International Trade & Commerce, Xi'an 712046, China
2.
School of Artificial Intelligence, Anhui Agricultural University, Hefei 230036, China
3.
Key Laboratory of Measurement and Control of Complex Systems of Engineering, Ministry of Education, Southeast University, Nanjing 210096, China
4.
School of Pharmacy, Shaanxi University of International Trade & Commerce, Xi'an 712046, China

Received: 28 February 2026 Revised: 27 April 2026 Accepted: 23 May 2026 Published: 09 September 2026

This paper investigates the dynamic behavior of a susceptible–infected–recovered–susceptible (SIRS) epidemic model, taking information intervention and vaccination measures into account. We find that, both in the deterministic model and the stochastic model, the basic reproduction number $ \mathcal{R} $ acts as a critical threshold for the disease's persistence and extinction. If $ \mathcal{R} < 1 $, the deterministic system has a disease–free equilibrium which is globally asymptotically stable. Moreover, the disease will eventually disappear with probability one in stochastic system, while if $ \mathcal{R} > 1 $, in deterministic model, the endemic equilibrium is globally asymptotically stable; similarly, within the stochastic system, a stationary distribution implies the disease will be prevailing.
- SIRS epidemic model,
- information intervention,
- extinction,
- persistence,
- threshold
Citation: Xiaoni Li, Qiang Li, Junxian Yang, Dongmei Fan, Hao Zhang. Threshold dynamics and global stability of a SIRS epidemic model with information intervention and vaccination[J]. Mathematical Modelling and Control, 2026, 6(3): 278-289. doi: 10.3934/mmc.2026021

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Abstract

This paper investigates the dynamic behavior of a susceptible–infected–recovered–susceptible (SIRS) epidemic model, taking information intervention and vaccination measures into account. We find that, both in the deterministic model and the stochastic model, the basic reproduction number $ \mathcal{R} $ acts as a critical threshold for the disease's persistence and extinction. If $ \mathcal{R} < 1 $, the deterministic system has a disease–free equilibrium which is globally asymptotically stable. Moreover, the disease will eventually disappear with probability one in stochastic system, while if $ \mathcal{R} > 1 $, in deterministic model, the endemic equilibrium is globally asymptotically stable; similarly, within the stochastic system, a stationary distribution implies the disease will be prevailing.

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