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中国工业与应用数学学会会刊
主管：中华人民共和国教育部
主办：西安交通大学
ISSN 1005-3085  CN 61-1269/O1
工程数学学报

工程数学学报 ›› 2024, Vol. 41 ›› Issue (3): 469-480.doi: 10.3969/j.issn.1005-3085.2024.03.007

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具有Logistic增长和心理作用的随机SIRS传染病模型定性分析

赵彦军1,  苏  丽1,  孙晓辉1,  李文轩2   

  1. 1. 吉林外国语大学国际商学院，长春 130117
    2. 吉林大学数学学院，长春 130012
  • 收稿日期:2021-08-19 接受日期:2022-11-18 出版日期:2024-06-15 发布日期:2024-08-15
  • 基金资助:
    国家自然科学基金 (11271154)；吉林省教育厅科学技术研究项目 (JJKH20231389KJ)；吉林省教育科学“十四五”规划2022年度课题 (GH22708).

Qualitative Analysis of Stochastic SIRS Epidemic Model with Logistic Growth and Psychological Effect

ZHAO Yanjun1,  SU Li1,  SUN Xiaohui1,  LI Wenxuan2   

  1. 1. International Business School, Jilin International Studies University, Changchun 130117
    2. College of Mathematics, Jilin University, Changchun 130012
  • Received:2021-08-19 Accepted:2022-11-18 Online:2024-06-15 Published:2024-08-15
  • Supported by:
    The National Natural Science Foundation of China (11271154); the Science and Technology Research Project of Education Department of Jilin Province (JJKH20231389KJ); the Education Science the Fourteenth Five-Year Plan Project of Jilin Province for 2022 (GH22708).

摘要： 基于流行病受环境噪声和心理作用的影响，建立了一类具有Logistic增长和心理作用的随机SIRS传染病模型，目的在于讨论Logistic增长和心理作用对模型全局动力学的影响。首先，通过构造Lyapunov函数并利用It$\hat{\rm o}$公式，证明了该模型全局正解的存在唯一性；然后，在适当的条件下，利用随机Lyapunov函数方法，应用LaSalle不变性原理得到该模型正解存在遍历平稳分布的充分条件。结果表明：环境和心理变化在一定条件下会对疾病起抑制作用。最后，通过数值模拟验证了理论结果的正确性。

关键词: SIRS传染病模型, Logistic增长, 心理作用, 平稳分布, 遍历性

Abstract: Based on the fact that epidemics are affected by environmental noise and psychological effects, a stochastic SIRS epidemic model with Logistic growth and psychological effects is established to discuss the effects of Logistic growth and psychological effects on the global dynamics of the model. Firstly, by constructing the Lyapunov function and using the It$\hat{\rm o}$ formula, the existence and uniqueness of the global positive solution of the model are proved, and then, under appropriate conditions, by using the random Lyapunov function method, the sufficient conditions for the existence of ergodic stationary distribution of the positive solution of the model are obtained by using the LaSalle invariance principle. The results show that environmental and psychological change can inhibit the disease under certain conditions. Finally, the correctness of the theoretical results is verified by numerical simulation.

Key words: SIRS epidemic model, Logistic growth, psychology effect, stationary distribution, ergodicity

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