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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 (1): 145-163.doi: 10.3969/j.issn.1005-3085.2024.01.009

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具有Logistic增长和Crowley-Martin型发生率的随机SIRS双流行病模型的动力学研究

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:2022-01-17 接受日期:2022-12-29 出版日期:2024-02-15 发布日期:2024-04-15
  • 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).

Dynamical Analysis of Stochastic SIRS Double Epidemic Model with Logistic Growth and Crowley-Martin Incidence Rate

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:2022-01-17 Accepted:2022-12-29 Online:2024-02-15 Published:2024-04-15
  • 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).

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摘要：

基于现实生活中多种疾病共存并且受环境噪声影响，建立了一个具有Logistic增长和Crowley-Martin型发生率的随机SIRS双流行传染病模型，目的在于讨论Logistic增长、Crowley-Martin型发生率和双流行传染病对模型全局动力学的影响。分析展示了模型的全局动力学由随机基本再生数决定。具体地，首先通过构造Lyapunov函数并利用Ito公式，证明了全局正解的存在唯一性；其次结合引入的随机基本再生数和构造的Lyapunov函数，应用LaSalle不变性原理得到决定疾病灭绝和持久的充分条件。结果表明：环境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: Logistic增长, Crowley-Martin型发生率, 双流行病, 灭绝, 持久

Abstract:

Based on the fact that many diseases coexist in real life and are affected by environmental noise, a stochastic SIRS double epidemic model with Logistic growth and Crowley-Martin type incidence is established to discuss the effects of Logistic growth, Crowley-Martin type incidence and double epidemic infectious diseases on the global dynamics of the model. It is obtained that the global dynamics of the model is determined by stochastic basic reproduction number. Firstly, by constructing the Lyapunov function and using the Ito's formula, the existence and uniqueness of the global positive solution are proved, and then, by combining the stochastic basic reproduction number and the constructed Lyapunov function, the sufficient conditions for determining the extinction and persistence of the disease are obtained by using the LaSalle invariance principle. The results show that the environmental change can inhibit the disease under certain conditions. Finally, the correctness of the theoretical results is verified by numerical simulation.

Key words: Logistic growth, Crowley-Martin incidence rate, double epidemic, extinction, permanence

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