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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 (4): 710-726.doi: 10.3969/j.issn.1005-3085.2024.04.008

• • 上一篇    下一篇

基于MSM群体的随机HIV/AIDS传染病模型分析

赵晓琦,  董玲珍   

  1. 太原理工大学数学学院，太原 030024
  • 收稿日期:2021-09-26 接受日期:2021-11-18 出版日期:2024-08-15
  • 通讯作者: 董玲珍 E-mail: linzhen_dong@aliyun.com

Dynamical Analysis of a Stochastic HIV/AIDS Epidemic Model in MSM Community

ZHAO Xiaoqi,  DONG Lingzhen   

  1. College of Mathematics, Taiyuan University of Technology, Taiyuan 030024
  • Received:2021-09-26 Accepted:2021-11-18 Online:2024-08-15
  • Contact: L. Dong. E-mail address: linzhen_dong@aliyun.com

摘要：

艾滋病 (AIDS) 是人类感染 HIV 病毒而导致免疫系统受到破坏的一种危害性极高的传染病，同性恋是传播HIV的重要途经之一。考虑到男男性行为 (MSM) 的传播特征，以及环境中随机因素的影响，一个基于MSM群体的随机HIV/AIDS传染病动力学模型被建立。利用随机微分方程的基本理论，分析了系统的动力学行为。首先，对于任意给定的正初值，证明了系统存在唯一的全局正解。从生物学的角度来看，这是必须成立的，这一结论确保了理论研究的合理性。进一步，鉴于在研究传染病模型的动态变化时，讨论疾病的消失与流行具有重要的应用价值。因此，对所建立的随机HIV/AIDS动力学模型中疾病的灭绝与流行进行了重点分析。特别地，利用伊藤公式，并通过构造一些特殊的函数，给出了疾病灭绝的充分条件；研究了系统唯一正的遍历平稳分布的存在性，即疾病盛行的存在性。最后，通过数值模拟，验证了疾病灭绝和盛行的理论结果，并通过与确定性系统的理论结果相比较，分析了随机扰动对系统动力学行为的影响。

关键词: AIDS模型, 全局正性, It$\hat{\rm o}$'s公式, 灭绝, 遍历平稳分布

Abstract:

AIDS is an extremely harmful infectious disease since HIV virus destroy the immune system of people. Homosexuality is one of the important ways to spread HIV. Considering the transmission characteristics of men who have sex with men (MSM) and the existence of random factors in the environment, a stochastic HIV/AIDS epidemic dynamic model based on MSM is established. Using the basic theory of stochastic differential equation, the dynamic behaviors of the system are analyzed. First, for any given positive initial value, it is proved that there is a unique and global positive solution. From a biological point of view, this conclusion must hold, which ensure it is value of studying such a system. When a virus dynamical model is investigated, the extinction and persistence of virus require the immediate attention. Therefore, by using It$\hat{\rm o}$'s formula and constructing some special functions, the sufficient conditions of disease extinction are given. Moreover, the existence of a unique positive ergodic stationary distribution of the system is discussed, which implies the prevalence of disease. Finally, the obtained theoretical results are verified by numerical simulation, and the influences of random disturbance on the dynamical behavior of the system are analyzed.

Key words: AIDS model, global positivity, It$\hat{\rm o}$'s formula, extinction, ergodic stationary distribution

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