一类总人口变化的随机传染病模型的阈值动力学行为
The Threshold Dynamic Behavior of a Stochastic Infectious Disease Model with Varying Total Population Size
DOI: 10.12677/PM.2022.126103, PDF, HTML, 下载: 221  浏览: 332 
作者: 任茹仪：兰州理工大学，理学院，甘肃 兰州
关键词: SIRS模型；疾病灭绝；解的存在唯一性；SIRS Model； Extinction of Disease； Existence and Uniqueness of Solutions
摘要: 为了研究死亡率、传染率和系统的随机扰动对疾病传播的影响，本文主要考虑了一个在变化的人口规模下，疾病的死亡率、传染率和系统都受到干扰的随机传染病模型。通过构造恰当的李雅普诺夫函数证明了解的存在唯一性。建立了决定疾病灭绝的阈值R0s，利用鞅的大数定理和It^o公式得到了传染病灭绝的充分和几乎必要条件。更具体地讲，如果R0s < 1，意味着疾病以指数方式灭绝。最终通过构造的阈值R0s可以发现，易感者的系统扰动会增强疾病的传播；而染病者的传染率、死亡率和系统扰动的随机波动会抑制疾病的传播。
Abstract: To study the effects of mortality, infection rates of random disturbances and random perturbation of the system on the spread of disease, in this paper, we consider a stochastic infectious disease model in which the mortality, transmission and system of the disease are disturbed with changing population size. The existence and uniqueness of knowledge is proved by constructing appropriate Lyapunov functions. Established thresholds R0s for determining disease extinction, applying Martingale’s theorem of large numbers and It^o. formulas, su.cient and almost necessary conditions have been obtained for the extinction of infectious diseases. More speci.cally, if R0s < 1, the disease will die out. Finally, by constructing threshold R0s, we find that the stochastic perturbations of the death rate and random perturbation of the system for susceptible population can enhance the spread of disease, while the stochastic perturbations of the death rate and random perturbation of the system for infectious population, as well as the transmission rate of the disease can suppress the spread of the disease.
文章引用：任茹仪. 一类总人口变化的随机传染病模型的阈值动力学行为[J]. 理论数学, 2022, 12(6): 938-951. https://doi.org/10.12677/PM.2022.126103

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