论文标题
具有隐藏状态的随机流行病模型
Stochastic epidemic SIR models with hidden states
论文作者
论文摘要
本文重点介绍并分析了考虑随机性的现实SIR模型。所提出的系统适用于文献中使用的大多数发病率，包括双线性发病率，Beddington-Deangelis发病率和II型功能响应。鉴于许多疾病会导致无症状的感染，因此我们研究了一个随机微分方程的系统，该系统还包括一类隐藏状态个体，为此，感染状态未知。我们假设未给出对隐藏的状态个体的直接观察，即$α（t）$，仅给出$α（t）$，并且只有噪声浪费的观察过程。本文使用非线性过滤技术与入侵类型分析（或使用Lyapunov指数分析），本文证明了该疾病的长期行为由阈值$λ\ in \ Mathbb {R} $控制，取决于模型参数。事实证明，如果$λ<0 $，被感染的个人的数字$ i（t）$会收敛到零快速的零，或者灭绝发生。相反，如果$λ> 0 $，则感染是地方性的，系统是永久性的。我们通过将结果应用于特定的照明示例中来展示我们的结果。还提供了数值模拟来说明我们的结果。
This paper focuses on and analyzes realistic SIR models that take stochasticity into account. The proposed systems are applicable to most incidence rates that are used in the literature including the bilinear incidence rate, the Beddington-DeAngelis incidence rate, and a Holling type II functional response. Given that many diseases can lead to asymptomatic infections, we look at a system of stochastic differential equations that also includes a class of hidden state individuals, for which the infection status is unknown. We assume that the direct observation of the percentage of hidden state individuals that are infected, $α(t)$, is not given and only a noise-corrupted observation process is available. Using the nonlinear filtering techniques in conjunction with an invasion type analysis (or analysis using Lyapunov exponents from the dynamical system point of view), this paper proves that the long-term behavior of the disease is governed by a threshold $λ\in \mathbb{R}$ that depends on the model parameters. It turns out that if $λ<0$ the number $I(t)$ of infected individuals converges to zero exponentially fast, or the extinction happens. In contrast, if $λ>0$, the infection is endemic and the system is permanent. We showcase our results by applying them in specific illuminating examples. Numerical simulations are also given to illustrate our results.
