Stochastic population processes: metastability, asymptotics and phase transitions

随机总体过程：亚稳态、渐近和相变

• 批准号: RGPIN-2018-04480
• 负责人: Foxall, Eric
• 金额: $ 1.53万
• 依托单位: University of British Columbia
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=678517
• 关键词: Stochastic; population; processes; metastability; asymptotics

The main goal of my research program is to develop and analyze stochastic models of interacting populations. A model of an interacting population comprises a set of individuals (people, animals, plants, etc.), a specification of the frequency of interaction between individuals (for example, a social network) and a set of rules for updating the state of individuals when they interact. An example is an epidemic model: there are a set of individuals, each either healthy or infected with the flu. Individuals that live or work close to one another interact more often than individuals that do not. When a healthy person encounters someone with the flu, with some probability, the healthy person contracts the flu. Note that non-interactive events can also be included: for example, a person with the flu may recover on their own after some time. The “stochastic” element means there is some randomness in the model. Thus in the above example, to simulate on a computer you would make a random choice (like flipping a coin) at the moment of interaction to determine whether the healthy person contracts the flu.******One way to study these models is by looking at their average behaviour. Even if each separate event is random, as long as we know the probabilities involved, and as long as individuals interact with each other in a fairly uniform way (i.e., the frequencies of interactions are close to the same for all pairs of individuals), when the model includes a large number of individuals it is reasonably straightforward to estimate (without actually running a simulation) what proportion of individuals are in each state at each moment in time. By appealing to these averages we obtain a simpler, so-called deterministic, system. However, the cost of this simplification is that important information concerning random fluctuations is lost. Fortunately, there are mathematical tools that can be used to study the fluctuations; the main idea is to describe the “rate” of these fluctuations, as a function of the system's average state at each moment in time. I intend to use these tools to study the fluctuations of these models directly, without needing recourse to computer simulation.******A phenomenon of particular interest is a phase transition, which is a change from one type of global behaviour to another, as parameters of the model are varied. For example, in the epidemic model, the infection can go from dying out quickly to causing a large outbreak, as the transmission rate of the infection is increased. Simple examples have shown that near the transition point, fluctuations tend to dominate the dynamics; in other words, they are most noticeable near a phase transition. If we can obtain a detailed description of these fluctuations, we may be able predict when a species is in danger of extinction, based on observations of its population over time. This is just one of many scenarios that we can study using stochastic models of interacting populations.

我的研究计划的主要目标是开发和分析相互作用群体的随机模型，相互作用群体的模型由一组个体（人、动物、植物等）组成，是个体之间相互作用频率的规范（一个例子是流行病模型：有一组个体，每个个体要么是健康的，要么是感染流感的个体。彼此亲近的人比不亲近的人更频繁地互动。健康人遇到流感患者，健康人有一定概率感染流感。请注意，也可以包括非交互事件：例如，流感患者可能会在一段时间后自行康复。 “随机”元素意味着模型中存在一定的随机性，因此在上面的示例中，要在计算机上进行模拟，您将在交互时做出随机选择（如掷硬币）来确定健康人是否感染流感。 ******研究这些模型的一种方法是查看即使每个单独的事件是随机的，只要我们知道所涉及的概率，并且只要个体以相当统一的方式彼此交互（即，所有对的交互频率接近相同）的个体），当模型包含大量个体时，可以相当直接地估计（无需实际运行模拟）每个时刻处于每种状态的个体比例。所谓确定性，系统不过成本。这种简化的缺点是丢失了有关随机波动的重要信息，幸运的是，有一些数学工具可以用来研究波动，其主要思想是描述这些波动的“速率”，作为系统平均状态的函数。我打算使用这些工具直接研究这些模型的波动，而不需要借助计算机模拟。****** 一个特别令人感兴趣的现象是相变，它是从一个状态到另一个状态的变化。全局行为的类型到另一个，作为参数例如，在流行病模型中，随着感染传播率的增加，感染可能会从迅速消失到引起大规模爆发，简单的例子表明，在过渡点附近，波动往往占主导地位。动态；换句话说，它们在相变附近最为明显。如果我们能够获得这些波动的详细描述，我们也许能够根据对物种种群随时间的观察来预测物种何时面临灭绝的危险。这只是我们可以使用随机模型研究的众多场景之一。相互作用的人群。