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: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=712934
• 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种群的随机模型。相互作用人群的模型包括一组个人（人，动物，植物等），个人之间相互作用频率（例如社交网络）的规范以及一组在互动时更新个人状态的规则。一个例子是一个流行病模型：有一组健康或感染流感的个体。与没有的个人相比，彼此亲近或彼此工作的个人相互作用更频繁。当一个健康的人遇到流感的人时，健康的人会感染流感。请注意，还可以包括非交互事件：例如，流感的人可以在一段时间后自行恢复。 “随机”元素意味着模型中存在一些随机性。在上面的示例中，要在计算机上模拟您将在互动时做出随机选择（例如翻转硬币），以确定健康的人是否会收缩流感。 研究这些模型的一种方法是查看其平均行为。即使每个单独的事件都是随机的，只要我们知道所涉及的可能性，并且只要个人以相当统一的方式相互互动（即，对于所有成对的个人，互动的频率接近相同的频率），当模型包括大量个人时，就可以合理地直接直接估算个人（实际运行一个模拟的情况），每个人都会在每个人的时间内使用什么比例。通过出现这些平均值，我们获得了一个更简单，所谓的确定系统。但是，这种简化的成本是丢失了有关随机波动的重要信息。幸运的是，有一些数学工具可用于研究波动。主要思想是将这些波动的“速率”描述为系统在每个时间时刻的平均状态的函数。我打算使用这些工具直接研究这些模型的波动，而无需识别计算机模拟。 特别感兴趣的现象是一种相变，这是从一种类型的全局行为到另一种类型的变化，因为模型的参数是不同的。例如，在流行病模型中，随着感染的传播速率增加，感染可能会从迅速死亡引起爆发。简单的例子表明，在过渡点附近，波动倾向于主导动力学。换句话说，它们在相过渡时最明显。如果我们能够根据对人口的观察随着时间的推移观察到的观察结果来预测某个物种何时有扩展的危险。这只是我们可以使用相互作用种群的随机模型研究的众多场景之一。