Stochastic Epidemic Models and Related Random Processes

随机流行病模型及相关随机过程

• 批准号: 1106669
• 负责人: Steven Lalley
• 金额: $ 36万
• 依托单位: University of Chicago
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-09-01 至 2014-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1106669&HistoricalAwards=false
• 关键词: Stochastic; Epidemic; Models; Related; Random

The primary focus of the project will be the study of stochastic models for epidemics and related random processes in which infection, information, or some other transmissible quantity is passed randomly among the nodes of a network. The dominant theme of the research will be the effect of the network geometry on the behavior of the epidemic processes, especially near critical values of the transmission rate parameter(s). Two large classes of network geometries, Euclidean and hyperbolic, will be studied. Euclidean geometries are appropriate for models of geographically structured populations, such as plants along a river bed; here, for instance, the network might consist of individual nodes arranged in communities placed at the vertices of a regular lattice, with interactions restricted to nodes in the same or neighboring communities. Hyperbolic geometries (such as those arising in expander graphs or ?small worlds? models) are in many instances more appropriate for human populations and computer networks, where interactions do not follow a regular geographic pattern. Both finite and infinite networks will be studied. The primary objective will be the description of large-population and/or large network limiting behavior.Stochastic epidemic models are closely related to a number of other large classes of stochastic processes, and these will play a large role in the research project. Epidemics of SIR type (susceptible, infected, removed) are at least in simple cases equivalent to bond percolation on the network. Epidemics of SIS type are contact processes. In Euclidean geometries, many of these processes have measure-valued scaling limits that obey stochastic partial differential equations of reaction-diffusion type. A secondary objective of the research will be to understand the threshold phenomena and phase transitions that occur in some of these related processes, and to explore how these are mirrored in epidemic models. Epidemics -- in human and animal populations, but also in computer and communications networks -- arise from chance events. These are, in many cases, simple but uncontrollable events: A passenger on a bus might or might not touch a guard rail that has been contaminated by MIRSA bacteria; a computer user might or might not click on an attachment to an email message informing him that a wealthy businessman in Nigeria is asking for his help in transferring 30 million dollars out of the country. These chance encounters are, however, repeated in large numbers, by thousands of people, hundreds of times, and so they occur with statistically predictable frequencies. This makes it possible -- at least in principle -- to predict the course of an epidemic in a large population.This research project is concerned with the study of simple mathematical models of epidemic processes. The particular emphasis of the study will be on understanding how the underlying geometry of the network of individuals or communications nodes through which the epidemic propagates affects its course.

该项目的主要重点是研究流行病和相关随机过程的随机模型，其中感染、信息或其他可传播量在网络节点之间随机传递。研究的主题将是网络几何对流行病过程行为的影响，特别是传播率参数的临界值附近。将研究两大类网络几何形状：欧几里得几何形状和双曲线形状。欧几里得几何适用于地理结构种群的模型，例如河床沿线的植物；例如，这里的网络可能由排列在规则网格顶点的社区中的各个节点组成，交互仅限于相同或相邻社区中的节点。双曲几何（例如在扩展图或“小世界”模型中出现的几何）在许多情况下更适合人类群体和计算机网络，其中相互作用不遵循规则的地理模式。将研究有限网络和无限网络。主要目标是描述大量人口和/或大型网络限制行为。随机流行病模型与许多其他大类随机过程密切相关，这些模型将在研究项目中发挥重要作用。 SIR 类型的流行病（易感、感染、移除）至少在简单情况下相当于网络上的债券渗透。 SIS类型的流行病是接触过程。在欧几里德几何中，许多过程都具有服从反应扩散类型的随机偏微分方程的测量值标度限制。该研究的第二个目标是了解一些相关过程中发生的阈值现象和相变，并探索这些现象如何反映在流行病模型中。流行病——在人类和动物种群中，以及在计算机和通信网络中——都是由偶然事件引起的。在许多情况下，这些都是简单但无法控制的事件：公共汽车上的乘客可能会也可能不会接触被 MIRSA 细菌污染的护栏；计算机用户可能会也可能不会点击一封电子邮件的附件，该附件通知他尼日利亚的一位富商正在寻求他的帮助，将 3000 万美元转出该国。然而，这些偶遇会被数千人、数百次大量重复，因此它们发生的频率在统计上是可预测的。这使得至少在原则上预测大量人群中流行病的进程成为可能。该研究项目涉及流行病过程的简单数学模型的研究。该研究的特别重点是了解流行病传播的个人或通信节点网络的基本几何结构如何影响其进程。