Stochastic Epidemic Models and Related Random Processes

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，在这些随机过程中，感染、信息或其他可传播的数量在网络的节点之间随机传递。研究的主要主题将是网络几何形状对流行病过程行为的影响，特别是在传播速率参数的临界值附近。两大类网络几何，欧几里得和双曲，将被研究。欧几里得几何适合于地理结构种群的模型，例如河床上的植物；例如，在这里，网络可能由单独的节点组成，这些节点以社区的形式排列在规则晶格的顶点上，相互作用仅限于相同或邻近社区的节点。双曲几何(如在展开图或？小世界?模型)在许多情况下更适合于人类种群和计算机网络，其中相互作用不遵循常规的地理模式。有限网络和无限网络都将被研究。主要目标将是描述大人口和/或大网络的限制行为。随机流行病模型与许多其他大类随机过程密切相关，这些随机过程将在研究项目中发挥重要作用。SIR类型的流行（易感、感染、移除）至少在简单的情况下相当于网络上的键渗透。SIS型流行病是接触过程。在欧几里得几何中，许多这些过程具有服从反应扩散型随机偏微分方程的测量值尺度极限。研究的第二个目标是了解在某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，也在计算机和通信网络中——是由偶然事件产生的。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触摸被MIRSA细菌污染的护栏；电脑用户可能会也可能不会点击一封电子邮件的附件，该电子邮件通知他，尼日利亚的一位富商正在请求他帮助将3000万美元转移到国外。然而，这些偶遇会被成千上万的人大量重复，数百次，因此它们的发生频率在统计学上是可以预测的。这使得——至少在原则上——在大量人口中预测流行病的进程成为可能。这个研究项目涉及流行病过程的简单数学模型的研究。这项研究的特别重点将是了解流行病传播所通过的个人或通信节点网络的基本几何结构如何影响其进程。

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.