Understanding the Dynamics of Stochastic Disease Spread in Metapopulations

确定传染病爆发、传播和灭绝的条件是一个重要的公共卫生问题。在全球消灭一种传染病的情况很少实现，但它仍然是许多疾病的一项公共卫生目标。更常见的是，人们可以观察到当地疾病灭绝或消失，然后通过迁移效应从其他地区重新引入疾病。一般来说，灭绝发生在由于随机转变而受到随机效应的种群中。随机性的起源可能来自系统内部，也可能来自外部环境。种群规模小、频率依赖传播的接触频率低、资源竞争和进化压力，以及种群和传播的异质性，都可能是灭绝发生的决定因素。灭绝事件的可能性受随机噪声的性质和强度以及其他因素的影响，包括爆发幅度和季节阶段的发生。对于大种群，种群内部噪声强度一般较小。然而，一个罕见的、大的波动可能会以非零概率发生，系统可能会达到灭绝状态。研究目的是利用由耦合区域或斑块组成的随机元种群模型研究疾病传播和灭绝的动力学。一个主方程的形式将被用来理解疾病的动态，并找到路径，最大限度地提高疾病灭绝的可能性。这一结果将使人们能够通过使用疫苗接种和检疫等控制方法来加速疾病的灭绝。这个建议是高度多学科的，涉及动力系统、随机过程、统计力学和控制理论。将要开发的数学工具将提供分析和确认数值结果的新方法。此外，分析将导致新的信息和系统行为的预测，并将提供对传染病爆发，传播和灭绝过程的更好理解。特别是，这种对疾病动态的新认识将有助于开发最优控制方法，以减少疾病的爆发和传播。该提案包括精心策划的项目，这些项目将涉及并支持本科生和研究生进行前沿研究。值得注意的是，蒙特克莱尔州立大学的学生，特别是数学科学系的学生中，有很大一部分是STEM学科中代表性不足的群体（包括女性和少数族裔）的成员，该研究项目将利用针对这些学生的现有项目。研究结果将通过研讨会、会议报告和同行评议期刊的出版物来传播。

Determining the conditions for the outbreak, spread, and extinction of an infectious disease is an important public health problem. Global eradication of an infectious disease has rarely been achieved, but it continues to be a public health goal for many diseases. More commonly, one can observe local disease extinction, or fade out, followed by a reintroduction of the disease from other regions through a migratory effect. In general, extinction occurs in populations undergoing stochastic effects owing to random transitions. The origins of stochasticity may be internal to the system or may arise from the external environment. Small population size, low contact frequency for frequency-dependent transmission, competition for resources and evolutionary pressure, as well as heterogeneity in populations and transmission, may all be determining factors for extinction to occur. The possibility of an extinction event is affected by the nature and strength of the stochastic noise, as well as other factors, including outbreak amplitude and seasonal phase occurrence. For large populations, the intensity of internal population noise is generally small. However, a rare, large fluctuation can occur with non-zero probability and the system may be able to reach the extinct state. The research objective is to study the dynamics of disease spread and extinction using stochastic metapopulation models that consist of coupled regions or patches. A master equation formalism will be used to understand the disease dynamics and to find the path that maximizes the probability of disease extinction. The results will enable one to speed up disease extinction through the use of control methods including vaccination and quarantine programs. The proposal is highly multidisciplinary, and involves dynamical systems, stochastic processes, statistical mechanics, and control theory. The mathematical tools that will be developed will provide new ways of analyzing and confirming numerical results. In addition, the analysis will lead to the prediction of novel information and system behavior, and will provide for improved understanding of infectious disease outbreak, spread, and extinction processes. In particular, this new understanding of disease dynamics will enable the development of optimal control methods to lessen disease outbreak and spread. The proposal includes carefully planned projects that will involve and support undergraduate and graduate students in leading-edge research. Significantly, the student population at Montclair State University, and in particular, the Department of Mathematical Sciences, includes a substantial proportion who are members of groups underrepresented in STEM disciplines (including women and minorities) and the research program will leverage existing programs directed to these students. The outcome of the research will be disseminated through seminars, presentations at meetings, and publications in peer-reviewed journals.