Spatial Population Games in Epidemiology and Ecology

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。在这个项目中，研究人员正在使用博弈论、动力系统和随机过程的数学方法来研究行为模式是如何在空间结构的种群中出现的，以帮助解释自然资源、健康和环境管理计划发展中的这些模式。博弈论方法应用于流行病学问题的新进展表明，大群体近似可以用来预测特定公共卫生措施的局限性。通过利用现有的大量关于空间分布的种群动态的研究，包括元种群和反应扩散方程，该项目应用并扩展了最近的研究，以解释个体日常活动的模式。特别是，研究人员正在解决流行病学问题，即地理上不同的感染风险如何影响预防措施。该团队试图了解空间显性混合如何改变纳什均衡策略，并决定风险热点的影响可以延伸到多远。我们也在进行研究，扩展艾萨克的经典理论，这样我们就可以研究少数人博弈中的基本行为，比如捕食回避，这在生态系统动力学中可能很重要。这些探索应该揭示关于问题规范及其解决方案的基本挑战，以及算法开发和数值近似等转化挑战。该项目为空间生态学和可持续管理领域的应用数学研究开辟了一条令人兴奋的新途径，它与生态学、经济学、社会学和流行病学等领域相交叉，可以吸引在分析、动力学和数值方面有才华的初级和高级数学家。该基金资助的研究生培训受到业界和政府的高度重视。资源、生态系统、社区和环境的可持续管理理论是复杂的，需要同时考虑经济过程、自然过程以及两者之间的反馈。可持续管理理论需要更多关注的一个方面是确定资源和风险的地理和空间分布如何调解经济过程和自然过程之间的反馈；从污染到供水再到农业，经济主体的行为可能对近邻和远邻产生意想不到的后果。由于在政策设计过程中经常缺少重要信息，地理效应可能使预测经济活动或监管行动的全部后果变得困难。正在开发的数学模型是对这些缺失信息的宝贵补充，并帮助我们有效地分配稀缺的社区和政府资源。该项目有助于对人口应如何应对地理上分散的风险作出一些基本预测，从而更合理地减轻风险。这些结果和它们所激发的研究使人们更好地理解生态过程，从而使国家能够面对目前存在的长期环境挑战。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).In this project, the investigators are using mathematical methods from game theory, dynamical systems, and stochastic processes to study how behavioral patterns emerge in spatially structured populations, to help account for these patterns in the development of natural-resource, health, and environmental management programs. New advances in the applications of game-theoretic methods to epidemiology problems show how large-population approximations can be used to predict the limitations of specific public health measures. By leveraging the substantial body of existing research on the dynamics of populations distributed in space, including metapopulations and reaction-diffusion equations, this project applies and extends the recent research to explain patterns the daily activities of individuals. In particular, the investigators are addressing epidemiological questions of how geographically heterogeneous risks of infection influence prophylactic measures. The team seeks to understand how spatially explicit mixing alters Nash equilibrium strategies and determines how far the influence from hot spots of risk can extend. We are also undertaking studies to expand classical theories like that of Isaacs so that we can study basic behaviors in few-player games like predation avoidance which are potentially important in the dynamics of ecosystems. These explorations should reveal foundational challenges regarding the specification of problems and their solution, and translational challenges like algorithm development and numerical approximation. The project as a whole opens up new exciting avenues of applied mathematics research in spatial ecology and sustainable management that intersects the fields of ecology, economics, sociology, and epidemiology, which can attract talented junior and senior mathematicians in analysis, dynamics, and numerics. The graduate student training funded by the grant is highly valued in both industry and government.The theory of the sustainable management of resources, ecosystems, communities, and our environment is complex, requiring the simultaneous accounting of economic processes, natural processes, and the feedbacks between the two. One aspect of sustainable management theory needing more attention is the identification of how geography and spatial distributions of resources and risks mediate the feedbacks between economic and natural processes; from pollution to water supplies to agriculture, the actions of economic agents can have unintended consequences for near-by and distant neighbors. Geographic effects can make it hard to anticipate the full consequences of an economic activity or regulatory action, to the extent that important information is often missing in the policy-design process. Mathematical models like the ones under development are valuable supplements to this missing information, and help us allocate scarce community and government resources efficiently. This project contributes to the development of some basic predictions of how populations should react to the geographically dispersed risks, allowing more rational risk mitigation. These results and the research they stimulate lead to a better understand ecological processes so that the nation may confront the long-term environmental challenges that now exist.