Floquet Theory for Stochastic Temporal Networks and Optimization Theory for the Design of Schedules for COVID-19

人类面临的问题没有比COVID-19更严重、更紧迫的了，而社交距离的就地庇护方法已经摧毁了美国经济。COVID-19社交距离可能需要一段长时间，原因有很多，包括疫苗对新出现菌株的有效性的不确定性。因此，希望通过开发“社会距离科学和工程”来确定数学原理的解决方案，这些解决方案在生产率和感染风险之间取得最佳平衡。因此，PI将开发（1）新的数学理论，以更好地和更严格地理解在经历每周和/或每日周期的社交接触网络上传播的流行病;以及（2）优化理论，以设计现实的社交接触网络（例如，对社区实行宵禁，对学术机构实行课程安排），既减轻了流行病的影响，又使社会/经济活动达到可接受的水平。这些技术将应用于与COVID-19相关的社交网络数据集，以验证理论并获得实践知识。这些数学和计算工具以及相关数据集将为教育机构、大公司和管理机构的政策制定者提供支持，帮助他们管理COVID-19以及未来流行病的经济和健康影响。该项目的第一个目标是扩展常微分方程的Floquet理论，以（a）描述随机、时变和周期性的社会接触网络上的流行病传播;（B）识别和分析由于社会网络的周期性和疾病进展在相似的时间尺度上变化而产生的疾病动力学的新行为;以及（c）研究有助于疾病定位的网络结构的影响（例如，在网络中的集线器节点和社区中）。对于动态变化的网络，这些问题仍然没有得到充分的研究。该项目的第二个目标是开发网络优化理论，以设计数学原则的社交距离协议，既可以抑制COVID-19传播，也可以保持可接受的经济和社会活动水平。具体而言，网络扰动和Floquet分解的优化技术将开发，然后将它们与非凸优化算法相结合。到目前为止，由于人类活动而动态变化的接触网络中的社会接触工程的数学甚至计算基础是缺乏的。最后，这些技术将被应用于与COVID-19相关的社交网络数据集，这些数据集提供了个人活动的摘要和匿名。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

No problem facing humanity is more severe and more urgent than COVID-19, and the shelter-in-place approach to social distancing has decimated the U.S. economy. COVID-19 social distancing may be required for a long period for a multitude of reasons including the uncertainty of the effectiveness of vaccines on emerging strains. Therefore, it is desired to identify mathematically principled solutions that strike an optimal balance between productivity and risk to infection through developing a “science and engineering for social distancing”. Thus motivated, the PIs will develop (1) novel mathematical theory to better and more rigorously understand epidemic spreading on social-contact networks that undergo weekly and/or daily cycles; and (2) optimization theory to design realistic social-contact networks (e.g., using curfews for community and course scheduling for academic institutions) that both mitigate pandemics and allow an acceptable level of social/economic activity. These techniques will be applied to social-network datasets related to COVID-19 to validate theory and obtain practical knowledge. These mathematical and computational tools and related datasets will support policy makers for educational institutions, large companies, and governing bodies as they manage the economic and health impacts of COVID-19 as well as future pandemics. The first goal of the project is to extend Floquet theory of ordinary differential equations to (a) characterize epidemic spreading on social-contact networks that are stochastic, time-varying, and periodic; (b) identify and analyze novel behaviors for disease dynamics that arise because the social-network’s periodicity and the disease progression change at similar time scales; and (c) study the effects of network structures that contribute to disease localization (e.g., in hub nodes and communities in the network). These questions remain underexplored for dynamically changing networks. The second goal of the project is to develop network optimization theory to design mathematically principled social-distancing protocols that can both suppress COVID-19 spreading and also maintain an acceptable level of economic and social activity. Specifically, network perturbation and optimization techniques for Floquet decompositions will be developed, and then they will be combined with non-convex optimization algorithms. To date, mathematical or even computational foundations for social-contact engineering in dynamically changing contact networks due to human activity are lacking. Finally, these techniques will be applied to social-network datasets related to COVID-19 which provide summarized and anonymized movements of individuals.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.