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

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.