Modelling the impact of social distancing and targeted vaccination on the spread of COVID-19 through a real city-scale contact network.

We use mobile device data to construct empirical interpersonal physical contact networks in the city of Portland, Oregon, both before and after social distancing measures were enacted during the COVID-19 pandemic. These networks reveal how social distancing measures and the public's reaction to the incipient pandemic affected the connectivity patterns within the city. We find that as the pandemic developed there was a substantial decrease in the number of individuals with many contacts. We further study the impact of these different network topologies on the spread of COVID-19 by simulating an SEIR epidemic model over these networks and find that the reduced connectivity greatly suppressed the epidemic. We then investigate how the epidemic responds when part of the population is vaccinated, and we compare two vaccination distribution strategies, both with and without social distancing. Our main result is that the heavy-tailed degree distribution of the contact networks causes a targeted vaccination strategy that prioritizes high-contact individuals to reduce the number of cases far more effectively than a strategy that vaccinates individuals at random. Combining both targeted vaccination and social distancing leads to the greatest reduction in cases, and we also find that the marginal benefit of a targeted strategy as compared to a random strategy exceeds the marginal benefit of social distancing for reducing the number of cases. These results have important implications for ongoing vaccine distribution efforts worldwide.

Replica symmetry breaking in bipartite spin glasses and neural networks.

Some interesting recent advances in the theoretical understanding of neural networks have been informed by results from the physics of disordered many-body systems. Motivated by these findings, this work uses the replica technique to study the mathematically tractable bipartite Sherrington-Kirkpatrick (SK) spin-glass model, which is formally similar to a restricted Boltzmann machine (RBM) neural network. The bipartite SK model has been previously studied assuming replica symmetry; here this assumption is relaxed and a replica symmetry breaking analysis is performed. The bipartite SK model is found to have many features in common with Parisi's solution of the original, unipartite SK model, including the existence of a multitude of pure states which are related in a hierarchical, ultrametric fashion. As an application of this analysis, the optimal cost for a graph partitioning problem is shown to be simply related to the ground state energy of the bipartite SK model. As a second application, empirical investigations reveal that the Gibbs sampled outputs of an RBM trained on the MNIST data set are more ultrametrically distributed than the input data themselves.