County-Level Social Distancing and Policy Impact in the United States: A Dynamical Systems Model

Social distancing and public policy have been crucial for minimizing the spread of SARS-CoV-2 in the United States. Publicly available, county-level time series data on mobility are derived from individual devices with global positioning systems, providing a variety of indices of social distancing behavior per day. Such indices allow a fine-grained approach to modeling public behavior during the pandemic. Previous studies of social distancing and policy have not accounted for the occurrence of pre-policy social distancing and other dynamics reflected in the long-term trajectories of public mobility data.

We propose a differential equation state-space model of county-level social distancing that accounts for distancing behavior leading up to the first official policies, equilibrium dynamics reflected in the long-term trajectories of mobility, and the specific impacts of four kinds of policy. The model is fit to each US county individually, producing a nationwide data set of novel estimated mobility indices.

A differential equation model was fit to three indicators of mobility for each of 3054 counties, with T=100 occasions per county of the following: distance traveled, visitations to key sites, and the log number of interpersonal encounters. The indicators were highly correlated and assumed to share common underlying latent trajectory, dynamics, and responses to policy. Maximum likelihood estimation with the Kalman-Bucy filter was used to estimate the model parameters. Bivariate distributional plots and descriptive statistics were used to examine the resulting county-level parameter estimates. The association of chronology with policy impact was also considered.

Mobility dynamics show moderate correlations with two census covariates: population density (Spearman r ranging from 0.11 to 0.31) and median household income (Spearman r ranging from -0.03 to 0.39). Stay-at-home order effects were negatively correlated with both (r=-0.37 and r=-0.38, respectively), while the effects of the ban on all gatherings were positively correlated with both (r=0.51, r=0.39). Chronological ordering of policies was a moderate to strong determinant of their effect per county (Spearman r ranging from -0.12 to -0.56), with earlier policies accounting for most of the change in mobility, and later policies having little or no additional effect.

Chronological ordering, population density, and median household income were all associated with policy impact. The stay-at-home order and the ban on gatherings had the largest impacts on mobility on average. The model is implemented in a graphical online app for exploring county-level statistics and running counterfactual simulations. Future studies can incorporate the model-derived indices of social distancing and policy impacts as important social determinants of COVID-19 health outcomes.