question: We combine COVID-19 case data with demographic and mobility data to estimate a modified susceptible-infected-recovered (SIR) model for the spread of this disease in the United States. We find that the incidence of infectious COVID-19 individuals has a concave effect on contagion, as would be expected if people have inter-related social networks. We also demonstrate that social distancing and population density have large effects on the rate of contagion. The social distancing in late March and April substantially reduced the number of COVID-19 cases. However, the concave contagion pattern means that when social distancing measures are lifted, the growth rate is considerable but will not be exponential as predicted by standard SIR models. Furthermore, counties with the lowest population density could likely avoid high levels of contagion even with no social distancing. We forecast rates of new cases for COVID-19 under different social distancing norms and find that if social distancing is eliminated there will be a massive increase in the cases of COVID-19, about double what would occur if the US only restored to 50% of the way to normalcy.