Before revision:  In addition, empirical data from the COVID-19 epidemic in New York City (NYC) and Chicago, as well as 50 US states provide a consistent characterization of the level of heterogeneity. Our estimates suggest that the hardest-hit areas, such as NYC, are close to the heterogeneity-modified herd immunity threshold following the first wave of the epidemic .
After revision:  In addition, empirical data from the COVID-19 epidemic in New York City (NYC) and Chicago, as well as from real-life face-to-face contact networks. These results suggest that the hardest-hit areas, such as NYC, are close to the heterogeneity-modified herd immunity threshold following the first wave of the epidemic .