Persistent heterogeneity not short-term overdispersion determines herd immunity to COVID-19

Tkachenko, Alexei V.; Maslov, Sergei; Elbanna, Ahmed; Wong, George N.; Weiner, Zachary J.; Goldenfeld, Nigel

It has become increasingly clear that the COVID-19 epidemic is characterized by overdispersion whereby the majority of the transmission is driven by a minority of infected individuals. Such a strong departure from the homogeneity assumptions of the traditional well-mixed compartment model is usually hypothesized to be the result of short-term super-spreader events, such as an individual's extreme rate of virus shedding at the peak of infectivity while attending a large gathering without appropriate mitigation. However, we demonstrate that the spread of epidemics is primarily sensitive to long-term, or persistent heterogeneity of individual susceptibility or infectivity. We demonstrate how to incorporate this heterogeneity into a wide class of epidemiological models, and derive a non-linear dependence of the effective reproduction number Re on the susceptible population fraction S. Persistent heterogeneity has three important consequences compared to the effects of short-term overdispersion: (1) It results in a major modification of the early epidemic dynamics; (2) It significantly suppresses the herd immunity threshold; (3) It also significantly reduces the final size of the epidemic. We estimate social and biological contributions to persistent heterogeneity using data on real-life face-to-face contact networks and age variation of the incidence rate during the COVID-19 epidemic. 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. However, this type of immunity is fragile as it wanes over time if the pattern of social interactions changes substantially.