Evidence that coronavirus superspreading is fat-tailed

Abstract

Superspreaders, infected individuals who result in an outsized number of secondary cases, are believed to underlie a significant fraction of total SARS-CoV-2 transmission. Here, we combine empirical observations of SARS-CoV and SARS-CoV-2 transmission and extreme value statistics to show that the distribution of secondary cases is consistent with being fat-tailed, implying that large superspreading events are extremal, yet probable, occurrences. We integrate these results with interaction-based network models of disease transmission and show that superspreading, when it is fat-tailed, leads to pronounced transmission by increasing dispersion. Our findings indicate that large superspreading events should be the targets of interventions that minimize tail exposure.

Keywords: COVID-19; SARS-CoV-2; extreme value theory; infectious disease; superspreading.

Fig. 1.

SARS-CoV and SARS-CoV-2 SSEs correspond to fat tails. (A) Histogram of Z for 60 SSEs. (B) Subsample of 20 diverse SARS-CoV and SARS-CoV-2 SSEs. *See Dataset S1 for details. (C) Zipf plots of SSEs (blue) and 10,000 samples of a negative binomial distribution with parameters (R₀,k) = (3,0.1), conditioned on Z > 6 (yellow). (D) Meplots corresponding to C. (E) Plots of $\widehat{\xi }$, the Hill estimator for ξ, for the samples in C. (F) Different extreme value distribution fits to the distribution of SSEs. (G) One-sample Kolmogorov–Smirnov and χ² goodness-of-fit test results for the fits in F. (H) Robustness of results, accounting for noise (Left) and incomplete data (Right). (I) Inconsistency of the maxima of 10,000 samples of a negative binomial distribution (yellow) with the SSEs in A, accounting for variability in (R₀,k) and data merging and imputation, in contrast to the maxima of 30 samples from a fat-tailed (Fréchet) distribution (blue) with tail parameter α = 1.7 and mean R₀ = 3. The numbers of samples in each case were determined so that the sample mean of maxima is equal to the sample mean from A. (J–K) Generality of inferred ξ to 14 additional SSEs from news sources (J) and a dataset of 1,347 secondary cases arising from 5,165 primary cases in South Korea (K) (Dataset S2).

Fig. 2.

Forward modeling of intervention strategies. (A) State transitions in a fine-grained network model of disease transmission. (B–E) Predicted total infected fraction for an intervention strategy that isolates a fraction φ of all individuals, namely those with degree greater than the threshold number, and yielding decreased mean connectivity of d and effective basic reproduction number of R₀. Here, R₀ depends on the coefficient of variation of the degree distribution, as detailed in Dataset S3. Trajectories from 100 simulations for BA random graphs (B and D) and WS random graphs (C and E) and their averages are shown, compared to the theoretical predictions for a well-mixed model.