Estimating the overdispersion in COVID-19 transmission using outbreak sizes outside China

Introduction

A novel coronavirus disease (COVID-19) outbreak, which is considered to be associated with a market in Wuhan, China, is now affecting a number of countries worldwide^1,2. A substantial number of human-to-human transmission has occurred; the basic reproduction number R₀ (the average number of secondary transmissions caused by a single primary case in a fully susceptible population) has been estimated around 2–3^3–5. More than 100 countries have observed confirmed cases of COVID-19. A few countries have already been shifting from the containment phase to the mitigation phase^6,7, with a substantial number of locally acquired cases (including those whose epidemiological link is untraceable). On the other hand, there are countries where a number of imported cases were ascertained but fewer secondary cases have been reported than might be expected with an estimated value of R₀ of 2–3.

This suggests that not all symptomatic cases cause a secondary transmission, which was also estimated to be the case for past coronavirus outbreaks (SARS/MERS)^8,9. High individual-level variation (i.e. overdispersion) in the distribution of the number of secondary transmissions, which can lead to so-called superspreading events, is crucial information for epidemic control⁹. High variation in the distribution of secondary cases suggests that most cases do not contribute to the expansion of the epidemic, which means that containment efforts that can prevent superspreading events have a disproportionate effect on the reduction of transmission.

We estimated the level of overdispersion in COVID-19 transmission by using a mathematical model that is characterised by R₀ and the overdispersion parameter k of a negative binomial branching process. We fit this model to worldwide data on COVID-19 cases to estimate k given the reported range of R₀ and interpret this in the context of superspreading.

Methods

Results

Our estimation suggested substantial overdispersion (k ≪ 1) in the offspring distribution of COVID-19 (Figure 1A and Figure 2). Within the current consensus range of R₀ (2–3), k was estimated to be around 0.1 (median estimate 0.1; 95% CrI: 0.05–0.2 for R₀ = 2.5). For the R₀ values of 2–3, the estimates suggested that 80% of secondary transmissions may have been caused by a small fraction of infectious individuals (~10%; Figure 1B).

Figure 1. MCMC estimates given assumed R₀ values.

(A) Estimated overdispersion parameter for various basic reproduction number R₀. (B) The proportion of infected individuals responsible for 80% of the total secondary transmissions (p_80%). The black lines show the median estimates given fixed R₀ values and the grey shaded areas indicate 95% CrIs. The regions corresponding to the likely range of R₀ (2–3) are indicated by colour.

Figure 2. Possible offspring distributions of COVID-19.

(A) Offspring distribution corresponding to R₀ = 2.5 and k = 0.1 (median estimate). (B) Offspring distribution corresponding to R₀ = 2.5 and k = 0.05 (95% CrI lower bound), 0.2 (upper bound). The probability mass functions of negative-binomial distributions are shown.

The result of the joint estimation suggested the likely bounds for R₀ and k (95% CrIs: R₀ 1.4–12; k 0.04–0.2). The upper bound of R₀ did not notably differ from that of the prior distribution (=13.5), suggesting that our model and the data only informed the lower bound of R₀. This was presumably because the contribution of R₀ to the shape of a negative-binomial distribution is marginal when k is small (Extended data, Figure S1)¹⁶. A scatterplot (Extended data, Figure S2)¹⁶ exhibited a moderate correlation between R₀ and k (correlation coefficient -0.4).

Model comparison between negative-binomial and Poisson branching process models suggested that a negative-binomial model better describes the observed data; WBIC strongly supported the negative-binomial model with a difference of 11.0 (Table 2). The simulation of the effect of underreporting suggested that possible underreporting is unlikely to cause underestimation of overdispersion parameter k (Extended data, Figure S3)¹⁶. A slight increase in the estimate of k was observed when the reproduction number for imported cases was assumed to be lower due to interventions (Extended data, Table S1).

Discussion

Our results suggested that the offspring distribution of COVID-19 is highly overdispersed. For the likely range of R₀ of 2–3, the overdispersion parameter k was estimated to be around 0.1, suggesting that the majority of secondary transmission may be caused by a very small fraction of individuals (80% of transmissions caused by ~10% of the total cases). These results are consistent with a number of observed superspreading events observed in the current COVID-19 outbreak¹⁷, and also in line with the estimates from the previous SARS/MERS outbreaks⁸.

The overdispersion parameter for the current COVID-19 outbreak has also been estimated by stochastic simulation¹⁸ and from contact tracing data in Shenzhen, China¹⁹. The former study did not yield an interpretable estimate of k due to the limited data input. In the latter study, the estimates of R_e (the effective reproduction number) and k were 0.4 (95% confidence interval: 0.3–0.5) and 0.58 (0.35–1.18), respectively, which did not agree with our findings. However, these estimates were obtained from pairs of cases with a clear epidemiological link and therefore may have been biased (downward for R₀ and upward for k) if superspreading events had been more likely to be missed during the contact tracing.

Although cluster size distributions based on a branching process model are useful in inference of the offspring distribution from limited data^12,20, they are not directly applicable to an ongoing outbreak because the final cluster size may not yet have been observed. In our analysis, we adopted an alternative approach which accounts for possible future growth of clusters to minimise the risk of underestimation. As of 27 February 2020, the majority of the countries in the dataset had ongoing outbreaks (36 out of 43 countries analysed, accounting for 2,788 cases of the total 2,816). Even though we used the case counts in those countries only as the lower bounds of future final cluster sizes, which might have only partially informed of the underlying branching process, our model yielded estimates with moderate uncertainty levels (at least sufficient to suggest that k may be below 1). Together with the previous finding suggesting that the overdispersion parameter is unlikely to be biased downwards²¹, we believe our analysis supports the possibility of highly-overdispersed transmission of COVID-19.

A number of limitations need to be noted in this study. We used the confirmed case counts reported to WHO and did not account for possible underreporting of cases. Heterogeneities between countries in surveillance and intervention capacities, which might also be contributing to the estimated overdispersion, were not considered (although we investigated such effects by simulations; see Extended data, Figure S3)¹⁶. Reported cases whose site of infection classified as unknown, which should in principle be counted as either imported or local cases, were excluded from analysis. Some cases with a known site of infection could also have been misclassified (e.g., cases with travel history may have been infected locally). The distinction between countries with and without ongoing outbreak (7 days without any new confirmation of cases) was arbitrary. However, we believe that our conclusion is robust because the distinction does not change with different thresholds (4–14 days), within which the serial interval of SARS-CoV-2 is likely to fall^22,23.

Our finding of a highly-overdispersed offspring distribution suggests that there is benefit to focusing intervention efforts on superspreading. As most infected individuals do not contribute to the expansion of transmission, the effective reproduction number could be drastically reduced by preventing relatively rare superspreading events. Identifying characteristics of settings that could lead to superspreading events will play a key role in designing effective control strategies.

Data availability