Ethics statement

This study only involved secondary analysis of publicly available contact survey data and SARS-CoV-2 viral load data. Secondary analysis of these data was approved by the London School of Hygiene and Tropical Medicine Observational Research Ethics Committee (ref 14400 and ref 21795).

Contact data

CoMix survey.

The CoMix survey was a behavioural survey launched on 24th of March 2020 to gather social and behavioural data to aid the response to the COVID-19 pandemic. The contact survey was based on the POLYMOD contact survey [10]. The sample was broadly representative of the UK adult population. Participants were invited to respond to the survey once every two weeks. Weekly data was collected by running two alternating panels. Parents completed the survey on behalf of children (17 years old or younger). Participants recorded direct, face-to-face contacts made on the previous day, specifying certain characteristics for each contact including the age and sex of the contact, whether contact was physical (skin-to-skin contact), and where contact occurred (e.g., at home, work, while undertaking leisure activities, etc.). Full details have been published elsewhere [11].

From the 24th May 2020 onwards, the CoMix survey included the ability to record an estimated mass contacts count in situations where it would be infeasible to record the detailed information of all contacts made, such as large gatherings. Since the BBC Pandemic survey did not include this option, and this potentially truncated the true contact distribution, we conducted a sensitivity analysis of the effect on R and k of imputing a heavier tail for the BBC Pandemic contact distribution. We did this by fitting a generalised Pareto distribution to the number of individuals reporting over 250 contacts in relaxed restrictions time periods (“Relaxed restrictions” and “School reopening” in Table 1) from the CoMix data and sampling proportionally from this distribution for the BBC Pandemic contacts.

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Table 1. Percentage of participants in the CoMix and BBC Pandemic surveys with more than a certain number of contacts in different time periods in the UK from March 2020 to May 2021, and pre-pandemic in 2017-2018.

https://doi.org/10.1371/journal.pcbi.1014715.t001

Analysis of social contact data

We compared contact distributions (overall and stratified by household and out-of-household) for the BBC Pandemic survey [7–9] and CoMix survey [11,12]. To account for differences in numbers of participants per time period, we calculated and plotted the percentage of participants reporting more than 5, 10, 20, 50, 100, and 200 contacts in the 24-hour period prior to filling in the survey for the BBC Pandemic survey (here referred to as Pre-pandemic) and nine indicative time periods between March 2020 and May 2021 from the CoMix survey representing different levels of restrictions, as defined in previous work [11,12] (Table 1). We fitted negative binomial distributions to numbers of daily contacts in the different time periods to estimate the mean and dispersion of the contact distributions. To assess whether the negative binomial adequately captured the contact distribution, or whether an excess of zero-contact individuals required a zero-inflated extension, we additionally fitted a zero-inflated negative binomial (ZINB) distribution to the per-individual daily contact counts for three indicative time periods (Pre-pandemic, 1st lockdown, School reopening) and compared models using the Akaike information criterion (AIC) and a likelihood ratio test (LRT). The negative binomial was found to be sufficient in all periods (zero-inflation parameter , AIC = +2.0, LRT p > 0.97; S2 Table).

Reconstruction of secondary infection distribution

We reconstructed the secondary infection distribution of SARS-CoV-2 in the UK over the course of 2020 as described below. We restricted our analysis to the first six time periods of the CoMix survey in 2020 (23rd March to 19th December) prior to the widespread emergence of more transmissible variants and mass vaccination.

Viral load trajectories

To predict the distribution of individual-level transmission at different points in time, we simulated 10,000 individual respiratory viral load trajectories of index cases over the course of infection. The viral load of each index case was constructed from a piecewise linear function of their Ct values defined by a proliferation phase (days from exposure to peak, ), a clearance phase (days from peak to cessation, ) and a peak Ct value, . The three parameters were drawn from distributions given in Kissler et al. [13] (Fig 1):

where denotes a truncated normal distribution with mean and variance , and lower and upper limits a and b. The Ct value at exposure and cessation of shedding was assumed to be 40 (negative).

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Fig 1. Schematic of the model.

The viral load and number of daily contacts (circles) varies from person to person and over time, influencing the number of secondary infections (stratified by household (outlined) and non-household (no outline)) they generate. The viral load at time of testing also determines the likelihood they will test positive and subsequently isolate.

https://doi.org/10.1371/journal.pcbi.1014715.g001

The Ct value was converted to viral load vl (in log10 RNA copies/ml) via the following formula [13]:

where and are the slope and intercept of a linear regression of Ct value on log10-transformed RNA concentration from a standard curve.

Infectiousness

The probability of infectiousness for a given viral load (in Ct) was estimated by fitting a logistic regression model to the probability of culturing virus at that viral load [14] (Fig 3B), :

where culture is the binary outcome of whether or not virus was cultured. The probability of infectiousness over the course of each individual’s viral load trajectory was then calculated using the estimated regression coefficients and to produce infectiousness trajectories (Fig 3C). Uncertainty in and was accounted for by drawing values for each individual from a bivariate normal distribution centred at the maximum likelihood estimate :

where is the estimated covariance matrix from the logistic regression. Drawing values for the regression coefficients in this way may be taken to represent a degree of variation in infectiousness between individuals for the same viral load, as Ke et al. [6] concluded was necessary to explain observed heterogeneity in infectiousness. We use viral culture positivity as our proxy for infectiousness because it indicates the presence of replication-competent virus and is therefore believed to provide a better indicator of transmission potential than Ct values from PCR measurements [15]. It also allows us to make use of empirical relationships linking viral load, viral culture positivity, and test detection: (i) viral load determines the probability of culturing viable virus, with the logistic relationship fitted directly from data in Pickering et al. [14] (Fig 3B); and (ii) LFT positivity closely tracks viable virus in controlled human challenge studies [5].

Contacts and contact durations

For the pre-pandemic time period and each of the pandemic time periods, we randomly sampled the number of household contacts and daily non-household contacts for index case i, where t is the time since exposure, from the empirical contact distribution for that time period from the BBC Pandemic contact survey [7–9] and CoMix contact survey [11,12]. The number of household and non-household contacts were sampled jointly by sampling a survey participant at random, to preserve correlation in individual-level numbers of household and non-household contacts. The numbers of non-household contacts on different days were sampled from the repeat survey responses of the chosen participant for the given time period, to maintain correlation in individual-level numbers of non-household contacts over time. Each household and non-household contact had a duration, and , defined as the proportion of a 24-hour period that the contact lasted (S1 Fig). For the pandemic time periods the duration was sampled from the contact durations for the same survey participant as the contact numbers, where these were available. This ensured that correlation between number of contacts and contact duration was maintained (S2 Fig). For the pre-pandemic time period, the duration was sampled from the contact durations in the POLYMOD survey, as contact duration data is not publicly available for the BBC Pandemic survey. Where the contact duration was missing, it was randomly sampled from the pre-pandemic or pandemic contact duration distribution according to whether it was a pre-pandemic or pandemic contact.

Infections

The infection process for contact between index case i and their contact j was modelled as Bernoulli for both household and non-household infections

with the probability of i infecting j on day t in location X, , where , given by

where is a scaling parameter which converts the product of the relative infectiousness of i on the day of contact, , and the duration of contact, , into a transmission rate. This process was iterated over each day of the index case’s infection up to a maximum infection duration of 44 days in the baseline analysis. While we modelled infection events for household members on each day of the index case’s infection, we only counted the first infection that occurred for each household member. In other words, household members could only be infected once. We assumed uniform susceptibility of individuals in the model, which does not vary by, for example, age.

The number of secondary infections for index case i (i.e., their individual-level reproduction number), , was therefore given by:

where is index case i’s duration of infection (defined as having infection detectable by PCR). We then estimated the mean, R, and overdispersion, k, of the number of secondary infections by fitting a negative binomial distribution to the values across all index cases. 95% percentile bootstrap confidence intervals (CIs) for the R and k estimates were calculated by bootstrapping the simulated numbers of secondary infections with 1000 bootstrap samples, calculating R and k for each bootstrap sample and then finding the 2.5 and 97.5 percentiles of the distributions of the bootstrap estimates. We calibrated such that the basic reproduction number R₀, the reproduction number for the pre-pandemic time period, was 2.5, based on estimates from the literature [12,16,17], for the baseline analysis and all sensitivity analyses varying heterogeneity in contact rates and viral load. We obtained values of between 1.1 and 4.7.

Contribution of contact and viral load heterogeneity to heterogeneity in secondary infections

To investigate the contribution of heterogeneity in contact rates and heterogeneity in viral load between individuals to variation in the secondary infection distribution, we calculated the overdispersion in secondary infections for different combinations of “homogeneous” and “heterogeneous” contact rates and viral load trajectories. In the “homogeneous” cases, contacts were Poisson distributed with mean equal to the mean daily contacts for each time period and all individuals had the median viral load trajectory (i.e., the trajectory with the median proliferation and clearance durations and median peak Ct value), while in the “heterogeneous” cases, contacts and viral load trajectories were allowed to vary according to their full distributions.

We performed a sensitivity analysis with greater variation in viral load trajectories across individuals, by doubling the standard deviations of the distributions for the proliferation time, clearance time and peak Ct value, to test the sensitivity of the results to the degree of viral load heterogeneity. For this sensitivity analysis, we considered onward transmissions up to 88 days post exposure.

Simulation of interventions

We assessed the hypothesis that if those with the highest viral loads are most likely to cause superspreading events, and LFTs are most sensitive for those with the highest viral loads, then lateral flow testing should be able to prevent superspreading events, which we defined as infecting over 10 contacts. We determined the probability of testing positive on an LFT at a given viral load, , by fitting a logistic regression model to data on LFT outcomes for isolates with different viral loads [14]. Detection was then modelled via a Bernoulli draw with this detection probability, that is,

where (95% CI: -13.3, 5.0) and (95% CI: 1.1, 2.6) were the estimated coefficients from the logistic regression. We then modelled index cases testing regularly with LFTs at different frequencies (every 1, 3, and 7 days) or only before events of a certain size (before meeting >10, > 20, >50 others) to determine the comparative effectiveness of rapid testing to control transmission. We assumed that a certain proportion of individuals self-isolated at home upon first testing positive, such that their non-household contacts were decreased to zero after the date of the positive test, but their number of household contacts was unchanged. The remaining individuals were assumed to continue making contacts as normal. We varied the proportion of individuals who self-isolated upon testing positive from 0-100% to model varying uptake/adherence. We assumed timing of testing was not affected by symptom onset and intensity, as we wished to model regular and event-triggered testing regardless of symptoms. We considered three indicative timepoints before (BBC Pandemic, 2018) and during the pandemic (1st lockdown (March-June 2020) and the reopening of schools (September 2020)). We reran the analysis for regular lateral flow testing with Poisson-distributed contacts instead of overdispersed contacts to look at the effect of contact heterogeneity on the impact of regular testing.

The code and data for this study can be found at https://github.com/bquilty25/superspreading_testing.

Changes in the distribution of social contacts with changes in the intensity of restrictions

The age distribution of all participants in the BBC Pandemic survey and of the participants in each of the nine periods of the CoMix survey is shown in S3 Fig. Relative to the mid-2020 age distribution of the UK population, children and older adults (70+ years) are under-represented in the BBC Pandemic data. The age distribution of CoMix participants is representative of the general population (with slight under-representation of children 4 years and over-representation of adults aged 50–69 years) and remains fairly stable across the different time periods.

Fig 2A and Table 1 show the proportions of survey participants in the UK reporting more than a certain number of contacts (5, 10, 20, 50, 100 and 200) in the previous day over time, both pre-pandemic (BBC Pandemic survey) and from the CoMix survey for the nine time periods during the survey with different levels of restrictions. The percentage of individuals reporting more than 20 contacts in a day was substantially lower during the pandemic compared to the pre-pandemic period (when it was 13.7%), varying from a low of 0.4% during the first lockdown from March to June 2020 to a peak of 6.1% in September 2020, when restrictions were most relaxed and schools reopened. The percentage of individuals reporting over 100 and 200 contacts was lower in the pre-pandemic BBC Pandemic contact survey compared to periods of relaxed restrictions due to differences in the reporting of high contact events. Overall, people reporting more than 50, 100, or 200 contacts made up less than 3% of the total CoMix survey sample.

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Fig 2. Changes in the distribution of social contacts in the UK from pre-pandemic to May 2021.

A. Percentage of participants in the BBC Pandemic and CoMix contact surveys reporting zero, over 5, 10, 20, 50, 100, and 200 daily contacts in 2018 (pre-pandemic) and nine time periods during the pandemic in the UK between March 2020 and May 2021. Median and 95% binomial confidence intervals shown. B. Distribution of the number of reported daily contacts for three indicative timepoints before and during the pandemic in the UK in 2020. C. Mean and overdispersion parameter, k, of a negative binomial distribution fitted to contacts during the different time periods. The heavier tail in the contact distribution for the School reopening period in panel B compared with the Pre-pandemic period is likely due to mass contacts not being recorded in the BBC Pandemic contact survey, which may have truncated the contact distribution. See S4 Fig for the BBC Pandemic contact distribution with an imputed heavier tail, which is used to perform a sensitivity analysis.

https://doi.org/10.1371/journal.pcbi.1014715.g002

The estimated mean and overdispersion parameter k, with lower values indicating more overdispersion in the distribution, of numbers of daily contacts in the UK were lower than that observed pre-pandemic, with mean daily contacts averaging 4.7 during the pandemic compared to 11.5 pre-pandemic, and k averaging 0.7 during the pandemic compared to 1.5 pre-pandemic, indicating individuals having on average lower, but more varied, numbers of daily contacts. These values also changed considerably across the different periods of restrictions in a similar pattern to the proportion of participants with high numbers of contacts, with the mean number of daily contacts ranging from 2.6 during the first lockdown to 6.9 when schools reopened in September 2020, with a similar drop during the third lockdown, but less of a drop during the second lockdown when schools remained open (Fig 2C). The overdispersion in contacts, k, was similar pre-pandemic and during the first lockdown (1.5 and 1.6 respectively) then became lower, indicating more variability, following the easing of the first lockdown, with k between 0.5 and 0.7 in the remaining 2020 periods (Fig 2C). Stratifying contacts by household and out-of-household revealed contact rates within the household remained stable throughout, with changing non-household contact rates driving much of the variation in the mean and k of daily contacts over the course of the pandemic (S5 Fig). Both the mean and k of out-of-household daily contacts remained lower than pre-pandemic contact rates. Contact durations in CoMix differed significantly between household and non-household contacts, with a median duration of 30 minutes (IQR: 5 – 180 minutes) for out-of-household contacts compared to 480 minutes (8 hours) (IQR: 180 – 1080 minutes) for household contacts (S1 Fig). S6 Fig, in which contacts are stratified by age as well household/out-of-household, shows that both household and out-of-household contacts in children (<18 years) remained higher than those in adults throughout 2020 and early 2021, with significant increases in out-of-household contacts of children during periods when schools were open. Contact levels were similar for males and females across all time periods.

Variation in viral load and infectivity over the course of infection

Analysis by Kissler et al. [13] of densely sampled viral load trajectories in individuals infected with SARS-CoV-2 suggested substantial variation between individuals in the duration of proliferation and clearance phases of infection, and lowest Ct value, inversely correlated with peak viral load (Fig 3A). Mapping viral load to infectivity via a logistic function based on viral load and culture from Pickering et al. [14] (Fig 3B), calculating the probability of infectivity by day (Fig 3C), then integrating under the infectivity curve to calculate total infectious virus shed, reproduces substantial heterogeneity in individual infectiousness as reported by Ke et al. [6]. We found an 73-fold difference in individual-level infectivity between the 2.5% and 97.5% percentiles of the distribution of the area under the infectivity curve (0.08 and 5.74 arbitrary units, respectively). Fitting a Gamma distribution to this distribution gave a shape parameter of 1.42 (1.6 for Ke et al.). A substantial proportion (19.4%) of individuals were estimated to be infectious for zero days, but the remainder were infectious for at least one day (Fig 3). Across all individuals, the median duration of infectiousness around peak viral load, that is, having high enough viral load to cause infections through culturable virus, was 2 days (95% prediction interval: 0, 6 days). The majority of individuals passed through at least one day of high infectiousness (peak culture probability greater than 0.6 for 68% of individuals, greater than 0.8 for 53%), and were therefore theoretically capable of causing superspreading events given sufficient contacts of sufficient duration. However, the percentage of infected individuals who can actually initiate a superspreading event is much lower than 50% because the time window for this level of infectivity is so narrow that it must coincide perfectly with a high number of contacts of sufficient duration for a high number of infections to occur.

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Fig 3. Variation in viral load progression and infectivity.

A. Individual-level viral load trajectories. B. Logistic model for probability of culturing virus given a certain viral load. C. Relative infectivity over time. D. Area under the infectivity curve (in arbitrary units, AU). E. Distribution of the number of days for which individuals were infectious. Solid lines in A and C represent median viral load and infectivity trajectories. The dashed line in A and B represents the RNA copies per ml required for a 50% probability of culturing virus or infectiousness. The shaded band in B represents the uncertainty in the probability of culturing virus given a certain viral load from the fitted logistic regression model from Pickering et al. [14]. S7 Fig shows the difference in viral load trajectories and infectivity for the sensitivity analysis with double the standard deviation in the viral load trajectory parameters.

https://doi.org/10.1371/journal.pcbi.1014715.g003

Predicting transmission dynamics from heterogeneity in contact rates and viral load progression

Simulating secondary infections using the stochastic model described above, we estimated the overdispersion, k, in the initial secondary infection distribution as 0.48 (95% CI 0.47 – 0.51) (Fig 4), in line with other estimates [18,19]. During the first national lockdown, R was estimated at 0.37 (95% CI 0.35 – 0.41) and k at 0.25 (95% CI 0.21 – 0.29). R rose as lockdown eased (R = 0.66, 95% CI 0.56 – 0.8) and restrictions were relaxed in the summer of 2020 (R = 1, 95% CI 0.92 – 1.1) and again as schools reopened in the autumn (R = 1.5, 95% CI 1.3 – 1.6). R then decreased again during the second lockdown to 1 (95% CI 0.92 – 1.2). As with contacts, k reduced as the pandemic began, indicating more variability in the secondary infection distribution, and varied between 0.12 and 0.17 after the first lockdown up to the end of 2020. A sensitivity analysis imputing a heavier tail for the BBC Pandemic contact distribution based on the tails of contact distributions from relaxed restriction periods during the CoMix survey led to a small increase in R₀ from 2.5 (95% CI 2.4 – 2.6) to 2.8 (95% CI 2.6 – 3) and a small reduction in k from 0.48 (95% CI 0.46 – 0.51) to 0.44 (95% CI 0.4 – 0.49) (S1 Table).

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Fig 4. Estimates of the mean and dispersion of negative binomial distributions fitted to simulated secondary infection distributions by time period in 2020.

The dispersion parameter k gives an indication of the variation in the secondary infection distribution, with smaller values corresponding to greater variation and values less than 1 corresponding to very large variation. Grey boxes in the R plot show the average upper and lower bounds (90% confidence interval) of the consensus estimates published by the Scientific Pandemic Influenza group on Modelling in the UK for the specified time periods [17]. Shaded bands show 95% bootstrap confidence intervals for the R and k estimates.

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Contribution to heterogeneity in secondary infections due to variation in contact rates and viral load progression

If both contact rates and viral load trajectories are treated as homogeneous, with Poisson-distributed contacts and no variation in viral load trajectory between individuals, then the secondary infection distribution is approximately Poisson with equal mean and variance, corresponding to very large values of k. If contacts are Poisson distributed but viral load trajectories are allowed to vary between individuals, k is 1.5 – 1.9. If viral load is fixed at the median trajectory, but contacts are overdispersed, values of k are lower (0.11 – 0.74) and approximately equal to that of overdispersed contacts and variable viral load trajectories (0.11 – 0.51), indicating that overdispersion in numbers of daily contacts, as the denominator in the infection process, contributes more to superspreading, meaning overdispersion in the secondary infection distribution, than variable viral load trajectories (Fig 5). The two overdispersed-contact scenarios, equal and variable viral load, differ only pre-pandemic (k = 0.74 versus 0.51). This is due to individuals in the upper tail of infectiousness generating disproportionately many secondary infections when contacts are plentiful, as a result of the right-skewed area-under-the-curve distribution, exceeding an 73-fold range (Fig 3D). During restriction periods this difference disappears, as compressed contact rates create a ceiling on secondary cases regardless of viral load, and the proportion infecting zero others and more than ten others converges across the two scenarios from the first lockdown onwards (Fig 5). In the sensitivity analysis with greater variation in individuals’ viral load trajectories, with double the standard deviation in the viral load trajectory parameters, the k estimates for the scenario with Poisson contacts (0.76 – 0.92) and variable viral loads were closer to those for overdispersed contacts and variable viral loads (0.08 – 0.33), but still higher (S8 Fig). The finding that contact heterogeneity plays a greater role in overdispersion in secondary infections than viral load heterogeneity therefore appears to be robust to much greater variation in peak viral load and duration of viral shedding than found in the cohort in Kissler et al. [13].

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Fig 5. Estimates of the dispersion of negative binomial distributions fitted to secondary infection distributions by period, with different combinations of homogeneous and heterogeneous contact rates and viral load trajectories, along with the proportion of individuals infecting zero others and more than ten others.

The dispersion parameter k gives an indication of the variation in numbers of secondary infections, with smaller values corresponding to greater variation and values less than 1 corresponding to very large variation. Values of k for equal viral load and equal contacts are not shown because they are very large, meaning the numbers of secondary infections are approximately Poisson distributed. Shaded bands show 95% bootstrap confidence intervals. The sensitivity analysis with greater variation in viral loads is shown in S8 Fig.

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Impact of rapid testing (regular testing vs. pre-event testing)

For pre-pandemic levels of contacts, uptake of regular LFT testing must exceed 70% to reduce R below 1 if testing daily and must be above 80% if testing every 3 days, but even 100% uptake will not reduce R below 1 if testing weekly (Fig 6A). For lockdown levels of contacts, testing does not appreciably reduce R below its already low level. For contact rates equivalent to those under relaxed restrictions with schools open, testing every 3 days will reduce R below 1 if uptake exceeds 70% (Fig 6A). Pre-event testing acts similarly, with high uptake (> 90%) and testing for events with more than 10 others necessary to reduce R below 1 for pre-pandemic levels of contact, but only testing when attending events with more than 20 others and 60% adherence required during relaxed restrictions with schools open (Fig 6B). Both regular testing and pre-event testing reduce the rate of superspreading as defined as the proportion of individuals infecting over 10 others. However, there is also an increase in the proportion of individuals that infect no one. In some cases, this results in a decrease in k (greater overdispersion in the secondary infection distribution) (Fig 6). From the sensitivity analysis comparing the impact of regular testing with and without overdispersion in contacts (S9 Fig), we can see that ignoring overdispersion in contacts reduces the estimated impact of testing in terms of reducing R and decreasing the proportion of superspreading events. This demonstrates the importance of accounting for contact overdispersion when estimating the impact of interventions on secondary infections.

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Fig 6. Effect of regular and pre-event lateral flow testing on R, k, and the proportion infecting over 10 or 0 others for varying levels of uptake or adherence (horizontal axes) and background contact rates (columns).

A. Regular testing. B. Pre-event testing.

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