The effect of variation of individual infectiousness on SARS-CoV-2 transmission in households

  1. Tim K Tsang  Is a corresponding author
  2. Xiaotong Huang
  3. Can Wang
  4. Sijie Chen
  5. Bingyi Yang
  6. Simon Cauchemez
  7. Benjamin John Cowling  Is a corresponding author
  1. WHO Collaborating Centre for Infectious Disease Epidemiology and Control, School of Public Health, Li Ka Shing Faculty of Medicine, The University of Hong Kong, China
  2. Laboratory of Data Discovery for Health, China
  3. Mathematical Modelling of Infectious Diseases Unit, Institut Pasteur, France

Abstract

Quantifying variation of individual infectiousness is critical to inform disease control. Previous studies reported substantial heterogeneity in transmission of many infectious diseases including SARS-CoV-2. However, those results are difficult to interpret since the number of contacts is rarely considered in such approaches. Here, we analyze data from 17 SARS-CoV-2 household transmission studies conducted in periods dominated by ancestral strains, in which the number of contacts was known. By fitting individual-based household transmission models to these data, accounting for number of contacts and baseline transmission probabilities, the pooled estimate suggests that the 20% most infectious cases have 3.1-fold (95% confidence interval: 2.2- to 4.2-fold) higher infectiousness than average cases, which is consistent with the observed heterogeneity in viral shedding. Household data can inform the estimation of transmission heterogeneity, which is important for epidemic management.

Editor's evaluation

While it has been demonstrated that for SARS-CoV-2, a small fraction of individuals contributes to the majority of onward transmission, this heterogeneity is driven by multiple factors that span both biological and behavioral causes. By performing a solid meta-analysis of household transmission studies, the authors fit a household transmission model to the curated data to estimate variation in infectiousness which provides a valuable contribution to the existing knowledge base. By collating data from multiple studies, they are able to more fully investigate individual variability.

https://doi.org/10.7554/eLife.82611.sa0

Introduction

Characterizing transmission is critical to control the spread of an emerging infectious disease. The reproductive number is the widely adopted measure of infectiousness. However, it only measures the average number of secondary cases infected by an infected person, not the heterogeneity in the number of transmissions. Variation of individual infectiousness is particularly highlighted by superspreading events (SSEs), in which a minority of cases are responsible for a majority of transmission events. Such phenomena, illustrated by the ‘80/20 rule’ (i.e., 20% of cases responsible for 80% transmission; Adam et al., 2020; Sun et al., 2021), have been observed in emerging infectious disease outbreaks (Lloyd-Smith et al., 2005), including severe acute respiratory syndrome (SARS) (Shen et al., 2004), Middle East respiratory syndrome (Cauchemez et al., 2016; Cowling et al., 2015), and most recently the COVID-19 pandemic (Adam et al., 2020; Lau et al., 2020; Sun et al., 2021; Wong and Collins, 2020; Zhao et al., 2021). In these outbreaks, the proportion of cases attributed to 80% transmission, and the dispersion parameter that is estimated by fitting the negative binomial distribution to the number of secondary cases (Lloyd-Smith et al., 2005), are considered as measures of transmission heterogeneity.

However, the number of contacts per index cases is often not reported in SSE studies, and hence not incorporated in the analyses. In addition, SSE studies usually analyze clusters from different settings, in which the baseline transmission risk and density of exposure could be different (Thompson, 2021). Finally, studies of transmission heterogeneity that focus on SSEs described in the literature may suffer from publication bias, with larger clusters having higher probability of being observed and reported (Zhao et al., 2021). Therefore, the observed heterogeneity in the number of secondary cases could be the result of large number of contacts in SSE settings, or confounding from these factors (Althouse et al., 2020; Bagdasarian and Fisher, 2020), instead of variation in individual infectiousness.

Households are one of the most important settings for SARS-CoV-2 transmission, with 4- to 10-fold higher transmission risk than other places (Thompson, 2021). Hence, household transmission studies provide an ideal setting to quantify variations in individual infectiousness. In a household transmission study, an index case is identified, and their household contacts are followed up for 1–2 weeks, during which there is high transmission potential (Tsang et al., 2016). Therefore, the number of contacts is known while transmission risks and reporting biases can be controlled. We aim to characterize the variation of individual infectiousness by analyzing data from household transmission studies.

Results

We conducted a systematic review to gather information on the number of secondary cases with the number of household contacts for each household, in the form of number of households with X cases among households of size Y. In total, we identified 17 studies, comprising 13,098 index cases and 31,359 household contacts (Appendix 1—figure 1, Appendix 1—table 1; Bernardes-Souza et al., 2021; Carazo et al., 2022; Dattner et al., 2021; Han et al., 2022; Hart et al., 2022; Hsu et al., 2021; Hubiche et al., 2021; Koureas et al., 2021; Laxminarayan et al., 2020; Layan et al., 2022; Lyngse et al., 2022; Méndez-Echevarría et al., 2021; Posfay-Barbe et al., 2020; Reukers et al., 2021; Wilkinson et al., 2021). Most studies covered a period from January to November 2020, which was dominated by ancestral strains, except for Layan et al. (from December 2020 to April 2021) and Hsu et al. (from January 2020 to February 2021), which covered both ancestral strains and the alpha variant.

We then developed a statistical model to quantify the degree and the impact of variation of infectiousness of cases on transmission dynamics. The individual-based household transmission model describes the probability of infection of household contacts as depending on the time since infection in other infected persons in the household, so that infections from outside the household (community infections), or infections via other household contacts rather than the index case (tertiary infections) are allowed (Cauchemez et al., 2009; Tsang et al., 2014; Tsang et al., 2021). Therefore, the model could estimate the per-contact hazard of infection, which implicitly controls for number of household contacts in households. We extend this model by adding a random effect (δi) on the individual infectiousness of cases. Here, the relative infectiousness of case i compared with case j is exp(δi)/exp(δj). The parameter for variation in individual infectiousness (hereafter denoted as infectiousness variation, σvar) is the standard deviation (SD) of the random effect characterizing individual infectiousness, so that δi follows a normal distribution with mean equal to 0 and SD equal to σvar .

We separately fit the models to 14 studies with more than 150 contacts (Carazo et al., 2022; Dattner et al., 2021; Dutta et al., 2020; Han et al., 2022; Hart et al., 2022; Hubiche et al., 2021; Koureas et al., 2021; Laxminarayan et al., 2020; Layan et al., 2022; Lyngse et al., 2022; Méndez-Echevarría et al., 2021; Reukers et al., 2021; Tsang et al., 2022; Wilkinson et al., 2021; Figure 1, Appendix 1—table 2). For 12 studies out of 14, models with infectiousness variation perform substantially better (range of ∆DIC: 5.8–268) (Figure 2, Appendix 1—table 2). From these 12 studies, the estimated infectiousness variation (σvar) ranged from 1.03 to 2.83. This suggests that, the 20% most infectious cases are 2.4- to 10-fold more infectious than the average case. Based on the two largest studies with 6782 and 3727 households (Carazo et al., 2022; Lyngse et al., 2022), the estimated infectiousness variation is 1.48 (95% credible interval [CrI]: 1.29, 1.7) and 1.41 (95% CrI: 1.19, 1.72), suggesting that, among all cases, the 20% most infectious are 3.5-fold (95% CrI: 3.0- to 4.2-fold) and 3.3-fold (95% CrI: 2.7- to 4.3-fold) more infectious than the average case. The estimated daily probability of infection from outside the household and estimated person-to-person transmission probability within households range from 0.0003 to 0.017, and from 0.06 to 0.51, respectively. The estimates of parameters for the relationship between number of contacts and transmission (larger value indicates stronger inverse association) range from 0.43 to 0.92, except for the study by Layan et al., 2022, where it is equal to 0.2. For all studies, the predicted final size distribution is consistent with the observed data and the model fit is judged adequate (Appendix 1—Tables 3–7, Appendix 1—figure 2). Simulation studies demonstrated that there is no important systematic bias, with 88–100% (depending on the parameter) of the 95% credible intervals covering the simulation value (Appendix 1—table 8). This suggests that the algorithm could estimate adequately the posterior distribution. We conduct a sensitivity analysis that assumes the individual infectiousness of cases follows a Gamma distribution, but the fit worsens substantially (Appendix 1—table 9).

Summary of statistics for 17 identified studies.

Figure shows the average number of contacts and standard deviation (SD) of number of contact, SD of number of secondary cases per index cases (σsec), and secondary attack rate (SAR) for 17 identified studies.

Modeling results of household transmission dynamics and infectiousness variation.

Figure shows the estimates of infectiousness variation (σvar), the estimated probability of infection from community and estimated probability of infection from households, and the reduction in deviance information criteriion (DIC) compared with the model without infectiousness variation. Models are fitted separately to 14 household transmission studies.

We conduct random effects meta-analyses on estimates of individual infectiousness from the 14 identified studies. The pooled estimate of infectiousness variation is 1.33 (95% confidence interval [CI]: 0.95, 1.70), suggesting that the 20% most infectious cases are 3.1-fold (95% CI: 2.2- to 4.2-fold) more infectious than the average case (Figure 3). Based on this fitted distribution, we estimate that 5.9% (95% CI: 1.4%, 11.1%) and 14.9% (95% CI: 7.2%, 20.7%) of cases could be at least 8- and 4-fold more infectious than average cases, respectively.

Estimate distribution of relative infectiousness based on the pooled estimate.

Red line indicates the estimated distribution and the gray lines indicate the associated uncertainty. Black dashed line indicates average infectiousness (relative infectiousness equal to 1), while the purple and blue dashed lines indicate top 20% and 10% infectiousness, respectively.

We further explore if the secondary attack rate (SAR, the proportion of infected contacts), and the SD of the distribution of number of secondary cases (σsec) (Figure 4; Appendix 1—table 10) may be correlated with the infectiousness variation. In meta-regression, we find the infectiousness variation is associated with SAR (Appendix 1—table 11). We estimate that doubling SAR are associated with 0.55 (95% CI: 0.21, 0.89) unit decrease in infectiousness variation, with R-squared equal to 67% respectively. In addition, we find that higher infectiousness variation is associated with only using PCR to ascertain secondary cases. Regarding other statistics, we find that σsec is positively associated with mean and SD of number of contacts. Other than these associations, we find no association between these statistics and implementation of lockdown, ascertainment method of index and secondary cases, and the circulating virus of SARS-CoV-2 in the study period.

Relationship between infectiousness variation and statistic.

In each panel, numbers represent the observed corresponding relationship for the identified studies. Panels A and B show the relationship between infectiousness variation (σvar) and secondary attack rate (SAR) and standard deviation (SD) of number of secondary cases per index cases (σsec). In the bottom, numbers in bracket indicate the number of household contacts in corresponding studies.

Discussion

In this study, we characterize the impact of variation of individual infectiousness on heterogeneity of transmission of COVID-19 in households. We demonstrate that it can be estimated from household data using a modeling approach. The pooled estimate of infectiousness variation from 14 studies suggests that the 20% most infectious cases have 3.1-fold (95% CI: 2.2- to 4.2-fold) higher infectiousness compared with average index cases. This implies there is substantial variation in individual infectiousness of cases in households. Given that we focused our analysis on households with known number of contacts in studies conducted in the early outbreaks caused by ancestral strains, the estimated infectiousness variations are corrected for the variations caused by number of contacts and transmission risks in different settings, difference in pre-existing immunity among contacts (almost everyone is naïve and unvaccinated), and the difference in transmissibility among variants. Hence, this estimated infectiousness variation measures the variation caused by difference in individuals, which may be contributed by both biological factors and host behaviors, but not other potential confounding factors mentioned above.

Regarding host behaviors, multiple contact patterns, particularly by age, could contribute to the variations in infectiousness of cases. For example, mother-child contacts are usually more intense than father-child contacts, and adult cases are more capable of self-isolation within a household compared with children. Furthermore, contact pattern studies suggested that school-age children and young adults tended to mix with people of the same age (Mossong et al., 2008; Mousa et al., 2021). Also, the duration of contact could vary, which could also contribute to the variations in infectiousness of cases (Toth et al., 2015). It should be noted that contacts have different features, including their number, frequency, and duration that may contribute to variations in infectiousness. Previous studies also suggest that the relative importance of contact characteristics on transmission may be different among viruses (De Cao et al., 2014). Individual behaviors may also be influenced by control measures and recommendation from public health agencies. For example, lockdown and stay-at-home orders may increase the time spending at home. Mask-wearing when in contact with other household members, using separate bedrooms and bathrooms, avoiding having meals together may reduce the transmission in households (Ng et al., 2021). Also, social disparity such as occupation may increase or reduce the risk of transmission in households, including the availability of personal protective equipment (PPE), or being healthcare worker (Yang et al., 2021). These factors may have impact of the transmission risk and hence SAR in households. In addition, heterogeneity in these different factors may have contributed infectiousness variation.

Biological factors may also contribute to such variations. For example, the SAR for index cases with fever and cough are 1.4- and 1.3-fold higher than index cases without fever and cough, respectively (Madewell et al., 2021). Viral shedding is used as a proxy measure of infectiousness, and consistently it also has substantial variations, as suggested by 11 systematic reviews (Appendix 1—Tables 12–14). Regarding the magnitude of viral shedding, studies report high variations of temporal viral shedding patterns among individuals (He et al., 2020; Jones et al., 2021; Sun et al., 2022). In addition, the duration of viral shedding can be highly heterogeneous, with pooled estimates of mean durations ranging from 11.1 to 30.3 days, and almost all reviews reporting high heterogeneity of estimates (Cevik et al., 2021; Chen et al., 2021b; Díaz et al., 2022; Fontana et al., 2021; Li et al., 2020b; Okita et al., 2022; Qutub et al., 2022; Rahmani et al., 2022; Xu et al., 2020; Yan et al., 2021; Zhang et al., 2021). Such heterogeneities still exist in subgroup analyses by age and severity (Cevik et al., 2021; Fontana et al., 2021; Okita et al., 2022; Qutub et al., 2022; Rahmani et al., 2022; Xu et al., 2020; Yan et al., 2021). The infectious period, proxied by duration of replicant competent virus isolation, is also heterogeneous (Rahmani et al., 2022; Appendix 1—table 14). One review also suggests that heterogeneity in viral shedding is an intrinsic virological factor facilitating higher dispersion parameter for SARS-CoV-2 if we compare it with the corresponding patterns in SARS-CoV-1 and pandemic influenza A(H1N1)pdm09 (Chen et al., 2021a).

The observed variation in individual infectiousness is consistent with past analyses of the dispersion parameter in a negative binomial distribution fitted to number of secondary cases per index case of COVID-19 (Sun et al., 2021; Wong et al., 2015). However, in these studies, the number of contacts was not considered. Therefore, the observed variations may not apply directly to households with limited number of contacts. It should be noted that the SD of the distribution of number of secondary cases (σsec) is only weakly correlated with infectiousness variation. This is because σsec is highly correlated with the mean and SD of the number of contacts, suggesting that it may depend on distribution of number of contacts, and hence may not be comparable among studies. Infectiousness variation is also correlated with the SAR. When SAR is higher, it is expected that more contacts are infected and therefore the observed number of secondary cases is less heterogeneous (Adam et al., 2022). We find higher infectiousness variation is associated with only using PCR to confirm secondary cases. One potential reason is that using other methods may lower the sensitivity of detecting infection, resulting in lower estimates of SAR and hence higher estimates of infectiousness variation. Further investigations are needed to explore roles of other potential factors affecting infectiousness variation, such as contact frequency among different regions (Mousa et al., 2021).

An important limitation of our study is that we do not have individual-level data. Therefore, we are unable to determine the impact of demographic factors like age and sex on infectiousness variation. Also, we cannot disentangle the host behaviors from biological factors. Previous analysis suggested no evidence of the impact of age on infectiousness of cases (Lau et al., 2020; Sun et al., 2021). In addition, we could not include factors affecting susceptibility to infection. Our estimates of infectiousness variation should be interpreted in light of these limitations: they capture heterogeneity in infectiousness due to demographic, host, and biological factors. However, in one study that included susceptibility component in the estimation of individual infectiousness, substantial heterogeneity remained with 20% of cases estimated to contribute to 80% of transmission (Tsang et al., 2022). Second, the recruitment methods among studies are different. This may affect the comparability of the results, although all index cases are laboratory-confirmed in all studies (Tsang et al., 2021). Third, we assumed that risks of infection from community for all households are the same, but there were different factors that may affecting this, including occupations, such as healthcare workers, social economic status that related to assess to PPEs (Yang et al., 2021). Finally, most of our identified studies were conducted in the period of circulation of ancestral strains, and therefore the identified infectiousness variation may not be directly applicable to other variants.

In conclusion, we developed a modeling approach to estimate variation in individual infectiousness from household data. Result indicates that there is substantial variation in individual infectiousness, which is important for epidemic management.

Materials and methods

Study design

The aim of this study was to develop a statistical model to quantify the variation of individual infectiousness in households, based on publicly available information. An index case was defined as the first detected case in a household, while secondary cases were defined as the identified infected household contacts of the index case. We conducted a systematic review to collect household studies with at least 30 households, reporting the number of secondary cases with number of household contacts for each household for COVID-19, in the form of number of households with X cases among households of size Y. For each study, we also extracted the study period, the coverage of tests of household contacts, the case ascertainment methods, the circulating virus of SARS-CoV-2, and the public health and social measures in the study period. This information was used as an input of modeling analyses in this study. Details of systematic review could be found in Appendix 1.

Estimation of variation of individual infectiousness in households

To determine if there are variations of individual infectiousness of cases, we used an individual-based household transmission model (; Tsang et al., 2014; Tsang et al., 2021). The model described the probability of infection of household contacts as depending on the time since infection in other infected people in the household, while infections from outside the household (community infections), or infections via other household contacts rather than the index case (tertiary infections) are allowed. We extended the model by adding a random effect parameter (δi) on the individual infectiousness of each case. In the model, the hazard of infection of individual j at time t from an infected household member i, with infection time ti in household k, was

λijt=λhXkβ *expδi*f(t-ti)

where λh was the baseline hazard, δi followed a normal distribution with mean 0 and SD σvar , which quantified the variation of individual infectiousness (hereafter denoted as infectiousness variation). The relative infectiousness of case i compared with case j was exp(δi)/exp(δj).

Xk was the number of household contacts. β was the parameter describing the relationship between number of household contacts and transmission rate. It ranged from 0 to 1, with 0 indicating that the transmission rate was independent of number of household contacts while 1 indicated that the transmission rate was inversely proportional to number of household contacts (i.e. dilution effect of the contact time per contact which was lower when the number of household contacts is higher). f(·) was the infectiousness profile since infection generated from the assumed incubation period (mean equal to 5 days) and infectious period (mean equal to 13) (He et al., 2020; Jing et al., 2020; Li et al., 2020a; Appendix 1—table 15). Extracted distribution were summarized in Appendix 1—table 15, and the shape of the assumed function was plotted on Appendix 1—figure 3.

Since the data were extracted from publication, the infection time of all cases was unavailable for all studies. Also, the individual infectiousness parameters δi for cases were augmented variables. Therefore, we conducted our inference under a Bayesian framework using data augmentation Markov chain Monte Carlo (MCMC) algorithm to joint estimate the model parameters, the missing infection time, and augmented variables using metropolis-hasting algorithm. We separately fitted this model to identified studies. We assessed the model adequacy by comparing the observed and expected number of infections in households by household size. We compared the model with or without the random effect for variation of individual infectiousness by using deviance information criterion (DIC) (Spiegelhalter et al., 2002). Smaller DIC indicated a better model fit. DIC difference >5 was considered as substantial improvement (Spiegelhalter et al., 2000). Details of the model and inference could be found in Appendix 1. We conducted a sensitivity analysis that used a Gamma distribution, instead of exponential of normal distribution (Appendix 1).

Meta-analysis and meta-regression

We conducted random effects meta-analyses on identified studies to obtain pooled estimates of individual infectiousness, using the inverse variance method and restricted maximum likelihood estimator for heterogeneity (Hedges and Vevea, 1998; Langan et al., 2019; Thompson and Sharp, 1999; Veroniki et al., 2019). Cochran Q test and the I2 statistic were used to identify and quantify heterogeneity among included studies (Cochran, 1954; Higgins et al., 2003). An I2 value more than 75% indicates high heterogeneity (Higgins et al., 2003). We conducted a meta-regression analysis to explore the association between infectiousness variation (σvar, SD) of the distribution of number of secondary cases (σsec, and SAR), and further including the following factors: the mean and SD of number of household contacts, implementation of lockdown, ascertainment method of index and secondary cases, only ancestral strains are circulating in study period, and all household contacts were tested.

Data availability

All data in this study are publicly available since they are extracted from published articles. Summarized data for analysis could be downloaded from https://github.com/timktsang/covid19_transmission_heterogeneity, (copy archived at swh:1:rev:634390fa66a4bfb998da691e7cd81cf45e38c6db; Tsang, 2022).

Appendix 1

Systematic review

We conducted a systematic review following the Preferred Reporting Items for Systematic Review and Meta-analysis (PRISMA) statement (Moher et al., 2009). A standardized search was done in PubMed, Embase, and Web of Science, using the search term ‘(“COVID-19” OR “SARS-CoV-2”) AND (“household” OR “family”) AND (“transmission”)’. The search was done on 22 June, 2022, with no language restrictions. Additional relevant articles from the reference sections were also reviewed.

Two authors (XH and CW) independently screened the titles and extracted data from the included studies, with disagreement resolved by consensus together with a third author (TKT). Studies identified from different databases were de-duplicated.

Household transmission model

Our aim was to explore the role of variation of individual infectiousness on transmission, with accounting for the variation and number of contacts, and the transmission probability. Therefore, we extended the household transmission model to account for variation of individual infectiousness of index cases (Cauchemez et al., 2009). We added a random effect on infectiousness of cases (Tsang et al., 2014). In the model, the hazard of infection of individual j at time t from an infected household member i, with infection time ti in household k, was

λijt=λhXkβ *expδi*f(t-ti)

where λh was the baseline hazard, δi was the relative individual infectiousness of cases, which followed a normal distribution with mean 0 and SD σvar .

Xk was the number of household contacts. β was the parameter describing the relationship between number of household contacts and transmission rate. It ranged from 0 to 1, with 0 indicating that the transmission rate is independent of number of household contacts while 1 indicates that the transmission rate is inversely proportional to number of household contacts (i.e. dilution effect of the contact time per contact which is lower when the number of household contacts is higher). f(·) was the infectiousness profile since infection generated from the assumed incubation period (mean equal to 5 days) and infectious period (mean equal to 13) (He et al., 2020; Jing et al., 2020; Li et al., 2020b).

The daily hazard of infection from outside the household is assumed to be λct=λc . Hence, the total hazard infection of individual j on day t was

λjt=λc+i:infectedλijt

Based on the transmission model, the probability that an individual i was infected with infection time ti0 was

P(yi=1,ti=ti0)=[1exp(λi(ti0))][exp(d=zi1ti01λi(d))]

where zi1 was the start of the follow-up day of individual i. Denote the total follow-up time for an individual i as zi2 , then the probability that an individual i did not get infected within the follow-up period was

P(yi=0,ti=ti0=zi2+1)=exp(d=zi1ti01λi(d))

Hence the log-likelihood function L was

L=i:yi=1log(1exp(λi(ti0))id=zi1ti01λi(d)

Index cases did not contribute to the likelihood and hence the summation was only on household contacts.

Inference, model comparison, and validation

The infection time for cases were unavailable in the data extracted from publication. Also, the individual infectiousness parameter δi was an augmented variable. Therefore, we conducted our inference under a Bayesian framework with using data augmentation MCMC algorithm to joint estimate the model parameters, the missing infection time, and the relative individual infectiousness with using metropolis-hasting algorithm. For the parameter that only takes positive value, we used a vague Uniform(0,10) prior. For the parameter for relationship between number of household contacts and transmission, we used Uniform(0,1). The algorithm runs for 50,000 iterations after a burn-in of 50,000 iterations. Converge was visually assessed. One run takes about 3 hr on our typical desktop computer. We fitted the model to all studies, but only 14 of them could converge and provide robust estimates of individual infectiousness.

We assessed the model adequacy by comparing the observed and expected number of infections in households by household size. We simulated 10,000 datasets with a structure identical to that of the observed data (in terms of number of household contacts), with parameters randomly drawn from the posterior distribution of model parameter.

We compared the model with or without the random effect for variation of individual infectiousness by using DIC (Spiegelhalter et al., 2002). Smaller DIC indicated a better model fit. DIC difference >5 was considered as substantial improvement (Spiegelhalter et al., 2000). Since the likelihood of observed data could not be directly evaluated for a given model (Celeux et al., 2006), we used an importance sampling approach to estimate the likelihood for the observed data and evaluate the DIC (Cauchemez et al., 2014; Liu, 2001). For each household, we simulated 2000 datasets, with parameters randomly drawn from the posterior distribution. Then we compared the observed data and simulated data. The contribution to the likelihood of a household was equal to the proportion of simulated data with infection status that exactly matched the observed data, for all household members. To avoid the problem of 0-valued likelihood, we used the approach developed by Cauchemez et al., 2014, and assumed the sensitivity and specificity for diagnosing a case were both 99.99%.

Comparison of modeling approach

Given that the infection time (or information that could inform infection time, such as symptom onset time) were not available for all studies, it may be possible to consider them as final size data, which may be analyzed by chain binomial model {Longini, 1988 #693;Longini, 1982 #335;Klick, 2011 #338;O'Neill, 2000 #368}. Including covariates affecting susceptibility is still possible, but showed to be difficult since the number of parameters increases exponentially by groups and type of infection {Klick, 2011 #338}. However, the model equation is about the escape probability of infection, which did not include the characteristic of infectee and therefore it is not possible to add covariates affecting infectiousness.

Sensitivity analysis

Our model assumed that the individual infectiousness parameter δi followed a normal distribution, and it was multiplied to the hazard in the form of exp(δi), therefore, the individual infectiousness essentially followed a lognormal distribution, which had a long tail. Therefore, we conducted a sensitivity analysis as follows:

λij(t)=λhXkβ δif(tti)

Here, δi followed a gamma distribution with mean 1, and SD σvar , which had a shorter tail compared with lognormal distribution. Model comparison suggested that assuming gamma distribution for individual infectiousness parameter performed substantially worse, compared with the lognormal distribution in the main analysis (Appendix 1—table 9).

Appendix 1—figure 1
Process of systematic review.
Appendix 1—figure 2
Correlation plots for the posterior distribution of model parameters in Lyngse et al., Carazo et al., Laxminarayan et al., and Dattner et al.
Appendix 1—figure 3
The density plot of the distribution of infectiousness profile since infections, used in the modeling analysis.
Appendix 1—table 1
Summary of characteristic of identified studies.
Author (year)LocationStudy periodCase ascertainment methodTest coverage of identified contactsSARS-COV-2 variantPublic health and social measures
Lyngse et al., 2022DenmarkFebruary 2020 to August 2020Index: RT-PCR
Secondary: RT-PCR
All contacts were testedAncestral strainsLockdown
Carazo et al., 2022CanadaMarch 2020 to June 2020Index: RT-PCR
Secondary: Symptom-based
NAAncestral strainsHandwashing, mask-wearing, and physical distancing
Laxminarayan et al., 2020IndiaMarch 2020 to July 2020Index: RT-PCR
Secondary: RT-PCR
All contacts were testedAncestral strainsLockdown, social distancing, contact tracing
Dattner et al., 2021IsraelMarch 2020 to June 2020Index: RT-PCR and Serology
Secondary: RT-PCR and Serology
All contacts were testedAncestral strainsLockdown
Layan et al., 2022IsraelDecember 2020 to April 2021Index: RT-PCR
Secondary: RT-PCR
All contacts were testedAncestral strains, alphaVaccination
Hart et al., 2022UKMarch 2020 to November 2020Index: RT-PCR
Secondary: RT-PCR and antibody test
All contacts were testedAncestral strainsIsolation
Hubiche et al., 2021CanadaApril 2020 to June 2020Index: Symptom-based and RT-PCR and serology
Secondary: Symptom-based and RT-PCR and serology
NAAncestral strainsClosure of school
Wilkinson et al., 2021Manitoba, CanadaMid-January 2020 to late
March 2020
Index: NAAT assay
Secondary: NAAT assay
Only symptomatic contacts were testedAncestral strainsIsolation and contact tracing
Tsang et al., 2022Shandong Province, ChinaJanuary 2020 to May 2020Index: RT-PCR
Secondary: RT-PCR
All contacts were testedAncestral strainsIsolation, mask-waring, social distancing
Reukers et al., 2021NetherlandsMarch 2020 to May 2020Index: RT-PCR
Secondary: RT-PCR
NAAncestral strainsSocial distancing
Han et al., 2022NetherlandsMarch 2020 to April 2020Index: RT-PCR and serology
Secondary: RT-PCR and serology
All contacts were testedAncestral strainsSocial distancing, self-quarantine and self-isolation orders, closure of schools, bars and restaurants, and urging people to work from home
Méndez-Echevarría et al., 2021Madrid, SpainMarch 2020 to May 2020Index: RT-PCR and serology
Secondary: RT-PCR and serology
All contacts were testedAncestral strainsLockdown
Dutta et al., 2020Rajasthan, IndiaMay 2020 to July 2020Index: RT‐PCR
Secondary: RT‐PCR
All contacts were testedAncestral strainsLockdown and stay-at-home orders, physical distancing
Koureas et al., 2021GreeceApril 2020 to June 2020Index: RT-PCR
Secondary: RT-PCR
All contacts were testedAncestral strainsQuarantine, screening, movement restrictions and gathering prohibition, isolation
Bernardes-Souza et al., 2021; Méndez-Echevarría et al., 2021BrazilMay 2020 to June
2020
Index: Serology and RT-PCR
Secondary: Serology
All contacts were testedAncestral strainsLockdown, gathering restrictions and face mask mandates
Posfay-Barbe et al., 2020SwitzerlandMarch 2020 to April 2020Index: RT-PCR
Secondary: RT-PCR
NAAncestral strainsClosure of schools, daycares, restaurants, bars, and shops, social distancing
Hsu et al., 2021Taiwan, ChinaJanuary 2020 to February 2021Index: RT-PCR
Secondary: RT-PCR
Only symptomatic contacts were testedAncestral strains, alphaMask-wearing, hand washing, and social distancing
Appendix 1—table 2
Summary of model estimates.
ArticleEstimates of infectiousness variationEstimates of probability of infection from community (10–2)Estimates of probability of infection from householdRelationship between transmission and number of contacts (β)
Lyngse et al.1.48 (1.29, 1.7)0.06 (0, 0.16)0.1 (0.08, 0.12)0.72 (0.59, 0.89)
Carazo et al.1.41 (1.19, 1.72)0.2 (0.01, 0.43)0.3 (0.24, 0.34)0.75 (0.62, 0.93)
Laxminarayan et al.2.44 (1.98, 3.23)0.03 (0, 0.11)0.04 (0.01, 0.07)0.92 (0.69, 1)
Dattner et al.1.12 (0.65, 1.69)0.63 (0.12, 1.06)0.31 (0.19, 0.42)0.65 (0.45, 0.91)
Layan et al.1.12 (0.74, 1.76)0.3 (0.02, 0.96)0.26 (0.15, 0.39)0.2 (0.01, 0.6)
Hart et al.0.35 (0.11, 0.96)0.74 (0.05, 2.12)0.51 (0.32, 0.65)0.72 (0.32, 0.98)
Hubiche et al.1.03 (0.45, 2.68)1.24 (0.14, 2.48)0.32 (0.06, 0.57)0.67 (0.13, 0.98)
Wilkinson et al.1.05 (0.12, 3.51)0.32 (0.02, 0.91)0.07 (0, 0.18)0.43 (0.02, 0.97)
Tsang et al.2.83 (1.48, 4.73)0.45 (0.05, 1.03)0.06 (0, 0.26)0.66 (0.06, 0.99)
Reukers et al.1.79 (0.72, 4.5)1.53 (0.18, 2.93)0.22 (0.01, 0.6)0.67 (0.07, 0.98)
Han et al.1.71 (0.51, 4.13)1.67 (0.19, 2.99)0.21 (0.02, 0.61)0.69 (0.07, 0.99)
Méndez-Echevarría et al.0.51 (0.13, 1.7)0.91 (0.05, 2.53)0.28 (0.06, 0.52)0.57 (0.04, 0.98)
Dutta et al.1.53 (0.5, 4.2)1.1 (0.12, 2.26)0.24 (0.01, 0.63)0.78 (0.15, 0.99)
Koureas et al.1.88 (0.6, 3.66)0.84 (0.1, 1.76)0.13 (0.01, 0.42)0.44 (0.02, 0.96)
Appendix 1—table 3
Model adequacy check for Lyngse et al.

Each element of the table has the format observed frequency – expected (posterior mean) frequency (95% credible interval).

Number of household contactsNumber of infected household contacts
012345
12366–2377 (2319, 2433)569–558 (502, 616)NANANANA
21117–1105 (1070, 1138)227–227 (198, 259)77–88 (69, 109)NANANA
31135–1118 (1077, 1156)214–235 (203, 267)89–86 (67, 106)41–41 (27, 56)NANA
4521–528 (501, 555)119–115 (94, 137)40–42 (30, 56)25–19 (11, 30)11–10 (4, 18)NA
5161–167 (152, 181)42–38 (27, 50)14–14 (7, 22)7–7 (2, 12)7–3 (0, 8)0–2 (0, 5)
Appendix 1—table 4
Model adequacy check for Carazo et al.

Each element of the table has the format observed frequency – expected (posterior mean) frequency (95% credible interval).

Number of household contactsNumber of infected household contacts
012345
1803–814 (765, 862)532–521 (473, 570)NANANANA
2476–454 (423, 485)179–186 (160, 212)154–169 (144, 195)NANANA
3518–520 (483, 557)201–202 (175, 232)132–130 (107, 155)133–131 (106, 158)NANA
4217–221 (197, 245)85–87 (70, 106)52–52 (39, 67)43–39 (27, 52)45–41 (28, 56)NA
576–74 (61, 88)30–30 (20, 41)20–18 (10, 26)13–12 (6, 20)7–10 (5, 17)10–11 (5, 18)
Appendix 1—table 5
Model adequacy check for Laxminarayan et al.

Each element of the table has the format observed frequency – expected (posterior mean) frequency (95% credible interval).

Number of household contactsNumber of infected household contacts
0123456789
1124–134 (122, 144)37–27 (17, 39)NANANANANANANANA
2135–137 (125, 147)12–18 (10, 27)20–12 (6, 20)NANANANANANANA
3188–176 (163, 188)13–22 (13, 33)11–9 (4, 16)5–9 (4, 17)NANANANANANA
4127–118 (108, 128)11–15 (8, 23)5–5 (1, 11)3–3 (0, 7)1–4 (1, 10)NANANANANA
575–76 (67, 84)12–10 (4, 16)3–3 (0, 8)1–2 (0, 5)1–1 (0, 4)3–2 (0, 6)NANANANA
641–40 (34, 46)7–5 (1, 10)2–2 (0, 5)0–1 (0, 3)0–0 (0, 3)0–0 (0, 2)1–1 (0, 3)NANANA
732–31 (25, 36)3–4 (1, 9)2–1 (0, 4)1–1 (0, 3)1–0 (0, 2)0–0 (0, 2)0–0 (0, 2)0–0 (0, 3)NANA
810–13 (10, 16)2–2 (0, 5)3–0 (0, 2)1–0 (0, 2)0–0 (0, 1)1–0 (0, 1)0–0 (0, 1)0–0 (0, 1)0–0 (0, 1)NA
914–16 (12, 20)2–2 (0, 6)1–1 (0, 3)2–0 (0, 2)1–0 (0, 2)0–0 (0, 1)0–0 (0, 1)0–0 (0, 1)0–0 (0, 1)1–0 (0, 2)
Appendix 1—table 6
Model adequacy check for Dattner et al.

Each element of the table has the format observed frequency – expected (posterior mean) frequency (95% credible interval).

Number of household contactsNumber of infected household contacts
0123456789
185–93 (77, 108)73–65 (50, 81)NANANANANANANANA
246–42 (32, 52)21–25 (17, 34)19–19 (11, 28)NANANANANANANA
338–29 (21, 38)13–18 (11, 25)9–12 (6, 19)8–9 (4, 15)NANANANANANA
424–24 (16, 32)19–15 (9, 23)11–10 (5, 17)8–7 (3, 13)1–5 (1, 11)NANANANANA
525–21 (14, 30)15–15 (9, 22)12–10 (5, 17)7–7 (3, 13)3–5 (2, 10)2–4 (1, 9)NANANANA
613–16 (9, 23)19–12 (6, 18)3–8 (3, 14)5–6 (2, 11)3–5 (1, 9)5–3 (0, 7)5–2 (0, 6)NANANA
75–9 (4, 15)6–7 (3, 13)5–5 (2, 10)7–4 (1, 8)4–3 (0, 7)4–2 (0, 6)4–2 (0, 5)0–1 (0, 4)NANA
810–8 (3, 14)7–7 (3, 12)3–5 (1, 10)4–4 (1, 8)0–3 (0, 7)3–2 (0, 6)3–2 (0, 5)4–1 (0, 4)0–1 (0, 4)NA
96–6 (2, 12)3–6 (2, 10)4–4 (1, 9)6–3 (0, 7)4–3 (0, 6)1–2 (0, 5)2–2 (0, 5)2–1 (0, 4)2–1 (0, 3)0–0 (0, 3)
Appendix 1—table 7
Comparison of model estimates from 100k Markov chain Monte Carlo (MCMC) iterations and 500k MCMC iterations.
ArticleNumber of MCMC iterationsEstimates of infectiousness variationEstimates of probability of infection from community (10–2)Estimates of probability of infection from householdRelationship between transmission and number of contacts (β)
Lyngse et al.100,0001.48 (1.29, 1.7)0.06 (0, 0.16)0.11 (0.08, 0.13)0.72 (0.59, 0.89)
500,0001.49 (1.29, 1.84)0.07 (0, 0.2)0.1 (0.07, 0.13)0.74 (0.6, 0.92)
Carazo et al.100,0001.41 (1.19, 1.72)0.2 (0.01, 0.43)0.35 (0.27, 0.42)0.75 (0.62, 0.93)
500,0001.44 (1.19, 1.78)0.21 (0.02, 0.46)0.34 (0.26, 0.41)0.77 (0.62, 0.95)
Laxminarayan et al.100,0002.44 (1.98, 3.23)0.03 (0, 0.11)0.04 (0.01, 0.08)0.92 (0.69, 1)
500,0002.47 (1.99, 3.23)0.03 (0, 0.11)0.04 (0.01, 0.07)0.92 (0.67, 1)
Dattner et al.100,0001.12 (0.65, 1.69)0.63 (0.12, 1.06)0.37 (0.21, 0.54)0.65 (0.45, 0.91)
500,0001.06 (0.65, 1.55)0.57 (0.1, 1)0.38 (0.24, 0.54)0.63 (0.43, 0.88)
Appendix 1—table 8
Simulation study for validating the estimates and the corresponding power.
ParameterSimulation valueMean estimateProportion covered (over 50)
Lyngse et al.
σvar: Infectiousness variation1.481.570.94
λc: hazard of infection from outside the household (10–2)0.060.120.88
λh: hazard of infection from an infected household member0.110.090.92
β: relationship between number of household contacts and transmission rate0.720.750.96
Carazo et al.
σvar: Infectiousness variation1.411.490.94
λc: hazard of infection from outside the household (10–2)0.20.240.96
λh: hazard of infection from an infected household member0.350.330.92
β: relationship between number of household contacts and transmission rate0.750.770.96
Laxminarayan et al.
σvar: Infectiousness variation2.442.60.94
λc: hazard of infection from outside the household (10–2)0.030.050.96
λh: hazard of infection from an infected household member0.040.030.9
β: relationship between number of household contacts and transmission rate0.920.821
Dattner et al.
σvar: Infectiousness variation1.121.180.92
λc: hazard of infection from outside the household (10–2)0.630.690.92
λh: hazard of infection from an infected household member0.370.360.96
β: relationship between number of household contacts and transmission rate0.650.680.98
Appendix 1—table 9
A sensitivity analysis of using normal distribution (main analysis) instead of lognormal distribution (sensitivity analysis) for individual infectiousness.
ArticleModelEstimates of infectiousness variationEstimates of probability of infection from community (10–2)Estimates of probability of infection from householdRelationship between transmission and number of contacts (β)Difference in DIC
Lyngse et al.Main analysis1.48 (1.29, 1.7)0.06 (0, 0.16)0.11 (0.08, 0.13)0.72 (0.59, 0.89)
Sensitivity analysis0.93 (0.84, 1.03)0.03 (0, 0.11)0.06 (0.05, 0.07)0.7 (0.58, 0.83)2821
Carazo et al.Main analysis1.41 (1.19, 1.72)0.2 (0.01, 0.43)0.35 (0.27, 0.42)0.75 (0.62, 0.93)
Sensitivity analysis1.05 (0.92, 1.21)0.06 (0, 0.24)0.18 (0.16, 0.2)0.71 (0.59, 0.84)2569
Laxminarayan et al.Main analysis2.44 (1.98, 3.23)0.03 (0, 0.11)0.04 (0.01, 0.08)0.92 (0.69, 1)
Sensitivity analysis1.31 (1.14, 1.48)0.02 (0, 0.08)0.05 (0.03, 0.06)0.91 (0.66, 1)818
Dattner et al.Main analysis1.12 (0.65, 1.69)0.63 (0.12, 1.06)0.37 (0.21, 0.54)0.65 (0.45, 0.91)
Sensitivity analysis0.66 (0.47, 0.88)0.3 (0.02, 0.71)0.18 (0.13, 0.22)0.56 (0.4, 0.75)908
Appendix 1—table 10
Summary of characteristic of identified studies.

SD: standard deviation.

ArticleNumber of householdsNumber of contactsNumber of secondary casesMean number of contactsSD of number of contactSecondary attack rate (SAR)SD of number of secondary cases (σsec)
Lyngse et al.678214,23319022.1 (2.07, 2.13)1.17 (1.15, 1.19)0.13 (0.13, 0.14)0.6 (0.59, 0.61)
Carazo et al.3727846025742.27 (2.23, 2.31)1.19 (1.16, 1.21)0.3 (0.29, 0.31)0.97 (0.95, 0.99)
Laxminarayan et al.91531132833.4 (3.28, 3.53)1.94 (1.86, 2.04)0.09 (0.08, 0.1)0.81 (0.77, 0.85)
Dattner et al.59122117203.74 (3.54, 3.94)2.5 (2.36, 2.65)0.33 (0.31, 0.35)1.6 (1.51, 1.7)
Layan et al.2086702643.22 (3.01, 3.44)1.58 (1.45, 1.75)0.39 (0.36, 0.43)1.59 (1.45, 1.76)
Hart et al.1724331942.52 (2.34, 2.69)1.16 (1.05, 1.29)0.45 (0.4, 0.5)1.11 (1, 1.24)
Hubiche et al.1032911192.83 (2.6, 3.05)1.18 (1.04, 1.37)0.41 (0.35, 0.47)1.14 (1, 1.32)
Wilkinson et al.95220262.32 (2.02, 2.61)1.47 (1.28, 1.71)0.12 (0.08, 0.17)0.66 (0.58, 0.77)
Tsang et al.47189384.02 (3.39, 4.66)2.22 (1.85, 2.79)0.2 (0.15, 0.27)1.31 (1.09, 1.65)
Reukers et al.55187783.4 (3.11, 3.69)1.1 (0.93, 1.35)0.42 (0.35, 0.49)1.33 (1.12, 1.64)
Han et al.55185783.36 (3.08, 3.65)1.08 (0.91, 1.33)0.42 (0.35, 0.5)1.27 (1.07, 1.57)
Méndez-Echevarría et al.63174572.76 (2.57, 2.95)0.78 (0.66, 0.94)0.33 (0.26, 0.4)0.96 (0.82, 1.17)
Dutta et al.32170555.31 (4.52, 6.1)2.28 (1.83, 3.03)0.32 (0.25, 0.4)1.59 (1.28, 2.12)
Koureas et al.32153504.78 (3.94, 5.62)2.43 (1.95, 3.23)0.33 (0.25, 0.41)2.14 (1.72, 2.84)
Bernardes-Souza et al.40112552.8 (2.33, 3.27)1.51 (1.23, 1.93)0.49 (0.4, 0.59)1.48 (1.21, 1.9)
Posfay-Barbe et al.39111462.85 (2.54, 3.16)0.99 (0.81, 1.27)0.41 (0.32, 0.51)0.94 (0.77, 1.21)
Hsu et al.3896492.53 (2.12, 2.93)1.27 (1.03, 1.64)0.51 (0.41, 0.61)0.87 (0.71, 1.12)
Appendix 1—table 11
Association between infectiousness variation estimated from household transmission models, and other statistics from 17 household studies, based on meta-regression.
StatisticInfectiousness variation (σvar*)Standard deviation (SD) of the distribution of number of secondary cases (σsec)Secondary attack rate (SAR)
Pooled estimates1.33 (0.95, 1.70)1.19 (1.03, 1.35)0.35 (0.28, 0.44)
I278.499.299.1
Factors
Estimateββexp (β)
Mean number of contacts0.31 (−0.16, 0.78)0.35 (0.22, 0.48)1.09 (0.83, 1.43)
SD number of contacts0.55 (−0.12, 1.21)0.42 (0.15, 0.69)0.80 (0.51, 1.25)
Implementation of lockdown0.16 (−0.61, 0.93)–0.06 (−0.40, 0.28)0.69 (0.43, 1.10)
Index cases are confirmed by PCR only0.48 (−0.32, 1.29)0.02 (−0.33, 0.38)1.09 (0.65, 1.85)
Secondary cases are confirmed by PCR only0.78 (0.13, 1.43)0.03 (−0.30, 0.35)0.88 (0.55, 1.41)
Only ancestral strains are circulating in study period0.76 (−0.03, 1.55)0.07 (−0.31, 0.45)0.75 (0.43, 1.30)
All household contacts were tested0.03 (−0.87, 0.93)0.28 (−0.04, 0.59)0.96 (0.58, 1.58)
  1. *

    Estimates based on results from 14 studies

Appendix 1—table 12
Pooled estimates for duration of viral shedding for COVID-19 from 11 systematic reviews.
StudySampling siteLaboratory methodPooled estimates (mean/median)95% CIRangeI2
CevikUpper respiratory tractPCR17.015.5–18.66.0–53·9
CevikLower respiratory tractPCR14.69.3–20.06.2–22.797%
CevikStoolViral culture17.214.4–20.19.8–27.996.6%
CevikSerum/bloodViral culture16.63.6–29.710.0–23.399%
FontanaRespiratory sourcesPCR18.4*15.5–21.35.5–53.598.87%
FontanaRectal/stoolPCR22.1*14.4–29.811–3395.86%
OkitaUpper respiratory tract (nasal swab+throat swab)PCR18.2917.00–19.898.33–39.9799%
OkitaSputumPCR23.7920.43–27.1615.50–32.0093%
OkitaBloodPCR14.6012.16–17.0511.00–17.5888%
OkitaStoolPCR22.3818.40–26.3510.67–51.4097%
QutubRespiratory tractViral culture28.75/11**8.5–14.5
RahmaniRespiratory tractPCR27.9023.27–32.527.40–132.0099.1%
XuRespiratory tract (symptomatic cases)PCR11.1±5.8***0–24
XuGastrointestinal tract (symptomatic cases)PCR23.6±8.8***10–33
YanUnrestrictedPCR16.814.8–19.499.56%
YanStoolPCR30.323.1–39.292.09%
YanRespiratory tractPCR17.514.9–20.699.67%
YanUpper respiratory tractPCR17.514.6–21.099.53%
DiazStoolPCR22*19–25
Chen(Asymptomatic infections)14.14*11.25–17.0411.00–17.25
LiUpper respiratory tract (nasopharyngeal/throat swabs)PCR11.4310.1–12.7775.3%
ZhangStoolPCR21.816.4–27.1
ZhangRespiratory tractPCR14.79.9–19.5
  1. *

    Median estimate, **median/grouped median, ***this study analyzed by cases, and reported mean ± SD.

Appendix 1—table 13
Pooled estimates for duration of viral shedding for COVID-19 in subgroups from seven systematic reviews.
StudySampling site (subgroups)Laboratory methodPooled estimates (mean/median)95% CIRangeI2
FontanaRespiratory sources (among severely ill patients)PCR19.8*16.2–23.511–3196.42%
FontanaRespiratory sources (in mild-to-moderate illness)PCR17.2*14.0–20.58–2595.64%
OkitaUpper respiratory tract (nasal swab)PCR19.3416.60–22.0799%
OkitaUpper respiratory tract (throat swab)PCR17.8516.43–19.2698%
OkitaUpper respiratory tract (age < 60)PCR16.9513.56–20.358.62–35.6798%
OkitaUpper respiratory tract (age ≥ 60)PCR21.2414.06–28.418.25–40.3399%
OkitaUpper respiratory tract (with comorbidities)PCR20.2617.60–22.929.67–34.0093%
OkitaUpper respiratory tract (without comorbidities)PCR14.6612.63–16.6910.82–27.2585%
OkitaUpper respiratory tract (severe patients)PCR20.7918.03–23.5514.00–38.3398%
OkitaUpper respiratory tract (nonsevere patients)PCR16.3614.07–18.668.00–37.3399%
OkitaUpper respiratory tract (severe patients) (for studies with mean age ≥ 40 + comorbidity > 30%)PCR21.5317.57–25.5014.00–29.5091%
OkitaUpper respiratory tract (severe patients) (for studies with mean age ≥ 40 + comorbidity >30%)PCR20.0815.87–24.2913.12–33.67100%
OkitaUpper respiratory tract (treated with glucocorticoid)PCR19.7217.92–21.5213.87–33.6792%
OkitaUpper respiratory tract (treated without glucocorticoid)PCR15.6414.18–17.108.33–31.6096%
OkitaUpper respiratory tract (treated with glucocorticoid) (for studies with mean age: 30–60+comorbidity > 50%)PCR21.9816.48–27.4814.25–33.6794%
OkitaUpper respiratory tract (treated without glucocorticoid) (for studies with mean age: 30–60+comorbidity > 50%)PCR16.1412.60–19.6813.22–24.4492%
OkitaUpper respiratory tract (Asian)PCR18.1016.95–19.258.33–34.7598%
OkitaUpper respiratory tract (European)PCR19.2711.59–26.958.50–39.97100%
OkitaUpper respiratory tract (Asian) (for studies with mean age ≥ 40 + comorbidity >40%)PCR20.6618.18–23.1412.00–32.0096%
OkitaUpper respiratory tract (European) (for studies with mean age ≥ 40 + comorbidity > 40%)PCR23.6810.85–36.5113.00–39.97100%
QutubRespiratory tract (severe patients)Viral culture47.5/20**9.0–53.0
QutubRespiratory tract (severe patients were not specified of excluded)Viral culture10/9**8.0–13.0
QutubRespiratory tract (immunocompromised patients)Viral culture54.36/20**9.0–85.98
QutubRespiratory tract (immunocompromised patients were not specified of excluded)Viral culture11.67/9**8.2–13.3
RahmaniNot specific (immunocompetent individuals)PCR26.5421.44–31.647.40–91.2099.3%
RahmaniNot specific (immunocompromised individuals)PCR36.2821.93–50.6315.90–132.0094.2%
XuRespiratory tract (asymptomatic cases)PCR9.4±5.1***
XuGastrointestinal tract (asymptomatic cases)PCR16.8±9.8***
YanUnrestricted (symptomatic cases)PCR19.717.2–22.799.34%
YanUnrestricted (asymptomatic cases)PCR10.98.3–14.398.89%
YanUnrestricted (severe patients)PCR24.318.9–31.191.88%
YanUnrestricted (nonsevere patients)PCR22.816.4–32.099.81%
YanUnrestricted (females)PCR19.49.5–39.493.93%
YanUnrestricted (males)PCR11.98.4–16.987.83%
YanUnrestricted (adults)PCR23.219.0–28.499.24%
YanUnrestricted (children)PCR9.98.1–12.285.74%
YanUnrestricted (with chronic diseases)PCR24.219.2–30.284.07%
YanUnrestricted (without chronic diseases)PCR11.55.3–25.082.11%
YanUnrestricted (treated with corticosteroid)PCR28.325.6–31.20.00%
YanUnrestricted (treated without corticosteroid)PCR16.211.5–22.592.27%
YanUnrestricted (antiviral treatment)PCR17.613.4–22.298.99%
YanUnrestricted (mono-antiviral treatment)PCR21.215.3–29.290.04%
YanUnrestricted (multiantiviral treatment)PCR20.313.7–30.399.46%
ChenUnrestricted (asymptomatic infections)PCR or serum antibody14.14*11.25–17.0411.00–17.25
  1. *

    Median estimate, **median/grouped median, ***this study analyzed by cases, and reported mean ± SD.

Appendix 1—table 14
Pooled estimates for duration of infectious period for COVID-19 in subgroups from two systematic reviews.
StudySampling site (subgroups)Laboratory methodPooled estimates95% CIRangeI2
RahmaniReplicant competent virus isolationViral culture7.275.70–8.843.40–89.0092.2%
WangNot specificNot specific6.255.09–7.51
RahmaniReplicant competent virus isolation (immunocompetent individuals)Viral culture6.334.92–7.753.00–13.0092.4%
RahmaniReplicant competent virus isolation (immunocompromised individuals)Viral culture29.5012.46–46.5313.80–89.0084.8%
Appendix 1—table 15
Probability distributions of the incubation period and relative infectivity levels during the infectious period.

For the infectious period, day 0 corresponds to the symptom onset day. These two distributions are used to generate the infectiousness profile since infections.

DayIncubation periodInfectious period
Mean = 5 daysDayMax = 13 days
10.058−51.0
20.11−41.0
30.14−31.0
40.16−21.0
50.15−11.0
60.1301.0
70.1011.0
80.06821.0
90.04430.8
100.02640.6
110.01450.4
120.007260.2
130.003470.1
140.00158

Data availability

The data and computer code (in R language) for conducting the data analysis can be downloaded from https://github.com/timktsang/covid19_transmission_heterogeneity (copy archived at swh:1:rev:634390fa66a4bfb998da691e7cd81cf45e38c6db).

References

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    1. Thompson HA
    (2021) SARS-cov-2 setting-specific transmission rates: a systematic review and meta-analysis
    Clinical Infectious Diseases: An Official Publication of the Infectious Diseases Society of America 73:e754–e764.
    https://doi.org/10.1093/cid/ciab100

Decision letter

  1. Amy Wesolowski
    Reviewing Editor; Johns Hopkins Bloomberg School of Public Health, United States
  2. Miles P Davenport
    Senior Editor; University of New South Wales, Australia

In the interests of transparency, eLife publishes the most substantive revision requests and the accompanying author responses.

Decision letter after peer review:

Thank you for submitting your article "The effect of variation of individual infectiousness on SARS-CoV-2 transmission in households" for consideration by eLife. Your article has been reviewed by 2 peer reviewers, and the evaluation has been overseen by a Reviewing Editor and Miles Davenport as the Senior Editor. The reviewers have opted to remain anonymous.

The reviewers have discussed their reviews with one another, and the Reviewing Editor has drafted this to help you prepare a revised submission.

Essential revisions:

1) Both reviewers have highlighted a number of areas where additional clarification (particularly for σ_var) which currently does not have sufficient detail/interpretation to warrant publication.

2) Questions about parameter identifiability and model validation were not sufficiently addressed – particularly in light of the many individual-level parameters estimated in the analysis.

3) Additional exploration of empirical relationships (such as individual infectiousness and household size, as an example) should be explored. For example, how much of the heterogeneity is driven by biological factors (such as being the primary case).

4) Additional work should be done to help disentangle differences between biological or behavioral factors since this will greatly change the interpretation of inferred parameters. In addition, more contextual information should be provided (socio-economic status and the implications of this as a confounder, a more detailed exploration of interventions that were in place and their implications) should be done.

Reviewer #1 (Recommendations for the authors):

Tsang et al. evaluate the degree of transmission heterogeneity among COVID-19 cases using data compiled from 17 household transmission studies. Transmission heterogeneity and super spreading are important but incompletely understood drivers of epidemic dynamics. While it is clear that substantial heterogeneity exists among infectors, but the relative contribution of behavior (e.g. the number of contacts), timing (e.g. large number of social interactions near the peak of infectiousness), and intrinsic biological differences in infectiousness are not resolved.

Tsang et al. compiled household transmission data from 17 published studies. They independently fit a model to each data set in order to quantify transmission heterogeneity arising from intrinsic biological differences in infectiousness, after controlling for differences in the number of contacts (i.e. household size), and timing (via the inferred time since infection). Repeated application of the same model to multiple datasets is a strength of this study because it allows the authors to assess associations between the inferred parameters and aspects of study design that can influence case ascertainment.

The authors estimate substantial intrinsic transmission heterogeneity via the parameter σ_var, which could be attributed to individual differences in viral loads or differences in the infector's contact intensity with others in the household. They show that intrinsic, within-household variation is associated with other population-level metrics of transmission heterogeneity, including p_80 and p_0.

Although the study focuses on the household size as a confounder of heterogeneity in infectiousness, the ways that the model controls for household size and the interpretation of the inferred heterogeneity parameter, σ_var could be explained more clearly. Furthermore, it seems that basic empirical relationships between individual infectiousness (δ_i) and household size are not explored. Finally, questions about parameter identifiability and model validation could be addressed more extensively, especially in light of the many individual-level parameters estimated in the analysis.

Overall, the study design is sound and the analysis of multiple datasets is a strength. However, the modeling approach and the biological interpretation/significance of the results could be more clearly explained. Also, given the large number of individual-level parameters estimated, it would be ideal to see a direct assessment of parameter identifiability (e.g. correlation plots), and model validation on simulated data (i.e. can the model estimate known parameters from simulated data?). The latter shouldn't be too difficult, given that the authors already have a simulation model for tests of model adequacy.

Specific comments:

1. Please explain assumptions about the individual infectiousness profile f(.) (from section 2 of the Materials and methods). What is the assumed shape of this function? How is it parameterized? How could the inclusion of this function in the model interact with inferred individual times of infection, especially if times of infection are inaccurate? On a related note, the authors might consider introducing the concept of contact timing as a factor that can influence heterogeneity in transmission.

2. The household size is a key conceptual focus of the study, but the main text methods/results don't explain clearly how the model accounts for household size or contact number. My interpretation is that the model accounts for household size in two ways. First. the model estimates the per-contact hazard of infection, which implicitly controls for household size/contact number. Second, the model includes a parameter, β, which represents a dilution effect, wherein the transmission hazard might be diluted in larger households due to less per-individual contact intensity. I'm not sure that the dilution effect is currently explained anywhere in the main text, and it could also be explained more clearly that the model is designed to estimate transmission heterogeneity after adjusting for household size.

2a. It would be helpful if the authors could discuss the distinction between contact number, contact frequency, and contact duration somewhere in the text.

3. Given the focus on household size and the number of contacts, it seems there are missed opportunities to explore how the inferred level of individual infectiousness (δ_i) co-varies with metrics like household size.

4. Please explain in more detail the meta-regression sensitivity analyses.

5. I'm not sure that tables S6-S9 are cross-referenced anywhere in the main text, and their legends aren't sufficiently detailed to fully communicate how to interpret these tables in the context of the broader study.

Reviewer #2 (Recommendations for the authors):

Transmission heterogeneity and superspreading of SARS-CoV-2 have been demonstrated repeatedly through real-world observational studies, with a small fraction of individuals accounting for the majority of onward transmission. However, this observed transmission heterogeneity is likely a superposition of accumulated variations from multiple factors, including but not limited to host behavioral factors such as the variation in contact numbers contact duration, and contact settings; variations in the adoption of preventive measures (mask-wearing, physical distancing, etc.); the effects of NPIs (case isolations and contact quarantines, population-level lockdowns); as well as biological factors including differences in shedding duration and intensity of the primary case, variation in susceptibility among close contacts. Fewer studies, however, attempted to isolate the effects of individual factors contributing to the overall transmission heterogeneity, while controlling for other factors. In the manuscript entitled "The effect of variation of individual infectiousness on SARS-CoV-2 transmission in households", the authors aim at characterizing the variation in individual infectiousness of SARS-CoV-2, controlling for other host factors. To achieve this, the authors performed a meta-analysis of household transmission studies conducted during or not too longer after the initial SARS-CoV-2 wave caused by the ancestral strain across the globe. The authors fitted a household transmission model to the curated data and estimated the variation in infectiousness by introducing random effects of individual infectiousness and its population-level distribution.

The study has several strengths. First, by choosing analyzing data from the household study, the authors were able to control for several key factors contributing to the overall transmission heterogeneity but not variation in individual infectiousness, including the number of contacts as well the setting of transmission (household). The authors also incorporated an additional parameter in the household transmission model to adjust for the impact of household size on household transmission risk, which has been identified as an important risk factor for SARS-CoV-2 transmission within the household across multiple studies. The authors also curated the studies during the early stage of the pandemic so that most household contacts would remain naïve during the study period, and the immune status of the household contact (either due to prior infection or vaccination) would be unlikely to confound the results of the study.

However, the study also has a few limitations. First, it is difficult to disentangle if the observed variation in infectiousness is due to biological factors or behavioral factors. During the study period of interest, public health agencies across the globe were recommending at-home isolation guidelines aiming at reducing transmission within the household, including mask-wearing when in contact with other household members, using separate bedrooms/bathrooms, avoiding having meals together, etc. The differences in the guidelines across nationals/regions as well as the level of compliance with guidelines at the household level would also impact individual household transmission risk. Second, the risk of acquiring infections from the community could be heavily influenced by the socio-economic status, since multiple studies have clearly demonstrated stark disparity of COVID-19 burden, factors such as occupation (essential workers tend to be low-wage jobs) assess to PPE and healthcare were likely to contribute to the observed disparity. Lastly, it is also difficult to entangle if the observed heterogeneity is due to the biological factors of the primary case (i.e., variation in shedding duration/intensity) or the contact (variation in susceptibility). The current formulation of the transmission model only addresses the former not the latter.

For the household transmission model concerning imputation of the timing of infection, it is unclear if the timing of infection is partially missing, or if the timing of infection were unavailable for all studies. If it is the former case, the authors should give a more detailed description of the imputation process in a study-by-study fashion and report the proportion of infection time that was imputed. If it is the latter case, I do not understand the benefit of explicitly modeling the temporal infectiousness profile, thus I would recommend the authors use a simpler chain-binomial model fitting to only the binary outcome (of infected or not) unless the authors make a convincing case otherwise.

Both p80 and p0 are poor statistics to characterize household transmission due to the discrete nature of household contact numbers. For example, it is unintuitive to interpret and compare p80 for a household size of 2 vs 10 (for a household size of 2 conditional on an index case within the household this could only be interpreted as the proportion of households with secondary infections, which is cross-household level statistics, while for a household size of 10, it can be interpreted as a meaningful statistical for each household). Furthermore, the interpretation of p80 as a metric for transmission heterogeneity becomes even murkier if multi-generation transmission within the household is considered, especially for a larger household. For p0, this metric is heavily dependent on household size. For example, one would expect the p0 for a household size of 10 (with on index) to be way smaller than p0 household size of 2 (assuming no dilution effects on transmission with the increase of household size), as in the former case p0 characterize the probability of all 9 uninfected contacts escaping infection from the index, while in the latter p0 characterizes the probability to escape infection by only 1 household contact. Thus, it is meaningless to compare p0 without controlling for household size across studies. For the arguments above, I recommend the authors remove both p80 and p0 from the paper.

Through the formulation of the household transmission model, the authors essentially assume a lognormal distribution of individual infectiousness, which is a long-tailed distribution by nature. I think this is an important point that needs to be highlighted. Has the author tried a normal distribution instead of a log-normal distribution? Would a normal distribution fit better or worse to the data than the log-normal one? I recommend the authors conduct a sensitivity analysis of a normal distribution of the individual infectiousness (i.e., not taking the exponential of the σ term for the hazard of infection function on page 7).

Table S5 is cropped.

https://doi.org/10.7554/eLife.82611.sa1

Author response

Essential revisions:

1) Both reviewers have highlighted a number of areas where additional clarification (particularly for σ_var) which currently does not have sufficient detail/interpretation to warrant publication.

We have modified the discussion to talk about the interpretation and limitation of infectiousness variation. We clarified that the estimated infectiousness variations measures the variation caused by difference in individuals, which may be contributed by both biological factors and host behaviors, but not other environmental factors. Please see Response 7 for more details.

We also added a limitation that our approach could not disentangle the observed heterogeneity from biological factors affecting infectiousness and variations in susceptibility of contacts. Please see Response 14 for more details.

2) Questions about parameter identifiability and model validation were not sufficiently addressed – particularly in light of the many individual-level parameters estimated in the analysis.

We have added the correlation plots for the posterior distribution of parameters, and also rerun a longer MCMC chain (with 500K iterations instead of 100K iterations) to ensure that our estimates of infectiousness variations are valid. Please see Response 7 for more details.

3) Additional exploration of empirical relationships (such as individual infectiousness and household size, as an example) should be explored. For example, how much of the heterogeneity is driven by biological factors (such as being the primary case).

The individual infectiousness parameters are nuisance parameter in our work, since there is no individual level information on the case and contacts. However, our model is designed in a way that covariates affecting susceptibility or infectiousness could be added if individual level information is available. Please see Response 11 for more details.

We also conduct meta-analysis and meta-regression to determine if any factors may be associated with the infectiousness variation. We found that this is not associated with the mean numbers of contacts and household size, and only associated with if the study used PCR to confirm secondary cases only (Table S11).

4) Additional work should be done to help disentangle differences between biological or behavioral factors since this will greatly change the interpretation of inferred parameters. In addition, more contextual information should be provided (socio-economic status and the implications of this as a confounder, a more detailed exploration of interventions that were in place and their implications) should be done.

We have extended our discussion on difference between biological and behavioral factors. In summary, behavioral factors including difference in frequency and duration of contacts caused by different factors, like the role in family, recommendation from public health agencies, among individuals. Biological factors include difference in viral shedding and infectious period among individuals. We also provide more information on other confounding factors, such as social-economic status, recommendation from public health agencies, the implementation of lockdowns that may increase the time spending at home. These may increase of decrease the SAR, which is shown to be negatively correlated with infectiousness variations. Please see Response 7 and 14 for more details.

Reviewer #1 (Recommendations for the authors):

Tsang et al. evaluate the degree of transmission heterogeneity among COVID-19 cases using data compiled from 17 household transmission studies. Transmission heterogeneity and super spreading are important but incompletely understood drivers of epidemic dynamics. While it is clear that substantial heterogeneity exists among infectors, but the relative contribution of behavior (e.g. the number of contacts), timing (e.g. large number of social interactions near the peak of infectiousness), and intrinsic biological differences in infectiousness are not resolved.

Tsang et al. compiled household transmission data from 17 published studies. They independently fit a model to each data set in order to quantify transmission heterogeneity arising from intrinsic biological differences in infectiousness, after controlling for differences in the number of contacts (i.e. household size), and timing (via the inferred time since infection). Repeated application of the same model to multiple datasets is a strength of this study because it allows the authors to assess associations between the inferred parameters and aspects of study design that can influence case ascertainment.

The authors estimate substantial intrinsic transmission heterogeneity via the parameter σ_var, which could be attributed to individual differences in viral loads or differences in the infector's contact intensity with others in the household. They show that intrinsic, within-household variation is associated with other population-level metrics of transmission heterogeneity, including p_80 and p_0.

Although the study focuses on the household size as a confounder of heterogeneity in infectiousness, the ways that the model controls for household size and the interpretation of the inferred heterogeneity parameter, σ_var could be explained more clearly. Furthermore, it seems that basic empirical relationships between individual infectiousness (δ_i) and household size are not explored. Finally, questions about parameter identifiability and model validation could be addressed more extensively, especially in light of the many individual-level parameters estimated in the analysis.

Overall, the study design is sound and the analysis of multiple datasets is a strength. However, the modeling approach and the biological interpretation/significance of the results could be more clearly explained. Also, given the large number of individual-level parameters estimated, it would be ideal to see a direct assessment of parameter identifiability (e.g. correlation plots), and model validation on simulated data (i.e. can the model estimate known parameters from simulated data?). The latter shouldn't be too difficult, given that the authors already have a simulation model for tests of model adequacy.

Thank you for your suggestions. In brief, we summarized that the estimated infectiousness variations measures the variation caused by difference in individuals, which may be affected by both biological factors and host behaviors but not other confounding factors, such as the difference in number of contacts, pre-existing immunity among contacts or difference in transmissibility among variants. We also added more details on the interpretation of the estimated infectiousness variation.

We added ‘Given that we focused our analysis on households with known number of contacts in studies conducted in the early outbreaks caused by ancestral strains, the estimated infectiousness variations are corrected for the variations caused by number of contacts and transmission risks in different settings, difference in pre-existing immunity among contacts (almost everyone is naïve and unvaccinated), and the difference in transmissibility among variants. Hence, this estimated infectiousness variation measures the variation caused by difference in individuals, which may be contributed by both biological factors and host behaviors, but not other confounding factors such as…. ‘ in the Discussion section in the revised manuscript.

We conducted a simulation study, and found there is no important systematic bias, and that 88-100% of 95% credible interval covered the simulation value. We also added the correlation plots in the revised manuscript. The correlation between parameters are moderate. We run long MCMC chain to ensure our results are valid. We conduct a sensitivity by running the MCMC chain with 500,000 iterations and the results are similar to the one with 100,000 iterations. This suggests that 100,000 iterations are enough (our main results).

We added ‘Simulation studies demonstrated there is no important systematic bias, with 88 to 100% of the 95% credible intervals covering the simulation value (Appendix 1 – Table 8). This suggests that the algorithm could estimate adequately the posterior distribution.’ In the Result section of the revised manuscript.

Specific comments:

1. Please explain assumptions about the individual infectiousness profile f(.) (from section 2 of the Materials and methods). What is the assumed shape of this function? How is it parameterized? How could the inclusion of this function in the model interact with inferred individual times of infection, especially if times of infection are inaccurate? On a related note, the authors might consider introducing the concept of contact timing as a factor that can influence heterogeneity in transmission.

We apologize for the confusion. We added a table to summarize the assumed values extracted from the reference papers, and a plot for the assumed individual infectiousness profile function f(.). We also moved the exact formula and more details of the model from appendix to the main text, to enhance the readability.

In the estimation, the infectiousness profile are used to determine the generation time, and it would influence the observed heterogeneity of distribution of number of secondary cases (including all generations). When there is no difference in infectiousness variation among cases, if the generation time is shorter, there will be more generations of cases, and hence the observed distribution of number of secondary cases would be more heterogeneous. Hence, this assumption is setting a ‘baseline’ of observed heterogeneity of observed distribution of number of secondary cases when there is no difference in infectiousness among cases, so that the model could estimate the extra heterogeneity due to variations of individual infectiousness.

We thank for the reviewer’s suggestion about contact timing, which is a good point and we also added in the Discussion. We added ‘the duration of contact could vary, which could also contribute to the variations in infectiousness of cases.’ in the Discussion section of the revised manuscript.

2. The household size is a key conceptual focus of the study, but the main text methods/results don't explain clearly how the model accounts for household size or contact number. My interpretation is that the model accounts for household size in two ways. First. the model estimates the per-contact hazard of infection, which implicitly controls for household size/contact number. Second, the model includes a parameter, β, which represents a dilution effect, wherein the transmission hazard might be diluted in larger households due to less per-individual contact intensity. I'm not sure that the dilution effect is currently explained anywhere in the main text, and it could also be explained more clearly that the model is designed to estimate transmission heterogeneity after adjusting for household size.

We thank the reviewer’s a comprehensive summary on the way that the model accounts for the household size or number of household contacts. We added this information, including the dilution effect, to the revised manuscript as follows:

We added ‘the model could estimate the per-contact hazard of infection, which implicitly controls for number of household contacts in households.’ In the Result section of revised manuscript.

We added ‘In the model, the hazard of infection of individual j at time t from an infected household member i, with infection time ti in household k, was λij(t)=λhXkβ exp(δi)f(tti) where λh was the baseline hazard, δi followed a Normal distribution with mean 0 and standard deviation σvar, which quantified the variation of individual infectiousness (hereafter denoted as infectiousness variation). The relative infectiousness of case i compared with case j was exp(δi)/exp(δj).

Xk was the number of household contacts. β was the parameter describing the relationship between number of household contacts and transmission rate. It ranged from 0 to 1, with 0 indicating that the transmission rate was independent of number of household contacts while 1 indicated that the transmission rate was inversely proportional to number of household contacts (i.e. dilution effect of the contact time per contact which was lower when the number of household contacts is higher).’ to the Method section of the revised manuscript.

2a. It would be helpful if the authors could discuss the distinction between contact number, contact frequency, and contact duration somewhere in the text.

Thank you for your suggestion. We agree this is an important point and added to the revised manuscript. We added ‘It should be noted that there are several components in the contacts, including the numbers, frequencies and duration of contacts that may contribute to the infectiousness variations. Previous studies also suggest that their relative importance in transmission may be different among viruses.’ In the Discussion in the revised manuscript.

3. Given the focus on household size and the number of contacts, it seems there are missed opportunities to explore how the inferred level of individual infectiousness (δ_i) co-varies with metrics like household size.

Thank you for your suggestion. We estimated that the posterior correlation between inferred level of individual infectiousness (δ_i) and household size are not significant ( the credible intervals are all covered one). This suggested the inferred individual infectiousness was not correlated with household size.

Author response table 1
StudyPearson’s correlationShearman’s correlation
Lyngse, et al.0.00 (-0.02, 0.03)0.00 (-0.03, 0.02)
Carazo, et al.0.00 (-0.03, 0.04)0.00 (-0.03, 0.03)
Laxminarayan, et al.0.00 (-0.07, 0.05)-0.02 (-0.08, 0.03)
Dattner, et al.0.01 (-0.07, 0.09)0.00 (-0.08, 0.09)

4. Please explain in more detail the meta-regression sensitivity analyses.

Sorry for the confusion. Meta-analysis and Meta-regression is conducted to pool the estimates of infectiousness variation, which was estimated separately for 14 identified studies. Furthermore, it allows us to determine the potential association between estimated infectiousness variation and study characteristics, including the following factors: the mean and SD of number of household contacts, implementation of lockdown, ascertainment method of index and secondary cases, only ancestral strains are circulating in study period, and all household contacts were tested. We have clarified in the Method section of the revised manuscript.

5. I'm not sure that tables S6-S9 are cross-referenced anywhere in the main text, and their legends aren't sufficiently detailed to fully communicate how to interpret these tables in the context of the broader study.

These tables summarize the systematic review and meta-analysis about viral shedding of COVID-19 cases, which is a proxy of individual infectiousness of cases. They showed heterogeneity in the level of viral shedding of COVID-19 cases, which support our results about variation of individual infectiousness.

Reviewer #2 (Recommendations for the authors):

Transmission heterogeneity and superspreading of SARS-CoV-2 have been demonstrated repeatedly through real-world observational studies, with a small fraction of individuals accounting for the majority of onward transmission. However, this observed transmission heterogeneity is likely a superposition of accumulated variations from multiple factors, including but not limited to host behavioral factors such as the variation in contact numbers contact duration, and contact settings; variations in the adoption of preventive measures (mask-wearing, physical distancing, etc.); the effects of NPIs (case isolations and contact quarantines, population-level lockdowns); as well as biological factors including differences in shedding duration and intensity of the primary case, variation in susceptibility among close contacts. Fewer studies, however, attempted to isolate the effects of individual factors contributing to the overall transmission heterogeneity, while controlling for other factors. In the manuscript entitled "The effect of variation of individual infectiousness on SARS-CoV-2 transmission in households", the authors aim at characterizing the variation in individual infectiousness of SARS-CoV-2, controlling for other host factors. To achieve this, the authors performed a meta-analysis of household transmission studies conducted during or not too longer after the initial SARS-CoV-2 wave caused by the ancestral strain across the globe. The authors fitted a household transmission model to the curated data and estimated the variation in infectiousness by introducing random effects of individual infectiousness and its population-level distribution.

The study has several strengths. First, by choosing analyzing data from the household study, the authors were able to control for several key factors contributing to the overall transmission heterogeneity but not variation in individual infectiousness, including the number of contacts as well the setting of transmission (household). The authors also incorporated an additional parameter in the household transmission model to adjust for the impact of household size on household transmission risk, which has been identified as an important risk factor for SARS-CoV-2 transmission within the household across multiple studies. The authors also curated the studies during the early stage of the pandemic so that most household contacts would remain naïve during the study period, and the immune status of the household contact (either due to prior infection or vaccination) would be unlikely to confound the results of the study.

However, the study also has a few limitations. First, it is difficult to disentangle if the observed variation in infectiousness is due to biological factors or behavioral factors. During the study period of interest, public health agencies across the globe were recommending at-home isolation guidelines aiming at reducing transmission within the household, including mask-wearing when in contact with other household members, using separate bedrooms/bathrooms, avoiding having meals together, etc. The differences in the guidelines across nationals/regions as well as the level of compliance with guidelines at the household level would also impact individual household transmission risk. Second, the risk of acquiring infections from the community could be heavily influenced by the socio-economic status, since multiple studies have clearly demonstrated stark disparity of COVID-19 burden, factors such as occupation (essential workers tend to be low-wage jobs) assess to PPE and healthcare were likely to contribute to the observed disparity. Lastly, it is also difficult to entangle if the observed heterogeneity is due to the biological factors of the primary case (i.e., variation in shedding duration/intensity) or the contact (variation in susceptibility). The current formulation of the transmission model only addresses the former not the latter.

We appreciate the reviewer’s very comprehensive summary of our work and unique insight of the interpretation of our results, and we added many of them in the revised manuscript. We expanded the section of our discussion listing the different limitations, in particular the fact that our approach can not disentangle the observed heterogeneity from biological factors affecting infectiousness and variations in susceptibility of contacts.

We added the followings in the Discussion in the revised manuscript:

“Given that we focused our analysis on households with known number of contacts in studies conducted in the early outbreaks caused by ancestral strains, the estimated infectiousness variations do not include the variations caused by number of contacts and transmission risks in different settings, difference in pre-existing immunity among contacts (almost everyone is naïve and unvaccinated), and the difference in transmissibility among variants. Hence, this estimated infectiousness variation measures the variation caused by difference in individuals, which may be contributed by both biological factors and host behaviors, but not other confounding factors.”

“It should be noted that there are several components in the contacts, including the numbers, frequencies and duration of contacts that may contribute to the infectiousness variations. Previous studies also suggest that their relative importance in transmission may be different among viruses. Individual behaviors may also be influenced by the implemented control measures and recommendation from public health agencies. For example, lockdown and stay-at-home orders may increase the time spending at home. Mask-wearing when in contact with other household members, using separate bedrooms and bathrooms, avoiding having meals together may reduce the transmission in households. Also, social disparity such as occupation may increase or reduce the risk of transmission in households, including the availability of personal protective equipment (PPE), or being healthcare worker. These may have impacted the SAR in households, but also estimated infectiousness variation in this setting.”

“In addition, we could not include factors affecting susceptibility to infection. Our estimates of infectiousness variation should be interpreted in light of these limitations: they capture heterogeneity in infectiousness due to demographic, host and biological factors. However, in one study that included susceptibility component in the estimation of individual infectiousness, substantial heterogeneity remained with 20% of cases estimated to contribute to 80% of transmission.”

“Third, we assumed that risks of infection from community for all households are the same, but there were different factors that may affecting this, including occupations, such as healthcare workers, social economic status that related to assess to personal protective equipment”

For the household transmission model concerning imputation of the timing of infection, it is unclear if the timing of infection is partially missing, or if the timing of infection were unavailable for all studies. If it is the former case, the authors should give a more detailed description of the imputation process in a study-by-study fashion and report the proportion of infection time that was imputed. If it is the latter case, I do not understand the benefit of explicitly modeling the temporal infectiousness profile, thus I would recommend the authors use a simpler chain-binomial model fitting to only the binary outcome (of infected or not) unless the authors make a convincing case otherwise.

Sorry of confusion. The timing of infection were unavailable for all studies. The reason of not using a simple chain-binomial model is that it is impossible to add the parameters characterizing individual infectiousness (δ_i) in the model. When there are different types of susceptible in data, chain-binomial model required adding separate probability parameters for each type of susceptible in the model, with each type of transmission (from household members or from outside households).

We have clarified this by adding ‘Since the data were extracted from publication, the infection time of all cases was unavailable for all studies.’ in the Method section in the revised manuscript. We also added a section in Appendix to explain this.

Both p80 and p0 are poor statistics to characterize household transmission due to the discrete nature of household contact numbers. For example, it is unintuitive to interpret and compare p80 for a household size of 2 vs 10 (for a household size of 2 conditional on an index case within the household this could only be interpreted as the proportion of households with secondary infections, which is cross-household level statistics, while for a household size of 10, it can be interpreted as a meaningful statistical for each household). Furthermore, the interpretation of p80 as a metric for transmission heterogeneity becomes even murkier if multi-generation transmission within the household is considered, especially for a larger household. For p0, this metric is heavily dependent on household size. For example, one would expect the p0 for a household size of 10 (with on index) to be way smaller than p0 household size of 2 (assuming no dilution effects on transmission with the increase of household size), as in the former case p0 characterize the probability of all 9 uninfected contacts escaping infection from the index, while in the latter p0 characterizes the probability to escape infection by only 1 household contact. Thus, it is meaningless to compare p0 without controlling for household size across studies. For the arguments above, I recommend the authors remove both p80 and p0 from the paper.

Thank you. We agree with the reviewer and removed p80 and p0 from the paper.

Through the formulation of the household transmission model, the authors essentially assume a lognormal distribution of individual infectiousness, which is a long-tailed distribution by nature. I think this is an important point that needs to be highlighted. Has the author tried a normal distribution instead of a log-normal distribution? Would a normal distribution fit better or worse to the data than the log-normal one? I recommend the authors conduct a sensitivity analysis of a normal distribution of the individual infectiousness (i.e., not taking the exponential of the σ term for the hazard of infection function on page 7).

Thank you for this suggestion. We conducted a sensitivity analysis for assuming the individual infectiousness followed a Γ distribution, which has a shorter tail compared with lognormal distribution. However, model comparison suggested that assuming γ distribution for individual infectiousness parameter performed substantially worse, compared with the lognormal distribution in the main analysis (Appendix 1 – Table 9).

Table S5 is cropped.

Sorry for the mistake, we revised the table.

https://doi.org/10.7554/eLife.82611.sa2

Article and author information

Author details

  1. Tim K Tsang

    1. WHO Collaborating Centre for Infectious Disease Epidemiology and Control, School of Public Health, Li Ka Shing Faculty of Medicine, The University of Hong Kong, Hong Kong, China
    2. Laboratory of Data Discovery for Health, Hong Kong, China
    Contribution
    Conceptualization, Data curation, Software, Formal analysis, Funding acquisition, Validation, Investigation, Visualization, Methodology, Writing - original draft, Writing – review and editing
    For correspondence
    timtsang@connect.hku.hk
    Competing interests
    No competing interests declared
    ORCID icon "This ORCID iD identifies the author of this article:" 0000-0001-5037-6776
  2. Xiaotong Huang

    WHO Collaborating Centre for Infectious Disease Epidemiology and Control, School of Public Health, Li Ka Shing Faculty of Medicine, The University of Hong Kong, Hong Kong, China
    Contribution
    Data curation, Investigation, Visualization
    Competing interests
    No competing interests declared
  3. Can Wang

    WHO Collaborating Centre for Infectious Disease Epidemiology and Control, School of Public Health, Li Ka Shing Faculty of Medicine, The University of Hong Kong, Hong Kong, China
    Contribution
    Data curation, Investigation
    Competing interests
    No competing interests declared
  4. Sijie Chen

    WHO Collaborating Centre for Infectious Disease Epidemiology and Control, School of Public Health, Li Ka Shing Faculty of Medicine, The University of Hong Kong, Hong Kong, China
    Contribution
    Investigation, Visualization
    Competing interests
    No competing interests declared
  5. Bingyi Yang

    WHO Collaborating Centre for Infectious Disease Epidemiology and Control, School of Public Health, Li Ka Shing Faculty of Medicine, The University of Hong Kong, Hong Kong, China
    Contribution
    Writing – review and editing
    Competing interests
    No competing interests declared
  6. Simon Cauchemez

    Mathematical Modelling of Infectious Diseases Unit, Institut Pasteur, Paris, France
    Contribution
    Conceptualization, Supervision, Funding acquisition, Methodology, Writing – review and editing
    Contributed equally with
    Benjamin John Cowling
    Competing interests
    No competing interests declared
    ORCID icon "This ORCID iD identifies the author of this article:" 0000-0001-9186-4549
  7. Benjamin John Cowling

    Laboratory of Data Discovery for Health, Hong Kong, China
    Contribution
    Conceptualization, Supervision, Funding acquisition, Methodology, Writing – review and editing
    Contributed equally with
    Simon Cauchemez
    For correspondence
    bcowling@hku.hk
    Competing interests
    consults for AstraZeneca, Fosun Pharma, GlaxoSmithKline, Moderna, Pfizer, Roche and Sanofi Pasteur. The authors report no other potential conflicts of interest
    ORCID icon "This ORCID iD identifies the author of this article:" 0000-0002-6297-7154

Funding

Food and Health Bureau (COVID190118)

  • Benjamin John Cowling

Research Grants Council, University Grants Committee (C7123-20G)

  • Benjamin John Cowling

Innovation and Technology Commission of Hong Kong Special Administrative Government (AIR@innoHK)

  • Benjamin John Cowling

Investissement d'Avenir (ANR-10-LABX-62-IBEID)

  • Simon Cauchemez

EMERGEN (ANRS0151)

  • Simon Cauchemez

INCEPTION (PIA/ANR-16-CONV-0005)

  • Simon Cauchemez

AXA Research Fund (101003589 (RECOVER))

  • Simon Cauchemez

Groupama (874735 (VEO))

  • Simon Cauchemez

The University of Hong Kong (202111159118)

  • Tim K Tsang

The funders had no role in study design, data collection and interpretation, or the decision to submit the work for publication.

Acknowledgements

The authors thank Maylis Layan for helpful discussion and Jiayi Zhou for technical assistance. This project was supported by the Health and Medical Research Fund, Food and Health Bureau, Government of the Hong Kong Special Administrative Region (grant no. COVID190118; BJC) and the Collaborative Research Fund (Project No. C7123-20G; BJC) of the Research Grants Council of the Hong Kong SAR Government. BJC is supported by the AIR@innoHK program of the Innovation and Technology Commission of the Hong Kong SAR Government. SC acknowledges financial support from the Investissement d’Avenir program, the Laboratoire d’Excellence Integrative Biology of Emerging Infectious Diseases program (grant ANR-10-LABX-62-IBEID), the EMERGEN project (ANRS0151), the INCEPTION project (PIA/ANR-16-CONV-0005), the European Union’s Horizon 2020 research and innovation program under grant 101003589 (RECOVER) and 874735 (VEO), AXA and Groupama. TKT acknowledges the Seed Fund for Basic Research (202111159118) from the University of Hong Kong.

Senior Editor

  1. Miles P Davenport, University of New South Wales, Australia

Reviewing Editor

  1. Amy Wesolowski, Johns Hopkins Bloomberg School of Public Health, United States

Publication history

  1. Received: August 11, 2022
  2. Preprint posted: August 30, 2022 (view preprint)
  3. Accepted: February 20, 2023
  4. Version of Record published: March 7, 2023 (version 1)

Copyright

© 2023, Tsang et al.

This article is distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use and redistribution provided that the original author and source are credited.

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  1. Tim K Tsang
  2. Xiaotong Huang
  3. Can Wang
  4. Sijie Chen
  5. Bingyi Yang
  6. Simon Cauchemez
  7. Benjamin John Cowling
(2023)
The effect of variation of individual infectiousness on SARS-CoV-2 transmission in households
eLife 12:e82611.
https://doi.org/10.7554/eLife.82611

Further reading

    1. Epidemiology and Global Health
    Peter Bruun-Rasmussen, Morten Hanefeld Dziegiel ... Søren Brunak
    Research Article Updated

    Background:

    Whether natural selection may have attributed to the observed blood group frequency differences between populations remains debatable. The ABO system has been associated with several diseases and recently also with susceptibility to COVID-19 infection. Associative studies of the RhD system and diseases are sparser. A large disease-wide risk analysis may further elucidate the relationship between the ABO/RhD blood groups and disease incidence.

    Methods:

    We performed a systematic log-linear quasi-Poisson regression analysis of the ABO/RhD blood groups across 1,312 phecode diagnoses. Unlike prior studies, we determined the incidence rate ratio for each individual ABO blood group relative to all other ABO blood groups as opposed to using blood group O as the reference. Moreover, we used up to 41 years of nationwide Danish follow-up data, and a disease categorization scheme specifically developed for diagnosis-wide analysis. Further, we determined associations between the ABO/RhD blood groups and the age at the first diagnosis. Estimates were adjusted for multiple testing.

    Results:

    The retrospective cohort included 482,914 Danish patients (60.4% females). The incidence rate ratios (IRRs) of 101 phecodes were found statistically significant between the ABO blood groups, while the IRRs of 28 phecodes were found statistically significant for the RhD blood group. The associations included cancers and musculoskeletal-, genitourinary-, endocrinal-, infectious-, cardiovascular-, and gastrointestinal diseases.

    Conclusions:

    We found associations of disease-wide susceptibility differences between the blood groups of the ABO and RhD systems, including cancer of the tongue, monocytic leukemia, cervical cancer, osteoarthrosis, asthma, and HIV- and hepatitis B infection. We found marginal evidence of associations between the blood groups and the age at first diagnosis.

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    Novo Nordisk Foundation and the Innovation Fund Denmark

    1. Epidemiology and Global Health
    2. Genetics and Genomics
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    Research Article

    Background: To evaluate the utility of polygenic risk scores (PRS) in identifying high-risk individuals, different publicly available PRS for breast (n=85), prostate (n=37), colorectal (n=22) and lung cancers (n=11) were examined in a prospective study of 21,694 Chinese adults.

    Methods: We constructed PRS using weights curated in the online PGS Catalog. PRS performance was evaluated by distribution, discrimination, predictive ability, and calibration. Hazard ratios (HR) and corresponding confidence intervals [CI] of the common cancers after 20 years of follow-up were estimated using Cox proportional hazard models for different levels of PRS.

    Results: A total of 495 breast, 308 prostate, 332 female-colorectal, 409 male-colorectal, 181 female-lung and 381 male-lung incident cancers were identified. The area under receiver operating characteristic curve for the best performing site-specific PRS were 0.61 (PGS000873, breast), 0.70 (PGS00662, prostate), 0.65 (PGS000055, female-colorectal), 0.60 (PGS000734, male-colorectal) and 0.56 (PGS000721, female-lung), and 0.58 (PGS000070, male-lung), respectively. Compared to the middle quintile, individuals in the highest cancer-specific PRS quintile were 64% more likely to develop cancers of the breast, prostate, and colorectal. For lung cancer, the lowest cancer-specific PRS quintile was associated with 28-34% decreased risk compared to the middle quintile. In contrast, the hazard ratios observed for quintiles 4 (female-lung: 0.95 [0.61-1.47]; male-lung: 1.14 [0.82-1.57]) and 5 (female-lung: 0.95 [0.61-1.47]) were not significantly different from that for the middle quintile.

    Conclusions: Site-specific PRSs can stratify the risk of developing breast, prostate, and colorectal cancers in this East Asian population. Appropriate correction factors may be required to improve calibration.

    Funding This work is supported by the National Research Foundation Singapore (NRF-NRFF2017-02), PRECISION Health Research, Singapore (PRECISE) and the Agency for Science, Technology and Research (A*STAR). WP Koh was supported by National Medical Research Council, Singapore (NMRC/CSA/0055/2013). CC Khor was supported by National Research Foundation Singapore (NRF-NRFI2018-01). Rajkumar Dorajoo received a grant from the Agency for Science, Technology and Research Career Development Award (A*STAR CDA - 202D8090), and from Ministry of Health Healthy Longevity Catalyst Award (HLCA20Jan-0022). The Singapore Chinese Health Study was supported by grants from the National Medical Research Council, Singapore (NMRC/CIRG/1456/2016) and the U.S. National Institutes of Health [NIH] (R01 CA144034 and UM1 CA182876).