Fig 1.
Lexis diagram of the study population.
Lines show households from the date of inclusion in the study to infection of the first household member (brown dots), completion of the inclusion period without infection (blue dots at time in study of 161 days) or dropout (blue dots with time in study shorter than 161 days). Date of inclusion of the first household was 24 August 2020, and last date of the study was 29 July 2021.
Fig 2.
Estimates of the household introduction hazard for adults.
Shown are the posterior median of the introduction hazard per person (blue line) with associated 95% credible envelope (gray area). Household introduction hazards of children and adolescents are obtained by multiplication of the hazard for adults with the relative introduction hazards for children and adolescents. Also presented are the daily number of hospitalisations (yellow dots). To remove weekday effects the number of hospitalisations are represented by a 7-day moving average centered around the current day.
Table 1.
Overview of household infection data.
Shown are household compositions (columns 1-3), the total number of households with a given compositions (column 4), and the total number of household infections within each group (columns 5-7). Full data are available in the online repository (https://github.com/mvboven/sars2-households).
Table 2.
Comparison of household models using information criteria.
Model selection is based on the information criteria LOO_IC and WBIC. Shown are results for models that do not stratify the population by age (‘no stratification’), and that stratify the population into children (0-12 years), adolescents (12-17 years), and adults (over 18 years). Stratified transmission models are considered in which susceptibility of different age groups is estimated while infectivity is assumed to be identical for different age groups (‘variable susceptibility’), in which infectivity is estimated while susceptibility is fixed (‘variable infectivity’), and in which both susceptibility and infectivity are estimated (‘proportionate mixing’). Within the ‘variable infectivity’ model we further consider sub-models with transmission rates for child-to-child, adolescent-to-adolescent, and adult-to-adult transmission. A saturated model with a separate parameter for each transmission rate is also considered (‘full model’). The number of within-household parameters is indicated by n. All models assume density-dependent transmission.
Fig 3.
Estimated person-to-person transmission probabilities.
Shown are posterior medians of the infectious contact probabilities, i.e. the probabilities that a transmission event would have occurred from an infected person over its infectious period if the contacted person had not already been infected by another person. Infectious contact probabilities are calculated from the person-to-person transmission rates per infectious period βij and the Laplace transform of the scaled infectious period distribution: , where α = 50 is the shape parameter of the infectious period probability distribution.
Table 3.
Estimates of within-household transmission parameters.
Parameter estimates are shown for the variable infectivity model with separate child-to-child transmission (cf. Table 2). Estimates are represented by posterior medians and 2.5% and 97.5% posterior quantiles, and are based on 1,000 samples from the posterior distribution. Notice that the introduction hazard parameters of children and adolescents are relative to the introduction hazard in adults (cf. Fig 2).
Table 4.
Estimates of within-household transmission parameters when coprimary cases are assumed to be infected within the household.
Parameter estimates are shown for the variable infectivity model with separate child-to-child transmission. See Table 3 for details and comparison with the main analysis.
Fig 4.
Posterior predictive checks in common households of size four, containing two adults and two children (cf. Table 1).
Shown are the fractions of the households with a given number of secondary infections (black dots) with exact binomial confidence ranges (lines) stratified by primary case (child or adult), together with the corresponding posterior probabilities of secondary cases.
Table 5.
Estimated secondary attack rates without and with vaccination.
The vaccine is assumed to be 90% effective (VES = 0.9) in preventing infection but not effective reducing infectiousness (VEI = 0). Shown are inferred secondary attack rates (SARs) for various household compositions. Estimates are represented by posterior medians and 2.5% and 97.5% posterior quantiles. Households consist of either a child and an adult (rows 1-2), an adolescent and an adult (rows 3-4), two children and two adults (rows 5-6), two adolescents and two adults (rows 7-8), or two children, two adolescents, and two adults (rows 9-11), thus including the most common household compositions with children and adolescents in the Netherlands (rows 1-8). For each household composition, SARs are calculated for all possible primary cases. Vaccination scenarios are considered in which adults are vaccinated, or in which both adults and adolescents are vaccinated. NA: households do not contain an adolescent. Estimates are based on 1,000 samples from the posterior distribution.
Fig 5.
Estimates of the probability of infection of an adult for different household compositions, primary infection (child, adolescent, adult), and vaccination strategies.
(A) no vaccination, (B) vaccination of adults, and (c) vaccination of adults and adolescents (C). Plots represent the posterior distribution (1,000 samples), and black dots indicate posterior medians. Vaccine efficacy for susceptibility is VES = 0.9. Notice the difference in scale on the y-axis between (A) and (B)-(C).
Table 6.
Estimated secondary attack rates without and with vaccination.
The vaccine is assumed to be 90% effective (VES = 0.9) in preventing infection and moderately effective reducing infectiousness (VEI = 0.5). Shown are inferred secondary attack rates (SARs) for various household compositions. Estimates are represented by posterior medians and 2.5% and 97.5% posterior quantiles. See Table 5 for details, and results without vaccination and with vaccination with a vaccine that does not reduce infectiousness.