### Rapid survey of social contacts and risk mitigation behaviour during December 2021

We developed an online survey about plans for Christmas 2021 to fill the gap in social contact and behavioural data. The survey covered the festive period from 20 December 2021 to 2 January 2022 and included questions about planned face-to-face interactions, numbers of households meeting indoors, vaccination and risk mitigation behaviours (complete list of questions in Additional file 1). The survey was advertised to the participants of three longitudinal cohorts: the Avon Longitudinal Study of Parents and Children (ALSPAC) [16,17,18], TwinsUK [19, 20] and the COVID Symptom Survey (CSS) Biobank [21]. TwinsUK and the CSS Biobank were managed by the same team and were treated as one combined cohort for this study.

ALSPAC is an intergenerational prospective birth cohort from the southwest of England. The study recruited 14,541 pregnant women with expected dates of delivery between 1 April 1991 and 31 December 1992 in the former county of Avon and has followed the women, their partners and children since. Full details of the cohort and study design have been described previously [16,17,18] and are available at www.bristol.ac.uk/alspac. The ALSPAC survey was deployed using Microsoft Forms. The survey was an anonymous, standalone survey and data were not linked to any other data on participants. The survey link went live on 9 December 2021 and was active until 22 December 2021. Participants of ALSPAC were invited to participate via a link in the annual newsletter which went to participants on 15 December 2021 and via social media posts, although anyone with the link could complete the survey.

TwinsUK is a UK registry of volunteer twins in the United Kingdom, with about 14,000 registered twins [19, 20]. The COVID Symptom Study (CSS) Biobank is a longitudinal study run by researchers at King’s College London with approximately 12,000 participants [21]. The TwinsUK/CSS Biobank survey was implemented in REDCap, accessible via an anonymous link advertised in the Christmas newsletter. The survey link was active from 15 to 20 December 2021.

Data from the surveys were analysed in R version 4.01. We calculated descriptive statistics by age, calculating the mean and 95% binomial confidence intervals. We used a logistic regression model to explore associations between risk mitigation behaviours.

### Modelling approach

We used an individual-based disease model based on social contact data, validated against the early growth of SARS-CoV-2 in England in 2020 [8]. The basic premise behind the approach is that we calculate a distribution of individual reproduction numbers for the entire population, based on individuals’ reported social contacts.

Say individual *i* has *k*_{i} social contacts on a given day. Each *k*th social contact involves *n*_{k} other individuals and it lasts for a time *d*_{k}, which acts to weight the number of contacts. Their personal individual reproduction number (i.e. the number of secondary cases they generate) is given by

$$R_i=tau sum_k=1^k_ileft[ SARright]_kn_kd_k,$$

where [*SAR*]_{k} is the Omicron-specific secondary attack rate (proportion of contacts that result in secondary infection) for the setting of the social contact, either household or non-household and *τ* is a constant calibrated to the reproduction number *R*_{} = 7 for the Delta variant (i.e. with Delta-specific secondary attack rates) in the absence of vaccination or natural immunity. We used social contact data from the social contact survey (SCS) [22, 23] and secondary attack rates estimated by UKHSA from positive tests in contacts named to NHS Test and Trace [14]. We use [*SAR*]_{k} to weight the type of contact (e.g. household contacts are more “intimate” than non-household ones) but assume that, for a given type of contact, all individuals have the same infectivity and susceptibility.

To calculate the population-level reproduction number from the individual reproduction numbers, we assume proportionate mixing between individuals, i.e. that the probability of contacting individual *j* is proportional to their number of contacts over the total number of contacts in the population, *R*_{j}/∑_{j}*R*_{j}. The population-level reproduction number therefore scales with the square of the individual-level reproduction numbers:

$$R_tsim fracsum_jleft(R_jright)^2sum_jR_j.$$

We use the individual reproduction numbers to calculate the cumulative numbers of cases, hospital admissions and deaths. We use the notation *σ*_{j} to denote the probability that individual *j* does not get infected during the ensuing epidemic wave. *σ*_{j} depends on the susceptibility of individual *j* but also on the infectiousness of all other individuals and the probability that they do/do not get infected; therefore, there is no closed form solution for calculating *σ*_{j}. Following [24], *σ*_{j} can be shown to be

$$log sigma_j=-R_jfracsum_kR_kleft(1-sigma_kright)sum_kR_k.$$

As there is no closed form solution for calculating *σ*_{j}, it is calculated by iteration, starting with *σ*_{j} = 0.5 for all *j*s and recalculating all final sizes, repeating until the estimates converge.

The cumulative number of cases is calculated from the individual *σ*_{j}s by multiplying by an individual-specific weight *w*_{j} based on the representativeness of the social contact survey that is used for the model. The total number of cases is ∑_{j}*w*_{j}(1 − *σ*_{j}). The number of deaths is calculated from the number of cases, multiplied by the age-specific infection fatality rate, ∑_{j}*w*_{j}(1 − *σ*_{j})*μ*_{j}.

### Modelling vaccination and natural immunity

We capture vaccination using the vaccination line list data provided by UKHSA on 26 November 2021. We aggregated the data to calculate the proportion by age that had received a single or double dose of each of the main available vaccines in England (AstraZeneca or Pfizer/Moderna), and a booster dose, leading to five categories. We estimated the proportion of individuals by age with immunity from a natural infection using the Pillar 2 data of test positive cases, assuming that 50% of infections were identified as cases. Therefore, we model the population using eight categories of vaccine/immune status: five vaccination states, immunity from vaccination and a natural infection, immunity from natural infection only and no immunity/unprotected. We made the simplifying assumption for the initial conditions that vaccine and infection status were independent—the proportions of each age group falling within the eight categories were calculated using data from 26 November 2021 (see Additional file 1, Table S1). We assumed immunity from vaccination and natural infection to provide superior protection against severe disease than vaccination alone.

The effect of vaccination is incorporated into the model via three mechanisms: by reducing the probability that an individual is infected (reduced susceptibility), reducing the probability that the individual will transmit to others (reduced transmissibility) and reducing the risk of severe disease and death. We use UKHSA estimates of vaccine effectiveness for the Delta variant [15] and use the approach in [25] to scale the effectiveness against infection and transmission down by taking the *v*^{th} power of the effectiveness expressed as a proportion—therefore, an effectiveness of 80% against Delta becomes 41% against Omicron [25]. We use a different exponent for the reduction in effectiveness against infection and against death.

The individual reproduction number is modified by vaccination by reducing the probability of transmission

$$R_i^(v)=left(1-varepsilon_t^(v)right)tau sum_k=1^k_ileft[ SARright]_kn_kd_k,$$

where (varepsilon_t^(v)) is the vaccine effectiveness against transmission for vaccine/immunity state *v*.

The population-level reproduction number is formed of all vaccine states

$$R_t^vacsim fracsum_ileft(sum_valpha_i^(v)R_i^(v)r_i^(v)right)sum_iR_i,$$

where (alpha_i^(v)) is the probability that individual *i* is in vaccination state *v* , such that (sum_valpha_i^(v)=1kern0.5em)and (r_i^(v)) is the ‘receiving’ risk of infection,

$$r_i^(v)=tau left(1-varepsilon_s^(v)right) sum_k=1^k_ileft[ SARright]_kn_kd_k,$$

with (varepsilon_s^(v)) being the vaccine effectiveness against infection for vaccine state *v* and *k* a constant calibrated to the initial reproduction number without vaccination.

The final size calculations are also modified by the action of the vaccine,

$$sigma_i^(v)=exp left(-r_i^(v)uptheta_iright),$$

where

$$uptheta_i=frac1sum_jR_jsum_jsum_valpha_j^(v)R_j^(v)left(1-sigma_j^(v)right).$$

The cumulative number of cases is calculated as

$$cases=sum_jw_jsum_valpha_j^(v)left(1-sigma_j^(v)right).$$

The cumulative number of hospital admissions is calculated with the individual-specific hospital admission rate *h*_{j} as:

$$hospital admissions=sum_jh_jw_jsum_valpha_j^(v)left(1-sigma_j^(v)right).$$

Finally, the cumulative number of deaths is calculated using the individual-specific mortality admission rate *μ*_{j} as:

$$deaths=sum_jmu_jw_jsum_valpha_j^(v)left(1-sigma_j^(v)right).$$

### Risk mitigation measures

This framework allows us to model risk mitigation measures at an individual level. In the model, an individual is associated with a probability of exhibiting risk mitigation behaviours, according to age, and determined by survey responses. For each model iteration, an individual is determined to practice that risk mitigation measure or not, based on a random number draw. For example, in persons aged 30–39, 67% report limiting in-person visits to shops, 59% report using a face mask, 51% report avoiding public transport, 47% report working from home and 81% report using home testing kits. So, for a single model run for an individual aged 35, we draw a random number, say *rand*_{1} = 0.69, in which case this individual would not limit visits to shops or public transport use, or use a face mask, or work from home, but would use home testing kits. We assume a single number determines all risk mitigation measures for an individual, rather than choosing each of them independently, reflecting the evidence that some individuals tend to be more cautious than others in all their activities.

Contact tracing is applied to symptomatic cases only (where the probability of symptoms is determined by the age of the individual), implemented by reducing the number of secondary cases by a proportion *CTF*, determined from the NHS Test and Trace statistics as approximately (proportion of cases reached and asked to provide details of recent close contacts) × (proportion who provided details for one or more close contact) × (proportion of contacts reached within 24 h) × (proportion of close contacts reached and asked to self-isolate) [26].

Lateral flow testing of asymptomatic cases is implemented in a similar way to contact tracing but originating from asymptomatic cases. Individuals were given an age-specific probability of using home testing kits based on the results of the ALSPAC survey. If a home testing kit was used and infection was identified, the number of secondary cases is reduced by the same proportion as for contact tracing, *CTF*. The sensitivity of lateral flow testing was taken as *s*_{L} = 50% [27].

Mask wearing was implemented by reducing the probability of transmission by a proportion *CS*. Our estimate for the current impact of mask-wearing on transmission is less than 25%. In a 2020 systematic review, Chu et al. reported a smaller risk reduction for face mask use in non-healthcare settings compared to for healthcare settings and a smaller reduction for single layer face masks as opposed to respirators and surgical masks [28, 29].

Working from home, limiting in-person shopping and avoiding public transport were implemented by eliminating contacts reported as occurring at work/in shops/on public transport. If a contact did not take place, we set *n*_{k} = 0 for that interaction. In addition, we simulated school holidays over Christmas by removing all contacts for children under 18 years old with “school” listed as the context, as these were assumed not to take place during the winter holidays.

### Changes to disease severity

We investigate four severity scenarios to illustrate the potential impact of the Omicron variant: (A) moderate severity and reduced vaccine effectiveness: a 20% reduction in mortality rates associated with Omicron infection and a reduction in vaccine effectiveness against severe disease compared to the Delta variant (*μ*_{Ο} = 0.8, *v*_{I} = 4, *v*_{D} = 4); (B) low severity and reduced vaccine effectiveness: a 50% reduction in mortality rates associated with Omicron infection and a reduction in vaccine effectiveness against severe disease compared to the Delta variant (*μ*_{Ο} = 0.5, *v*_{I} = 4, *v*_{D} = 4); (c) moderate severity and no reduction in vaccine effectiveness: a 20% reduction in mortality rates associated with Omicron infection with no reduction in vaccine effectiveness against severe disease (*μ*_{Ο} = 0.8, *v*_{I} = 4, *v*_{D} = 1); (d) low severity and no reduction in vaccine effectiveness: a 40% reduction in mortality rates associated with Omicron infection with a small reduction in vaccine effectiveness against severe disease (*μ*_{Ο} = 0.5, *v*_{I} = 4, *v*_{D} = 2).

### Model implementation

The model was written in R version 4.01. The population of individuals was simulated 100 times and results aggregated. A summary of parameter values and interpretations is given in Table 1. The model code is available at https://github.com/ellen-is/Reckoners-Xmas21 [32].