Quantifying the relationship between swab positivity and severe outcomes

The time-lag between swab positivity and severe outcomes decreased over the duration of the study. Fitting the time-delay model (see Methods) to rounds 1 to 7 (1 May to 3 December 2020) of REACT-1, we estimated a discrete time-lag of 19 (18, 20) days to hospitalisations, and a time-lag of 26 (25, 27) days to deaths (Fig 1, S1 Table). These estimates are in line with those obtained by fitting the same model to all 19 rounds of the data. Fitting the model to rounds 14 to 19 (9 September 2021 to 31 March 2022) of the study, we estimated much shorter time-lags of 7 (7, 8) days to hospitalisations and 18 (18, 18) days to deaths. Models fit to subsets of the data by age group (64 years and under, 65 years and over) showed similar time-lags between age-groups, though the time-lags tended to be slightly shorter for those aged 64 and under. In contrast, fitting the models to rounds 8 to 13 (30 December 2020 to 12 July 2021) of REACT-1 identified very different estimates between age groups and some extremely long time-lags. However, this was over a period of time when large proportions of the population were being vaccinated against Coronavirus Disease 2019 (COVID-19), with different rates of vaccination by age group (Fig 2). This could have led to continuously changing severity, severely biasing estimates obtained from the model. Due to the likely high degree of bias during rounds 8 to 13, below we present only the models obtained from rounds 1 to 7 and rounds 14 to 19.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 1. Comparison of parameters obtained by fitting to different subsets of the REACT-1 data.

(A) Best-fitting time-lag parameter and 95% credible intervals for models fit to deaths (cross, time-lag between swab positivity and deaths) and hospitalisations (vertical line, time-lag between swab positivity and hospitalisations), for models fit to all age-groups (green), those aged 64 years and under (orange), and those aged 65 years and over (purple). (B) Best-fitting scaling parameter and 95% credible intervals on a log₁₀ x-axis for models fit to deaths (cross) and hospitalisations (vertical line), for models fit to all age-groups (green), those aged 64 years and under (orange), and those aged 65 years and over (purple). All numerical values are provided in S1 Table.

https://doi.org/10.1371/journal.pbio.3002118.g001

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 2. Vaccination coverage and variants responsible for infections in England, as inferred from public data sets.

(A, B) Proportion of individuals who have been vaccinated with at least a single dose (green), proportion of individuals who have been vaccinated with at least 2 doses (orange), and the proportion of individuals who have received a third “booster” dose (purple), for those aged 64 years and under (A) and those aged 65 years and over (B). (C) Proportion of infections in England, for which lineages were detected, which were identified as the Omicron variant (purple), the Delta variant (orange), the Alpha variant (green), and any other lineage (pink). Data supporting this figure can be found in S1 Data.

https://doi.org/10.1371/journal.pbio.3002118.g002

There was a high degree of correlation between smoothed estimates of swab positivity and the time-delayed signal of severe outcomes, though over the course of the study, the 2 signals substantially diverged (Figs 3–5, S2 Table). Assuming the estimated time-lag for the time-delay models fit to rounds 1 to 7 of REACT-1, we found a Pearson correlation coefficient for modelled swab positivity over rounds 1 to 7 against death and hospitalisations of 0.985 and 0.978, respectively (p-values < 0.001). Similarly high levels of correlation were found for rounds 8 to 13 assuming the same time-lag. Correlation over rounds 14 to 19 was lower at 0.732 and 0.757 (for death and hospitalisations, respectively) when assuming a time-lag from models fit to rounds 1 to 7 (though still significant, p-values < 0.001), but only slightly lower at 0.895 and 0.955 (for death and hospitalisations, respectively) when assuming the time-lags from models fit to rounds 14 to 19. Broadly similar patterns in correlations were found when looking at both age-groups. However, for those aged 64 and under during rounds 14 to 19, correlation was greatly reduced with a value of 0.169 (p-value = 0.01), even when estimating using the time-lag from the model fit to rounds 14 to 19.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 3. A comparison of daily deaths and hospitalisations to swab positivity as measured by REACT-1.

Daily swab positivity for all 19 rounds of the REACT-1 study (black points with 95% credible intervals, left hand y-axis) with P-spline estimates for swab positivity (solid black line, shaded area is 95% credible interval). (A) Daily deaths in England (red points, right hand y-axis) and P-spline model estimates for expected daily deaths in England (solid red line, shaded area is 95% credible interval, right hand y-axis). The black vertical dashed line on 10 August 2021 splits the data into 2 periods: rounds 1–13 and rounds 14–19 of REACT-1. During rounds 1–13, daily deaths have been shifted by 26 days backwards in time along the x-axis. During rounds 14–19, daily deaths have been shifted by 18 days backwards in time along the x-axis. The 2 y-axes have been scaled using the population size and best-fit scaling parameter from the time-delay model fit to rounds 1–7. (B) Daily hospitalisations in England (blue points, right hand y-axis) and P-spline model estimates for expected daily hospitalisations in England (solid blue line, shaded area is 95% credible interval, right hand y-axis). The black vertical dashed line on 10 August 2021 splits the data into 2 periods: rounds 1–13 and rounds 14–19 of REACT-1. During rounds 1–13, daily hospitalisations have been shifted by 19 days backwards in time along the x-axis. During rounds 14–19, daily hospitalisations have been shifted by 7 days backwards in time along the x-axis. The 2 y-axes have been scaled using the population size and best-fit scaling parameter from the time-delay model fit to rounds 1–7 of REACT-1. Data supporting this figure can be found in S2 Data.

https://doi.org/10.1371/journal.pbio.3002118.g003

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 4. A comparison of daily deaths to swab positivity as measured by REACT-1, by age group.

Daily swab positivity for all 19 rounds of the REACT-1 study (black points with 95% credible intervals, left hand y-axis) with P-spline estimates for swab positivity (solid black line, shaded area is 95% credible interval) for (A) those aged 64 years and under, and (B) those aged 65 years and over. (A) Daily deaths for those aged 64 years and under in England (red points, right hand y-axis) and corresponding P-spline model estimates for the expected number of deaths (solid red line, shaded area is 95% credible interval, right hand y-axis). The black vertical dashed line on 10 August 2021 splits the data into 2 periods: rounds 1–13 and rounds 14–19 of REACT-1. During rounds 1–13, daily deaths have been shifted by 24 days backwards in time along the x-axis. During rounds 14–19, daily deaths have been shifted by 16 days backwards in time along the x-axis. The 2 y-axes have been scaled using the population size and best-fit scaling parameter from the time-delay model fit to rounds 1–7 of REACT-1. (B) Daily deaths for those aged 65 years and over in England (red points, right hand y-axis) and corresponding P-spline model estimates for the expected number of deaths (solid red line, shaded area is 95% credible interval, right hand y-axis). The black vertical dashed line on 10 August 2021 splits the data into 2 periods: rounds 1–13 and rounds 14–19 of REACT-1. During rounds 1–13, daily deaths have been shifted by 24 days backwards in time along the x-axis. During rounds 14–19, daily deaths have been shifted by 19 days backwards in time along the x-axis. The 2 y-axes have been scaled using the population size and best-fit scaling parameter from the time-delay model fit to rounds 1–7 of REACT-1. Data supporting this figure can be found in S2 Data.

https://doi.org/10.1371/journal.pbio.3002118.g004

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 5.

A comparison of daily hospitalisations to swab positivity as measured by REACT-1, by age group. Daily swab positivity for all 19 rounds of the REACT-1 study (black points with 95% credible intervals, left hand y-axis) with P-spline estimates for swab positivity (solid black line, shaded area is 95% credible interval) for (A) those aged 64 years and under, and (B) those aged 65 years and over. (A) Daily hospitalisations for those aged 64 years and under in England (blue points, right hand y-axis) and corresponding P-spline model estimates for the expected number of hospitalisations (solid blue line, shaded area is 95% credible interval, right hand y-axis). The black vertical dashed line on 10 August 2021 splits the data into 2 periods: rounds 1–13 and rounds 14–19 of REACT-1. During rounds 1–13, daily hospitalisations have been shifted by 17 days backwards in time along the x-axis. During rounds 14–19, daily hospitalisations have been shifted by 6 days backwards in time along the x-axis. The 2 y-axes have been scaled using the population size and best-fit scaling parameter from the time-delay model fit to rounds 1–7 of REACT-1. (B) Daily hospitalisations for those aged 65 years and over in England (blue points, right hand y-axis) and corresponding P-spline model estimates for the expected number of hospitalisations (solid blue line, shaded area is 95% credible interval, right hand y-axis). Daily hospitalisations have been shifted by 18 days backwards in time along the x-axis. The black vertical dashed line on 10 August 2021 splits the data into 2 periods: rounds 1–13 and rounds 14–19 of REACT-1. During rounds 1–13, daily hospitalisations have been shifted by 18 days backwards in time along the x-axis. During rounds 14–19, daily hospitalisations have been shifted by 9 days backwards in time along the x-axis. The 2 y-axes have been scaled using the population size and best-fit scaling parameter from the time-delay model fit to rounds 1–7 of REACT-1. Data supporting this figure can be found in S2 Data.

https://doi.org/10.1371/journal.pbio.3002118.g005

The estimated severity of infection was found to decrease over the duration of the study. Using the time-delay model, we were able to estimate the IFR and IHR (see Methods for assumptions). Fitting the model to rounds 1 to 7 (1 May to 3 December 2020), we estimated the IHR to be 2.6% (2.5%, 2.7%), and the IFR to be 0.67% (0.65%, 0.70%). Fitting the model instead to rounds 14 to 19 (9 September 2021 to 31 March 2022), we estimated the IHR to be approximately 3.5-fold lower at 0.76% (0.75%, 0.77%), and the IFR to be approximately 7-fold lower at 0.097% (0.096%, 0.099%).

The severity of the virus, as measured per infected individual, was far lower in those aged 64 and under, relative to those aged 65 and over. From the models fitted to rounds 1 to 7 (1 May to 3 December 2020), we estimated the IHR to be 0.96% (0.93%, 1.00%) and the IFR to be 0.059% (0.057%, 0.061%) for those aged 64 and under. In comparison, the IHR and IFR for those aged 65 and over were approximately 16-fold and 91-fold higher, at 15% (14%, 17%) and 5.4% (4.9%, 5.9%), respectively, for the same time period. As before, estimates of the IHR and IFR were found to be lower when the model was fitted to rounds 14 to 19.

Changes in severity during mass vaccination and the emergence of Alpha, Delta

We detected an increase in the severity of infection into the winter of 2020. Using the best-fit time-lag obtained from fitting the time-delay model to rounds 1 to 7 (1 May to 3 December 2020) of REACT-1, we estimated the daily IFR and IHR from the modelled estimates of swab positivity and the time-lag adjusted signals of hospitalisations and deaths over rounds 1 to 13 (1 May 2020 to 12 July 2021) (Fig 6). An increase in the daily IFR was observed in late November 2020, with the increase lasting until late January 2021. We compared the mean IFRs and IHRs over approximately 4-week periods to a baseline period from 1 May 2020 to 11 November 2020 (S3 Table), a period before vaccinations when wild-type variants dominated. The mean IFR in late November 2020 (8 November to 5 December) was 1.68 (1.39, 1.93) times greater than baseline, and in January 2021 (3 January to 30 January) was 1.31 (1.11, 1.56) times greater than baseline; during these 2 periods, the Alpha variant was responsible for 15% and 86% of infections, respectively. The increase in the IFR was observed in both age-groups, but no increase was observed in the IHR.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 6. Estimates of the IFR and IHR over 19 rounds of REACT-1.

IFR and IHR (solid black line, grey shaded region is 95% credible interval) estimated from the multiplicative difference between the REACT-1 P-splines for swab positivity and the time-delay adjusted death or hospitalisation P-splines, accounting for population size, mean duration of positivity, and test sensitivity. The 95% credible intervals of the best-fitting average IFR and IHR over rounds 1–7 (red shaded area) and rounds 14–19 (blue shaded area) estimated using time-delay models are shown for comparison (available in S1 Table). The black vertical dashed line on 10 August 2021 splits the data into 2 periods: rounds 1–13 and rounds 14–19 of REACT-1. (A) IFR across all age groups assuming a time-lag of 26 days during rounds 1–13 and 18 days during rounds 14–19. (B) IHR across all age groups assuming a time-lag of 19 days during rounds 1–13 and 7 days during rounds 14–19. (C) IFR in those aged 64 years and under assuming a time-lag of 24 days during rounds 1–13 and 16 days during rounds 14–19. (D) IHR in those aged 64 years and under assuming a time-lag of 17 days during rounds 1–13 and 6 days during rounds 14–19. (E) IFR in those aged 65 years and over assuming a time-lag of 24 days during rounds 1–13 and 19 days during rounds 14–19. (F) IHR in those aged 65 years and over assuming a time-lag of 18 days during rounds 1–13 and 9 days during rounds 14–19. Data supporting this figure can be found in S3 Data.

https://doi.org/10.1371/journal.pbio.3002118.g006

The severity of the virus, as measured by the IFR and IHR, decreased from late January 2021 until April 2021, after which it increased until 12 July 2021 (the end of round 13). Both the daily IFR and daily IHR decreased until April. The mean IFR in April 2021 (28 March to 24 April) reduced to 0.25 (0.17, 0.34) of baseline, and the mean IHR to 0.51 (0.35, 0.68) of baseline. During this period, the Alpha variant represented 96% of infections, 47% of the population had received at least 1 dose, and 11% 2 doses of a vaccine. By June/July 2021 (20 June to 17 July), the mean IFR increased to 0.43 (0.37, 0.53) of baseline, and the mean IHR increased to 0.84 (0.72, 1.03) of baseline. At this point, while the proportion vaccinated had increased to 66% with at least 1 dose and 50% with 2 doses, the Delta variant was now the dominant variant representing 99% of infections over this period.

There was a more substantial and quicker decrease in severity for those aged 65 and over, compared to those aged 64 and under. In April 2021 (28 March to 24 April), for those aged 65 and over, the mean IFR and IHR were 0.28 (0.18, 0.49) and 0.42 (0.27, 0.74) of baseline, respectively. For those aged 64 and under, the mean IFR and IHR were comparatively higher (though with overlapping credible intervals) at 0.48 (0.33, 0.71) and 0.72 (0.52, 0.97) of their baseline, respectively. By June/July 2021 (20 June to 17 July), the mean IFR had increased to 0.98 (0.80, 1.15) of baseline for those aged 64 and under, but only to 0.63 (0.44, 1.06) for those aged 65 and over. The mean IHR in June/July 2021 had only increased to 0.76 (0.53, 1.24) for those aged 65 and over, but it was now significantly higher than baseline in those aged 64 and under, at 1.32 (1.14, 1.55) times baseline. There were large differences in the proportion vaccinated by this period; 98% of those aged 65 and over had received 2 doses of vaccine, whereas only 39% of those aged 64 and under had received 2 doses of vaccine.

Changes in severity during booster vaccination and the emergence of Omicron

From September 2021 to April 2022, the severity of infection decreased. Using the best-fit time-lag obtained from fitting the time-delay model to rounds 14 to 19, we estimated the daily IFR and IHR from the modelled estimates of swab positivity and the time-lag adjusted signals of hospitalisations and deaths for rounds 14 to 19 of REACT-1 (Fig 6). The daily IFR and IHR decreased steadily from September 2021 onwards with a sharp and rapid reduction in late December 2021. Over this period, the proportion of the population that had received a third dose of vaccine (“booster” dose) steadily increased, saturating at about 54% by mid-January 2021 (S4 Table). Observed trends in the daily IFR and IHR were broadly similar across both age groups, despite approximately double the proportion of the population aged 65 and over having received a booster dose, compared to the population aged 64 and under, by the end of March 2022. The rapid reduction of the daily scaling parameters in late December 2021 coincided with the rapid increase in the proportion of infections caused by the Omicron variant. We compared the mean IFRs and IHRs over approximately 4-week periods to a baseline period from 4 September 2021 to 16 October 2021 (S4 Table), a period before Omicron’s emergence and with low proportions of the population having received booster doses. By March 2022 (6 March to 2 April), when Omicron had reached near total coverage (responsible for 99% of infections), the mean IFR was 0.069% (0.066%, 0.072%) at 0.36 (0.31, 0.42) of baseline, and the mean IHR was 0.62% (0.58%, 0.65%) at 0.50 (0.44, 0.59) of baseline.

Changes in the case ascertainment rate of England’s mass testing programme

The time-lag between swab positivity and daily case numbers (identified through England’s mass testing) varied over the duration of the study. Fitting the time-delay model (see Methods) for daily case numbers to rounds 1 to 7 (1 May to 3 December 2020) of REACT-1 we estimated a discrete time-lag of 3 (3, 4) days (Fig 7, S5 Table). Over rounds 8 to 13 (30 December 2020 to 12 July 2021), we estimated a discrete time-lag of −7 (−8, −7) days (the time-series of daily cases led the time-series of swab positivity), and over rounds 14 to 19 (9 September 2021 to 31 March 2022), we estimated a discrete time-lag of 1 (1, 1) days. There was a high degree of correlation between the smoothed estimates of swab positivity and the time-adjusted signals of daily case numbers, but this correlation was lowest for rounds 14 to 19, which saw some divergences in the 2 signals (Fig 7, S6 Table).

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 7. A comparison of daily cases to swab positivity as measured by REACT-1.

(A, B, C) Daily swab positivity for all 19 rounds of the REACT-1 study (black points with 95% credible intervals, left hand y-axis) with P-spline estimates for swab positivity (solid black line, shaded area is 95% credible interval). Daily cases in England (green points, right hand y-axis) and P-spline model for expected daily cases in England (solid green line, shaded area is 95% credible interval, right hand y-axis). The 2 y-axes have been scaled using the population size and best-fit scaling parameter from the time-delay model fit to the rounds shown in each subfigure (available in S5 Table). (A) During round 1–7, daily cases have been shifted by 3 days backwards in time along the x-axis. (B) During round 8–13, daily cases have been shifted by 7 days forwards (−7 days backwards) in time along the x-axis. (C) During round 14–19, daily cases have been shifted by 1 day backwards in time along the x-axis. (D) Estimates of the case ascertainment rate over 19 rounds of REACT-1. Case ascertainment (solid black line, grey shaded region is 95% credible interval) estimated from the multiplicative difference between the REACT-1 P-spline for swab positivity and the time-delay adjusted P-spline for daily cases, accounting for population size, mean duration of positivity, and test sensitivity. The 95% credible intervals of the best-fitting average case ascertainment rates (red shaded area) over rounds 1–7, rounds 8–13, and rounds 14–19 estimated using separate time-delay models are shown for comparison (available in S5 Table). The black vertical dashed lines split the data into 3 periods: round 1–7, rounds 8–13, and rounds 14–19 of REACT-1. During rounds 1–7, the case ascertainment was estimated assuming a time-lag of 3 days. During rounds 8–13, the case ascertainment was estimated assuming a time-lag of −7 days. During rounds 14–19, the case ascertainment was estimated assuming a time-lag of 1 days. Data supporting this figure can be found in S4 Data.

https://doi.org/10.1371/journal.pbio.3002118.g007

The estimated case ascertainment rate (the proportion of infections identified with a positive test by England’s mass testing programmes [see Methods]) varied over the duration of the study. Similarly to before (when estimating IFR and IHR), we were able to use the time-delay model to estimate the case ascertainment rate (see Methods for assumptions). Fitting the model to all 19 rounds, we estimated the case ascertainment rate to be 36.1% (35.7%, 36.5%). However, fitting the model instead to subsets of rounds, we identified changes in the case ascertainment rate. Fitting the model to rounds 1 to 7, we estimated the case ascertainment rate to be 28.3% (27.5%, 29.2%). Fitting the model to rounds 8 to 13, we estimated the case ascertainment rate to be higher at 57.6% (56.0%, 59.4%). Finally, fitting the model to rounds 14 to 19, we estimated the case ascertainment rate to have decreased to 33.8% (33.3%, 34.3%).

There were some clear temporal trends in the daily case ascertainment rate at the timing of key changes. Using the best-fit time-lags obtained from fitting the time-delay model to each period of the REACT-1 study (rounds 1 to 7, 8 to 13, and 14 to 19), we estimated the daily case ascertainment rates from the modelled estimates of swab positivity and the time-lag adjusted daily case numbers (Fig 7). The case ascertainment rate increased in July 2020 from approximately 20% (for the period of May to June 2020) to approximately 30% (for the period of August to December 2020), in line with the rollout of mass community testing in England on 2020 July 2 (previously testing was clinically focused) [6]. There was a sharp increase in the case ascertainment rate from May 2021 to July 2021, a period in which Delta had just replaced Alpha as the dominant variant. In contrast, there was a sharp decrease in the case ascertainment rate from December 2021 to March 2022, the period in which Omicron replaced Delta as the dominant variant and resulted in 2 large waves of infection prevalence in England.

REACT-1 data

The REACT-1 methods are published [17,29]. At each round, a random subset of the population aged 5 years and over was contacted by letter using the list of general practitioner patients in England held by the NHS. Those who agreed to participate were then sent a self-administered swab test (for 5- to 12-year-olds, a parent/guardian administered the test). The participants completed a questionnaire providing sociodemographic information such as age, gender, ethnicity, and occupation. Swab tests were collected by a courier and sent via cold chain (latterly by post) to a commercial laboratory for a reverse transcription polymerase chain reaction (RT-PCR) test for SARS-CoV-2. A test was swab positive if either both N- and E-gene were detected, or N-gene detected with a Ct value less than 37. The REACT-1 study received research ethics approval from the South Central-Berkshire BResearch Ethics Committee (IRAS ID: 283787). Consent was obtained from all participants or their parent/guardian for minors. During initial registration for the study, participants were asked “Are you willing to take part in this study?/Are you willing for your child to take part in this study?” with possible answers being “1. Yes, I want to take part in this study” or “2. No, I do not want to take part.”. Those who answered “2. No, I do not want to take part.” were not sent testing kits and did not participate further in the study.

Public data

Data for deaths, hospitalisations, and cases were obtained from “the official UK government website for data and insights on coronavirus” [6]. Daily hospitalisation figures include all people in England who are admitted to hospital within 14 days of testing positive for SARS-CoV-2 and those who test positive after admission. Daily death figures include the number of people in England that died within 28 days of their first positive test for COVID-19 being reported (by date of death not date of reporting). Daily case numbers include the number of people in England who tested positive for SARS-CoV-2 by the date in which the sample was taken from the individual being tested. Hospitalisation and death data were available by age group; from this, we created 3 time series for each data set: the total number of events (death or hospitalisation), the number in those aged 64 years and under, and the number in those aged 65 years and over. Death, hospitalisation, and case data were downloaded on 20 May 2022, and only data up to 15 May 2022 were included in the analysis.

The daily cumulative number of individuals who had been vaccinated with a single dose and with 2 doses (any vaccine approved in the UK), by age group, was again downloaded from “the official UK government website for data and insights on coronavirus” [6]. Data by age group were collated into those aged 64 years and under, those aged 65 years and over, and at all ages. The cumulative number of vaccinations given (first and second doses) was then converted into the cumulative proportion of the population vaccinated using population estimates of England by age group [44]. Note that current population estimates are unavailable and are most likely greater than the values used; estimates of the proportion of the population vaccinated are thus not exact.

The weekly numbers of each SARS-CoV-2 lineage detected in routine surveillance data by lower tier local authority (LTLA) was downloaded from the “Lineage in Space and Time website” [45]. Data by LTLA were aggregated in order to give the daily number of each lineage detected in England as a whole. The weekly proportion of lineages that were the Alpha variant (or an Alpha sublineage), the Delta variant (or a Delta sublineage), and the Omicron variant (or an Omicron sublineage) was then straightforwardly calculated.

P-spline model + mixed-effects P-spline

Smoothed models were fit to each time series in order to get an estimate of the expected number of outcomes (deaths, hospitalisations, and cases) or the expected prevalence (REACT-1 swab positivity). Bayesian P-spline models [46,47] were fit to the hospitalisation and death data for each age group (64 years or under, 65 years and over, and all ages), daily case data (all ages), and to the overall swab positivity for REACT-1. To maintain statistical power, a single mixed-effects Bayesian P-spline model was fit to the swab positivity in REACT-1 for the 2 age groups. The models consist of a system of basis splines defined over the window of the study period with approximately 1 basis spline every 5 days. The P-spline model is then a linear combination of these basis splines: where π_j is the outcome variable of interest on the j^th day, g() is the link function (logit for Binomial swab positivity data, log for count data), b_i is the coefficient for the i^th basis spline, and B_ij is the value of the i^th basis spline on the j^th day. Overfitting of the model was prevented through the inclusion of a second-order random walk prior distribution on the basis spline coefficients, b_i: where

This prior distribution penalises any changes in the first derivative of the response function, reflecting the expected trend of an epidemic over a small time period (constant growth rate). The degree to which changes in the first derivative are penalised is controlled by ρ, which is a further parameter of the model and takes a loose but proper inverse gamma prior distribution ρ~IG(0.001,0.001). The first 2 basis spline coefficients are given an uninformative prior distribution, b₁, b₂~Constant.

In the mixed-effects version of the model in which the model fits to the time series for 2 age-groups simultaneously, the second-order random-walk prior distribution is modified slightly: where and where u_i,k is the mean value of u_i,k averaged over k age-groups. For the most part, this is the same as before, but now parameters are defined for k age-groups. The difference lies in that as well as penalising changes in the first derivative of the response function; differences between age groups in the changes of the first derivative are also penalised. This has the effect of syncing the model across age-groups while also allowing divergences to occur when there is sufficient evidence. The degree of penalisation is controlled by a further parameter ζ, which we give a loose but proper inverse-gamma prior distribution as before, ζ~IG(0.001, 0.001). All models are fit to data using a No-U-Turn Sampler (NUTS) [48] implemented in STAN [49]. For the REACT-1 data, we fit the daily weighted number of positive and negative tests assuming a Binomial likelihood. When fitting to the daily number of deaths, hospitalisations, and cases, we assume a Negative-Binomial likelihood with an overdispersion parameter that is treated as an additional parameter of the model with an uninformative constant prior distribution. The model fitting returns a full posterior distribution of all parameters from which the mean response function and credible intervals can be calculated. Estimates of swab positivity for the period between REACT-1 rounds 7 and 8 are not included in any analysis, as between the rounds, there was a large peak in infections that we cannot estimate using the REACT-1 data. Similarly, we did not include estimates of swab positivity for the period between rounds 13 and 14 in any analysis; this is because there was a substantial break in the study (approximately 2 months) for which dynamics cannot be inferred accurately. Estimates of swab positivity between other rounds are included as there was only a small break between most rounds (approximately 2 weeks).

Time-delay model

In order to investigate the relationship between the REACT-1 data and hospitalisations, deaths, and cases, we define a simple model consisting of 2 parameters. The first parameter, the time-lag τ, sets the discrete number of days between the REACT-1 data and the time series of interest. The second parameter, the scaling factor ϵ, sets the multiplicative difference between REACT-1 data and the time series of interest correcting for population size. The estimated proportion swab positive on day i is then related to the time series of interest on day i+τ by:

Written this way, the scaling parameter, ϵ, represents the percentage of those in the population that are swab positive on day i that will be admitted to hospital, have died, or have tested positive for SARS-CoV-2 through mass testing on day i+τ. The modelled proportion swab positive on day i, can then be fitted to the REACT-1 daily weighted number of positive and negative tests, assuming a weighted Binomial likelihood. If this was done using the raw data for hospitalisations, deaths, or cases, then there would be errors for any days in which zero counts occurred. There would also be the possibility for overfitting due to well-aligned noise between the raw time series and the REACT-1 data. In order to avoid these pitfalls, instead of using the raw data as the time series, the P-spline model estimates are used instead. To take into account the uncertainty in the P-spline model, a random subset of 1,000 parameter combinations is selected from the posterior distribution. For each set of parameter combinations, a time-series can be calculated. The average log-likelihood, fitting to the REACT-1 data, over all 1,000 random draws of the P-spline’s posterior distribution is then used to fit the time-lag and scaling parameter using an MCMC. The time-delay models were fit to all rounds of REACT-1 and to subsets of rounds: rounds 1 to 7, rounds 8 to 13, and rounds 14 to 19. Further analyses for deaths and hospitalisations were only performed using the time-lags from models fit to rounds 1 to 7 and rounds 14 to 19. Further analyses for daily cases were performed using the time-lags from models fit to rounds 1 to 7, rounds 8 to 13, and rounds 14 to 19. This was because drastic changes in severity over rounds 8 to 13 (due to vaccination) should not have had a large impact on case ascertainment over this period but would have had a large impact on the IFR and IHR.

Correlation between time series

Under the assumption of a particular time-lag, the Pearson correlation was measured between the mean daily swab positivity (as estimated using the P-spline model) and the time-delayed mean daily outcomes (deaths, hospitalisations, or cases) (as estimated using the P-spline model). Correlation between swab positivity and daily outcomes was estimated for the periods of rounds 1 to 7, rounds 8 to 13, and rounds 14 to 19. For deaths and hospitalisations, the time-lags used were those estimated from the time-delay model fit to rounds 1 to 7 and fit to rounds 14 to 19. For daily cases, the time-lags used were those estimated from the time-delay model fit to rounds 1 to 7, rounds 8 to 13, and rounds 14 to 19.

Variation in scaling parameter over time

With the assumption of a particular time-lag, and that it does not change over time, variation in the scaling parameter over time between REACT-1 data and hospitalisation, death, or case data can straightforwardly be estimated. For each day, the posterior distribution for the estimate of swab positivity that day, inferred from the P-spline model fit to REACT-1, can be extracted. Similarly, the posterior distribution for the estimate of hospitalisations, deaths, and cases on a fixed number of days later (determined by time-lag) can also be extracted. The daily scaling parameter, and its uncertainty, between the 2 P-spline estimates can then be calculated for the whole study period.

We calculated the average scaling parameter over a set period of time for deaths and hospitalisations. We split the time period of rounds 1 to 13 of REACT-1 into approximately 4-week periods and 1 initial baseline period running from 1 May 2020 to 7 November 2020. This period was chosen as the baseline as it was before any vaccination had occurred and before the Alpha variant had increased to a substantial proportion. Similarly, we split the time period of rounds 14 to 19 of REACT-1 into approximately 4-week periods and 1 initial baseline period running from 4 September 2021 to 16 October 2021. This period was chosen as the baseline as it occurred before Omicron emerged and before a substantial proportion of the population had received a third vaccine dose.

For each period, we extracted the posterior distribution for the estimates of swab positivity over the period, inferred from the P-spline model fit to REACT-1, and calculated the posterior distribution of the mean swab positivity over the period. Similarly, we extracted the posterior distribution for the estimate of hospitalisations and deaths for the equivalent period a fixed number of days later (determined by the time-lag) and calculated the posterior distribution of the mean. We then estimated the mean scaling parameter (the difference between mean swab positivity and mean hospitalisations or deaths) over the period, together with its uncertainty. Additionally, we estimated the multiplicative difference between the mean scaling parameter for a specific period and the baseline period and its uncertainty.

Converting the scaling parameters to the IFR, IHR, and case ascertainment rate

The scaling parameters above correspond to the percentage of those who are swab positive in a population on a particular day who will be hospitalised/have died/tested positive (as a result of mass testing) on a day in the future set by the time-lag parameter. By converting swab positivity to the incidence of infection, we can use the scaling parameters to estimate the IFR, IHR, and case ascertainment rate. The time-series of incidence and swab positivity are similar—though with a time-delay due to the finite duration, individuals remain swab positive after infection [42]. Under the simplifying assumption that swab positivity can be converted to incidence by a multiplicative constant, the IFR, IHR, and case ascertainment rate can be estimated by multiplying the scaling parameter by the same constant; we use the mean duration that an individual remains swab positive, estimated at 14.0 days, and sensitivity to detect a positive swab, estimated at 0.79 [42]. These 2 numbers allow all scaling parameter estimates to be converted to IFR (for deaths) and IHR (for hospitalisations) by multiplying the scaling parameter by 14.0×0.79 = 11.06.