key: cord-0946395-5i746v80
authors: Hart, W. S.; Abbott, S.; Endo, A.; Hellewell, J.; Miller, E.; Andrews, N.; Maini, P. K.; Thompson, R. N.
title: Inference of SARS-CoV-2 generation times using UK household data
date: 2021-05-30
journal: nan
DOI: 10.1101/2021.05.27.21257936
sha: 1369d4812698d112910fc209ecbe903d0cdaa99a
doc_id: 946395
cord_uid: 5i746v80

The distribution of the generation time (the interval between individuals becoming infected and passing on the virus) characterises changes in the transmission risk during SARS-CoV-2 infections. Inferring the generation time distribution is essential to plan and assess public health measures. We previously developed a mechanistic approach for estimating the generation time, which provided an improved fit to SARS-CoV-2 data from January-March 2020 compared to existing models. However, few estimates of the generation time exist based on data from later in the pandemic. Here, using data from a household study conducted from March-November 2020 in the UK, we provide updated estimates of the generation time. We consider both a commonly used approach in which the transmission risk is assumed to be independent of when symptoms develop, and our mechanistic model in which transmission and symptoms are linked explicitly. Assuming independent transmission and symptoms, we estimated a mean generation time (4.2 days, 95% CrI 3.3-5.3 days) similar to previous estimates from other countries, but with a higher standard deviation (4.9 days, 3.0-8.3 days). Using our mechanistic approach, we estimated a longer mean generation time (6.0 days, 5.2-7.0 days) and a similar standard deviation (4.9 days, 4.0-6.3 days). Both models suggest a shorter mean generation time in September-November 2020 compared to earlier months. Since the SARS-CoV-2 generation time appears to be changing, continued data collection and analysis is necessary to inform future public health policy decisions.

based on data from later in the pandemic. Here, using data from a household study conducted 24 from March-November 2020 in the UK, we provide updated estimates of the generation time. 25 We consider both a commonly used approach in which the transmission risk is assumed to be 26 independent of when symptoms develop, and our mechanistic model in which transmission 27 and symptoms are linked explicitly. Assuming independent transmission and symptoms, we 28 estimated a mean generation time (4.2 days, 95% CrI 3.3-5.3 days) similar to previous 29 estimates from other countries, but with a higher standard deviation (4.9 days, 3.0-8.3 days). 30 Using our mechanistic approach, we estimated a longer mean generation time (6.0 days, 5.2-31 7.0 days) and a similar standard deviation (4.9 days, 4.0-6.3 days). Both models suggest a 32 shorter mean generation time in September-November 2020 compared to earlier months. 36 The generation time of a SARS-CoV-2 infector-infectee pair is defined as the period of time 37 between the infector and infectee each becoming infected [1] [2] [3] [4] [5] . The distribution of the 38 generation times of many infector-infectee pairs characterises the temporal profile of 39 infectiousness of an infected host (averaged over all hosts and normalised so that it represents 40 a valid probability distribution) [6] . Inferring the generation time distribution of SARS-CoV-41 2 is important in order to predict the effects of non-pharmaceutical interventions such as 42 contact tracing and quarantine [7, 8] . In addition, the generation time distribution is widely 43 used in epidemiological models for estimating the reproduction number from case 44 notification data [6, [9] [10] [11] and is crucial for understanding the relationship between the 45 reproduction number and the epidemic growth rate [3, 6] .

The SARS-CoV-2 generation time distribution has previously been estimated using data from 48 known infector-infectee transmission pairs [8, 12, 13] or entire clusters of cases [14] [15] [16] . 49 These studies involved data [8, 14, [17] [18] [19] [20] Despite estimation of the SARS-CoV-2 generation time in Asia early in the pandemic, 62 relatively little is known about the generation time distribution outside Asia, and whether or 63 not any changes have occurred in the generation time since the early months of the pandemic. 64 In particular, we are aware of only one study in which the generation time was estimated 65 using data from the UK [23] . In that study [23] , data describing symptom onset dates for 50 66 infector-infectee pairs, collected by Public Health England between January and March 2020 67 as part of the "First Few Hundred" case protocol [24, 25] , were used to infer the generation (Table 1 ). In the second model (the "mechanistic model"), we 81 use a mechanistic approach in which potential infectors progress through different stages of 82 infection, first becoming infectious before developing symptoms [12] . Infectiousness is 83 therefore explicitly linked to symptoms in the mechanistic model. By fitting separately to 84 data from three different time intervals within the study period, we explore whether or not 85 All rights reserved. No reuse allowed without permission.

(which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. fitted models, we calculated posterior estimates of the mean ( Figure 1A ) and standard 100 All rights reserved. No reuse allowed without permission.

(which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (Table 1) , although those estimates lie within our credible interval. On the 108 other hand, our estimated standard deviation of 4.9 days (95% CrI 3.0-8.3 days; Figure 1B , 109 blue violin) is substantially higher than previous estimates (Table 1) . Using our mechanistic 110 model, we obtained a higher estimate for the mean generation time of 6.0 days (95% CrI 5.2-111 7.0 days; Figure 1A , red violin), and a similar estimate for the standard deviation (4.9 days, 112 95% CrI 4.0-6.3 days; Figure 1B (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. comparisons of goodness of fit between the models were not readily available. However, 131 comparing model predictions of the difference between successive symptom onset dates with 132 the UK household data indicated that both models provided a similar fit to the data ( Figure   133 S3).

As described in Methods, the generation time distribution that we considered in Figure 1 households. This is due to depletion of susceptible household members in small households 140 before longer generation times can be attained [6, 27] . As a result, we also predicted the mean 141 and standard deviation of realised generation times within the study households ( Figure S4A Figure 2B ) and the serial interval ( Figure 2C ). The TOST distribution 154 (which characterises the relative expected infectiousness of a host at each time from symptom 155 All rights reserved. No reuse allowed without permission.

(which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. model was more concentrated around the time of symptom onset compared to that obtained 157 using the independent transmission and symptoms model ( Figure 2B ), as was found in [12] . 158 In contrast, the estimated serial interval distributions were similar for the two models ( Figure   159 2C). (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. 

The results shown in Figure 3A suggest a lower mean generation time in September-180 November (2.9 days, 95% CrI 1.8-4.3 days) compared to earlier months (4.9 days, 95% CrI (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted May 30, 2021.

followed a lognormal distribution that was obtained in a previous meta-analysis [31] . In 195 contrast, we assumed in our mechanistic approach that each infection could be decomposed 196 into three gamma distributed stages (latent, presymptomatic infectious and symptomatic 197 infectious), so that the incubation period was also gamma distributed (with the same mean distributed. In both cases, we obtained similar results to those shown for that model in Figure   204 1.

In Figure S9 , we relaxed the assumption of a fixed incubation period distribution, using the In our main analyses, we assumed that household transmission was frequency-dependent, so 214 that the force of infection exerted by an infected household member on each susceptible 215 household member scaled with 1/ , where is the household size. In order to explore the 216 robustness of our results to this assumption, we considered alternative possibilities where 217 infectiousness scaled with "# , for = 0 (density-dependent transmission) and = 0.5 218 ( Figure S10A-B) . We also conducted an analysis in which the dependency, , was estimated 219 All rights reserved. No reuse allowed without permission.

(which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. We also considered the sensitivity of our results to the assumed relative infectiousness of 230 asymptomatic infected hosts ( Figure S11 ). In most of our analyses, we assumed that the (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. In this study, we estimated the generation time distribution of SARS-CoV-2 in the UK by 247 fitting two different models to data describing the infection status and symptom onset dates of 248 individuals in 172 households. The first of these models was predicated on an assumption 249 that transmission and symptoms are independent. While this assumption has often been made 3.0 days. One potential cause of this disparity is the difference in isolation policies for 268 symptomatic hosts between countries. In particular, the UK's policy of self-isolation may be 269 All rights reserved. No reuse allowed without permission.

(which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. Our results did not indicate any clear difference in goodness of fit between the two models 295 ( Figure S3) . A range of factors should therefore be considered when deciding which of our 296 estimates of epidemiological parameters to use in subsequent analyses. Although any model 297 requires simplifying assumptions to be made, our mechanistic approach allows the standard 298 assumption of independent transmission and symptoms to be relaxed by providing a 299 mechanistic underpinning to the relationship between the times at which individuals display 300 symptoms and become infectious. Furthermore, as described above, this model was shown in

[12] to provide a better fit to another SARS-CoV-2 dataset than a model assuming (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. Figure S7 ). (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. One advantage of our approach is that we are able to include the contribution of 347 asymptomatic infected hosts to household transmission chains in our analyses. We showed 348 that our estimated generation time distribution was robust to the assumed relative Our study has some limitations. Since we used household transmission data in our analyses, 360 the generation time for transmission outside the household may differ from our estimates.

Future extensions to our approach may account for the possibility that more than one (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. In the study, all household members who tested positive in either a PCR or antibody test were 393 assumed to have been infected. Conversely, all individuals who tested negative for antibodies 394 (and where the two PCR tests were either negative or were not carried out) were assumed to 395 have remained uninfected, irrespective of symptom status. For 6% of the study cohort, no 396 antibody test was carried out and any PCR tests were negative. These hosts with unknown 397 infection status were excluded from our main analyses (but were counted in the household 398 size), although we also considered the sensitivity of our results to this assumption. (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted May 30, 2021. (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. independent latent (E), presymptomatic infectious (P) and symptomatic infectious (I) stages.

The duration of each stage was assumed to be gamma distributed, and infectiousness was (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. We denote the expected infectiousness of household member , at time since infection, by (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. where )*+ is the probability density function of the incubation period (this was assumed to be 530 known; the exact incubation period distribution we used is given below).

Each household was assumed to be independent, so that the overall likelihood was given by 533 the product of the contributions from each household.

All rights reserved. No reuse allowed without permission.

(which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. For the independent transmission and symptoms model, we assumed a lognormal incubation 538 period distribution with mean 5.8 days and standard deviation 3.1 days [31] . For the 539 mechanistic model, we assumed a gamma distributed incubation period with the same mean 540 and standard deviation; this was for mathematical convenience, since the incubation period 541 was decomposed into the sum of independent gamma distributed latent and presymptomatic 542 infectious periods. Results for the independent transmission and symptoms model using a 543 gamma distributed incubation period are shown in Figure S8 , and we account for uncertainty 544 in the exact parameters of the incubation period distribution in Figure S9 . Lognormal priors were assumed for fitted model parameters (these parameters were the mean 559 and standard deviation of the generation time distribution, in addition to the overall 560 All rights reserved. No reuse allowed without permission.

(which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted May 30, 2021. ; https://doi.org/10.1101/2021.05.27.21257936 doi: medRxiv preprint infectiousness, ! ). The priors for the mean and standard deviation of the generation time 561 distribution had medians of 5 days and 2 days, respectively (these choices were informed by 562 previous estimates of the SARS-CoV-2 generation time distribution [8, 13, 14]), and were 563 chosen to ensure a prior probability of only 0.025 that these parameters exceeded very high 564 values of 10 days and 7 days, respectively. The exact priors we used are detailed in Table S2 . where the likelihood contribution, (9) 6 ; (9) 8, from each household, , was computed as 572 described in the previous section, and denote the prior density of by ( ).

In each step of the chain, we carried out (in turn) one of the following: (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The chain was run for 10,000,000 iterations; the first 2,000,000 iterations were discarded as 607 burn-in. Posteriors were obtained by recording only every 100 iterations of the chain. (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. 

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No reuse allowed without permission. (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity

day of symptom of onset for each host). For each household, , accept the 585 proposed augmented data, :;<: (9) , from that household with probabilitywhere <=> (9) denotes the corresponding augmented data from the previous step