transmissions is higher than estimated using standard approaches, with a substantial 1 proportion of transmissions occurring in a short time window prior to symptom onset. 2
Finally, we consider the implications of our results for contact tracing. 3 4 Transmission pair data (Fig. 1A) generally comprise symptom onset dates for known source-5 recipient pairs. These data may be supplemented with partial information about infection 6 times, consisting of a range of possible exposure dates for the source and/or recipient (3). 7
While the serial interval for each pair can be calculated directly from the data (with some 8 uncertainty, given the unknown precise times of symptom appearance on the onset dates 9
(17)), other time intervals, including the generation time and TOST (which is negative for 10 presymptomatic transmissions), are unobserved. In almost all previous approaches that have 11 been used to estimate the generation time distribution of SARS-CoV-2 from transmission 12 pair data (Fig. 1B, left panel) , the infectiousness of the source at a given time since infection 13 is assumed to be independent of their incubation period (3, 7, 8, 14) . In contrast, in our 14 mechanistic approach (Fig. 1B, right panel) , which is based on compartmental modelling, 15 each infected host passes through three stages of infection -latent (E), presymptomatic 16 infectious (P), and symptomatic infectious (I). Infectiousness is assumed to be constant 17 during each stage but may vary between presymptomatic and symptomatic infectious hosts. 18 All rights reserved. No reuse allowed without permission. perpetuity. preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in The copyright holder for this this version posted November 23, 2020. infectiousness and symptoms within individuals. In standard approaches (left panel), the infectiousness of a host 5 at a given time since infection is independent of their incubation period. In our approach (right panel), we 6 assume that individuals are not infectious during the latent (E) period, and that infectiousness may vary between 7 the presymptomatic infectious (P) and symptomatic infectious (I) periods, for example due to changing 8 behaviour in response to symptoms (18) .
We considered four different models of infectiousness: 11 All rights reserved. No reuse allowed without permission. perpetuity. preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in The copyright holder for this this version posted November 23, 2020. ; https://doi.org/10.1101/2020.11.20.20235754 doi: medRxiv preprint variability in the generation time between individuals was lower for the independent 23 transmission and symptoms model compared to the other three models ( Fig. 2A) . On the 24 other hand, the TOST distribution was most concentrated around the time of symptom onset 25 All rights reserved. No reuse allowed without permission. perpetuity. preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in The copyright holder for this this version posted November 23, 2020. ; https://doi.org/10.1101/2020.11.20.20235754 doi: medRxiv preprint for the best-fitting variable infectiousness model, and least concentrated for the independent 1 transmission and symptoms model (Fig. 2B ). In our best-fitting model, infectiousness was 2 found to decrease immediately after onset, likely due to behavioural factors that reduce the 3 transmission risk following symptom onset (18) . 
In all panels, lines represent: variable infectiousness model (blue), constant infectiousness model (red), Ferretti 13 model (orange), and independent transmission and symptoms model (purple).
Using the posterior distributions of model parameters that were obtained when we fitted the 16 models to data, we calculated the posterior distribution of the proportion of transmissions 17 occurring prior to symptom onset for each model (Fig. 3A) . The median (95% CI) Ferretti 20 All rights reserved. No reuse allowed without permission. perpetuity. preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in The copyright holder for this this version posted November 23, 2020. ; https://doi.org/10.1101/2020.11.20.20235754 doi: medRxiv preprint model, and independent transmission and symptoms model, respectively. Our central 1 estimate of 65% of transmissions occurring prior to symptom onset using our best-fitting 2 model is higher than estimated in previous studies that have fitted statistical models of 3 infectiousness using transmission pair data (3, 4, 9) -for example, estimates of 37% and 55% 4
were obtained under an assumption of independent transmission and symptoms in (3). 
The estimates in Fig. 3A describe the proportion of transmissions that occur prior to symptom 15 onset, and therefore apply only to individuals who go on to develop symptoms. However, 16 these estimates can be combined with the results of a previous study (1) in which the extent 17 of asymptomatic transmission (i.e., transmissions from individuals who never display 18 symptoms) was characterised (Fig. S2) , to obtain estimates for the total proportion of non-19 All rights reserved. No reuse allowed without permission. perpetuity. preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in The copyright holder for this this version posted November 23, 2020. ; https://doi.org/10.1101/2020.11.20.20235754 doi: medRxiv preprint symptomatic (i.e., either presymptomatic or asymptomatic) transmissions in an entire 1 population of hosts for the different models (Fig. 3B) . Again, the non-symptomatic 2 proportion was highest for the variable infectiousness model, and lowest for the independent 3 transmission and symptoms model. 4 5 Finally, we considered the implications of these results for contact tracing (Fig. 4) . In Fig.  6 4A, we show the proportion of infectious contacts of hosts who go on to develop symptoms 7 that are identified if contacts are traced up to different times before the symptom onset time 8 of the index case (i.e., for different contact elicitation windows). Long duration contact 9 elicitation windows are impractical and place significant strain on contact tracing systems, 10 leading to contact elicitation windows of two days being used in countries such as the UK 11 (24) and USA (25). In the best-fitting variable infectiousness model, 84% of infectious 12
contacts are estimated to be identified when tracing up to 2 days before symptom onset ( Fig.  13 4A, blue dashed). This is due to the clustering of transmission events around the symptom 14 onset time (cf. Fig. 2B ) and compares to a lower estimate of only 74% if the standard 15 assumption of independence between transmission and symptoms is made (Fig. 4A , purple). 16
We also explored the effect of the timing of isolation of infected individuals identified 17 through contact tracing, and estimated the reduction in onwards transmissions from infected 18 contacts if contact tracing is conducted quickly (Fig. 4B ). Compared to the best-fitting 19 variable infectiousness model, the standard independent transmission and symptoms model 20 estimated a similar reduction in transmission when a host was isolated within 4 days of 21 exposure, but under-predicted the efficacy of isolation later in infection. We assumed that 22 contact identification and isolation is perfectly effective in this analysis, but we also explored 23 the sensitivity of our results to this assumption in the Supplementary Materials. In each case 24 All rights reserved. No reuse allowed without permission. perpetuity. preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in The copyright holder for this this version posted November 23, 2020. ; https://doi.org/10.1101/2020.11.20.20235754 doi: medRxiv preprint that we considered, under the best-fitting model a two-day contact elicitation window is more 1 effective than estimated using existing approaches (Fig. S3) . 
As we have shown, our mechanistic approach provides an improved fit to transmission pair 11 data, and predicts a higher proportion of transmissions occurring in a short time window 12 before symptom onset, compared to previous approaches. These conclusions are robust to the 13 exact incubation period distribution that we assumed (26) when fitting the different models to 14 transmission pair data (Fig. S4 ). Our best-fitting model outperforms a model predicated on a 15
critical assumption -that infectiousness is independent of symptom status -which underlies 16 most previous studies in which the generation time distribution of COVID-19 is estimated (3, 17 14) . That assumption neglects potential relationships between symptoms and viral 18 All rights reserved. No reuse allowed without permission. perpetuity. preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in The copyright holder for this this version posted November 23, 2020. ; https://doi.org/10.1101/2020.11.20.20235754 doi: medRxiv preprint shedding, as well as behavioural changes in response to symptom onset (18) . Some 1 alternative assumptions have also been considered when fitting to transmission pair data, 2 such as the possibility that infectiousness depends only on the time since symptom onset, 3 independent of the time of infection (4, 27). If the serial interval is always positive, which is 4 not always the case for COVID-19 (11) , this is equivalent to assuming that the serial interval 5 and generation time distributions are identical (15, 23, 28). In a previous study (4), another 6 model (the Ferretti model) was developed in which a host's infectiousness could depend on 7 both the time since infection and the time since symptom onset, and was found to outperform 8 models in which their infectiousness depends on either one of these two times alone. 9
However, as we have demonstrated, our mechanistic approach provides an improved fit to 10 data compared to the statistical Ferretti model. In addition, our method has the advantage of 11 being useful for parameterising population-scale compartmental epidemic forecasting 12 models, since the time periods in our approach correspond naturally to compartments (29). 13
In summary, using a mechanistic approach to infer key epidemiological quantities from 15 transmission pair data indicates that a higher proportion of SARS-CoV-2 transmissions occur 16 prior to symptoms than previously thought. Furthermore, a significant proportion of 17 transmissions arise shortly before symptom onset, indicating that contact tracing is beneficial 18 even if the contact elicitation window is short. Continued use and refinement of contact 19
tracing programmes in countries worldwide is therefore of clear public health importance. 20 21 All rights reserved. No reuse allowed without permission. perpetuity. preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in The copyright holder for this this version posted November 23, 2020. ; https://doi.org/10.1101/2020.11.20.20235754 doi: medRxiv preprint
Notation and general details 2
Here, we outline the notation used in this section when describing the different models that 3 we considered. For a given transmission pair, we label the source as 1 and the recipient 2, and 4 define: where , is the basic reproduction number (for hosts that develop symptoms at some stage 22 during infection). We also let ( | !$% ) and ( | !$% ) be the expected infectiousness at 23 time since infection and at time since onset, respectively, conditional on an incubation 24 All rights reserved. No reuse allowed without permission. perpetuity. preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in The copyright holder for this this version posted November 23, 2020. ; https://doi.org/10.1101/2020.11.20.20235754 doi: medRxiv preprint period of !$% (note that ( | !$% ) = ( − !$% | | !$% ) and ( | !$% ) = 1
We considered several different models for infectiousness (details of individual models are 4 given below). In each model, the conditional infectiousness, ( | !$% ), or equivalently, 5 ( | !$% ), is specified. The distributions of the generation time and TOST can be recovered 6 from this conditional infectiousness by averaging over the incubation period distribution 7 (which is assumed to be known): 8
Alternative (equivalent) expressions for the generation time and TOST distributions are 11 available for some of the models considered (these are detailed in the "Models of 12 infectiousness" subsection below). 13
To obtain an expression for the serial interval distribution, we note that 15 #(+ = )*#) + !$%,. . 16
We assume throughout that )*#) and !$%,. are independent, so that the serial interval 17 distribution is given by the convolution 18
The proportion of presymptomatic transmissions (out of all transmissions generated by 20 individuals who develop symptoms) can be calculated as 21 perpetuity. preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in The copyright holder for this this version posted November 23, 2020. ; https://doi.org/10.1101/2020.11.20.20235754 doi: medRxiv preprint 13 1 Data 2
Following (4), we considered COVID-19 transmission pair data from five different studies 3 (3, 9, [19] [20] [21] , totalling 191 source-recipient pairs. In all 191 transmission pairs, both the 4 source and the recipient developed symptoms, and the symptom onset date of each host was 5 recorded. In four of the five studies (3, 9, 19, 20) , intervals of exposure were available for 6 either the source or recipient (or both), whereas in the other (21) , only symptom onset dates 7 were recorded. 8 9
In the main text, the incubation period was assumed to follow a Gamma distribution with 11 shape parameter 5.807 and scale parameter 0.948 (26). This corresponds to a mean 12 incubation period of 5.5 days and a standard deviation of 2.3 days. However, to demonstrate 13 that our main conclusions are robust to the exact incubation period distribution used, we also 14 repeated our analyses using an alternative, more dispersed, Gamma distributed incubation 15 period with a mean of 5.3 days and a standard deviation of 3.2 days (30) (Fig. S4) . 16 17
In this model, the infectiousness of each host at a given time since infection is assumed to be 20 independent of their incubation period, so that 21
where the generation time distribution, '($ , is prescribed. We assumed (3, 8) that 23 '($ ∼ Gamma( , ). 24 All rights reserved. No reuse allowed without permission. perpetuity. preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in The copyright holder for this this version posted November 23, 2020. ; https://doi.org/10.1101/2020.11.20.20235754 doi: medRxiv preprint
The shape parameter ( ) and scale parameter ( ) were estimated when we fitted the model to 1 transmission pair data. The TOST distribution for this model is given by 4
whilst the proportion of presymptomatic transmissions is 6 (1 + 0(206)/8 ) 9:; , ≥ 0.
Here, !$% is the mean incubation period, , and are estimated parameters, and we set 14 perpetuity. preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in The copyright holder for this this version posted November 23, 2020. ; https://doi.org/10.1101/2020.11.20.20235754 doi: medRxiv preprint
In our mechanistic approach, we divided each infection into three stages: latent (E), 2 presymptomatic infectious (P), and symptomatic infectious (I). The stage durations were 3 assumed to be independent, and infectiousness was assumed to be constant over the duration 4 of each stage. We denote the stage durations by <///= , their density and cumulative 5 distribution functions by <///= and <///= , and the infectiousness of hosts in the P and I 6 stages by //= , respectively. We also define the ratio 7
Note that in this model, the basic reproduction number is 9 , = / / + = = , 10
where //= are the respective mean durations of the P and I stages. 11 12 We further assumed that the durations of each stage followed Gamma distributions, with where 17
In particular, the scale parameters of < and / were both assumed to be equal to 1 ( !$% ⁄ ), 19
in order to ensure a Gamma distributed incubation period, 20
We fixed !$% = 5.807 and = 1/(5.807 × 0.948), in order to obtain the specified 22 incubation period distribution (see "Incubation period" subsection above). When we fitted the 23 All rights reserved. No reuse allowed without permission. perpetuity. preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in The copyright holder for this this version posted November 23, 2020. ; https://doi.org/10.1101/2020.11.20.20235754 doi: medRxiv preprint model to data, we assumed that = = 1, so that the symptomatic infectious period followed an 1 exponential distribution. The parameters < and were estimated in the fitting procedure. 2
We considered two versions of the model: one in which we assumed = 1 (the constant 3 infectiousness model), and one in which was also estimated (the variable infectiousness 4 model). incubation period of !$% , can be calculated to be 8
where ?()@ ( ; , ) is the cumulative distribution function of a Beta distributed random 10 variable with parameters and , and 11
The TOST distribution is given by 13 )*#) ( )*#) ) = m (1 − / (− )*#) )), )*#) < 0, (1 − = ( )*#) )), )*#) ≥ 0.
14 The generation time can be written as 15
where * is the time between the start of the P stage and the transmission occurring, and 17 therefore the generation time distribution is given by the convolution 18
where the density, * , of * satisfies 20 * ( * ) = p _1 − / ( * )`
The proportion of presymptomatic transmissions is 22 All rights reserved. No reuse allowed without permission. perpetuity. preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in The copyright holder for this this version posted November 23, 2020. 
where the dependence of the conditional expected infectiousness, ( | !$% , ), on the model 11 parameters, , is indicated explicitly. A derivation of this expression is given in the 12
Supplementary Text. Assuming that each transmission pair in our dataset was independent, 13 the overall likelihood was therefore given by the product of the contributions, ($) ( ), from 14 each individual transmission pair, i.e., 15
where is the total number of transmission pairs. 17
In order to account for uncertainty in the exact symptom onset times within the day of onset, 19
we fitted the models to the data using data augmentation MCMC. In particular, in alternating 20 steps of the chain, we updated either the vector of model parameters, , or the symptom onset 21 times of each source and recipient. The chain was run for 2.5 million steps, of which the first 22 All rights reserved. No reuse allowed without permission. perpetuity. preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in The copyright holder for this this version posted November 23, 2020. ; https://doi.org/10.1101/2020.11.20.20235754 doi: medRxiv preprint 500,000 were discarded as burn-in. Posterior distributions of parameters were obtained by 1 recording only every 100 iterations of the chain (assuming a uniform prior distribution for 2 each model parameter), whilst the parameter values that maximised the likelihood were taken 3 as point estimates. Full details of the model fitting procedure are given in the Supplementary 4
Text. 5 6 Distributions of the presymptomatic and total non-symptomatic proportion of transmissions 7
Expressions for the proportion of transmissions, / , generated prior to symptom onset, are 8
given for the individual models above. Once asymptomatic cases are accounted for, the 9 overall non-symptomatic proportion of transmissions can be written as 10
where G is the proportion of infected individuals who remain asymptomatic, and G is the 12 ratio between the average number of secondary cases generated by an asymptomatic host and which are consistent with estimates in (1). These distributions are shown in Fig. S2 . We then 22
combined samples from the assumed distributions of G and G with the sample that we 23 generated from the posterior distribution of / to obtain a distribution for the total proportion 24 of non-symptomatic transmissions. 25 All rights reserved. No reuse allowed without permission. perpetuity. preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in The copyright holder for this this version posted November 23, 2020. ; https://doi.org/10.1101/2020.11.20.20235754 doi: medRxiv preprint 1
First, we considered the proportion of the infectious contacts of a symptomatic index case 3 that will be found, if contacts are traced up to ; days before the time of symptom onset (of 4 the index host). In this case, assuming that it is possible to trace a fraction ; of the host's 5 total contacts between times − ; and ∞ since symptom onset, then the proportion of 6 infectious contacts found is equal to 7
; (1 − )*#) (− ; )). 8
Note that we assumed that the TOST distribution for a detected index case does not differ 9 from that of a random infected individual. In the main text, we assumed that ; = . = 1 (i.e., contact identification and isolation are 19 both 100% effective), although values of ; and . below 1 are considered in the 20
Supplementary Materials (Fig. S3) . 21 22 All rights reserved. No reuse allowed without permission. perpetuity. preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in The copyright holder for this this version posted November 23, 2020. ; https://doi.org/10.1101/2020.11.20.20235754 doi: medRxiv preprint 20, 793-802 (2020). perpetuity. preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in The copyright holder for this this version posted November 23, 2020. ; https://doi.org/10.1101/2020.11.20.20235754 doi: medRxiv preprint