where W (p), the "weight" of the primary cohort, represents the average number of forward 159 intervals that an individual in cohort P(p) produces over the course of their infection. When 160 we measure within-individual delays, we expect W (p) ≤ 1 because only a subset of indi-161 viduals who experience the primary event (e.g., infection) will eventually experience the 162 secondary event (e.g., symptom onset). For between-individual delays, we expect W (p) to 163 change throughout an epidemic, because individuals infected earlier in an epidemic will infect 164 more individuals on average than those infected later. 165 Substituting p = s − τ , it follows that
If we are considering incubation periods, the left hand side of this equation is the probability 167 density that an individual who became symptomatic at time s had an incubation period 168 of length τ . From the right hand side, we see that this probability density depends on the . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) individuals infected at time s − τ ), and the forward delay distribution f s−τ (τ ) (in this case, the probability density that an incubation period that starts at time s − τ ends at time s).
Several different mechanisms drive the changes in forward and backward delay distri- Eq.
(2) suggests that backward delay distributions change over time even if their corresponding forward delay distribution does not change. Backward delay distributions depend on changes in the primary cohort size over time due to conditionality of observations: Conditioning on individuals whose secondary events have occurred at the same time means that we tend to observe shorter (or longer) inter-event delays when cohort size has been increasing (decreasing) through time. When incidence is growing exponentially, we can calculate the amount of bias exactly. Assuming that the forward delay distribution (f p (τ ) ≈ f 0 (τ )) and the weight parameter (W (p) ≈ W (0)) remain constant during the exponential growth phase, we can substitute P(t) = P(0) exp(rt) in Eq. (2) to obtain: The serial interval is defined as the time between when an infector becomes symptomatic 197 and when an infectee becomes symptomatic (Svensson, 2007) . Previous studies have often 198 expressed serial intervals τ s in the form (Fig. 1A) :
where τ i1 and τ i2 represent incubation periods of an infector and an infectee, respectively, 200 and τ g represents the generation interval between the infector and the infectee. These studies 201 assume that τ i1 and τ i2 are drawn from the same distributions and concluded that the serial 202 and generation intervals have the same mean (Svensson, 2007 is based on the implicit assumption that distributions of realized incubation periods, τ i1 and 205 τ i2 , as well as generation interval, τ g , are intrinsic to individuals (and not dependent on 206 epidemic dynamics at the population-level) -something that is generally true of forward 207 but not backward incubation-period distributions. We refer to the definition Eq. (4) as the 208 intrinsic serial interval (Fig. 1A) .
To correctly link the realized serial-interval distribution to the renewal process between 210 cases based on symptom onset dates, we must use the forward serial interval (i.e., use the 211 perspective of a cohort of infectors that share the same symptom onset time). Given that an 212 infector became symptomatic at time p, to calculate the forward serial interval we first go 213 backward in time to when the infector was infected, and then forward in time to when the 214 infectee was infected, and then forward to when the infectee became symptomatic. In Fig. 1B , 215 we see that τ i1 is drawn from the backward incubation period distribution of the cohort of 216 infectors who became symptomatic at time p; τ g is drawn from the forward generation-217 interval distribution of the cohort of infectors who became infected at time p − τ i1 ; and 218 τ i2 is drawn from the forward incubation period distribution of the cohort of infectees who 219 became infected at time p − τ i1 + τ g . Likewise, we can define the backward serial-interval 220 distribution for a cohort of infectees who became symptomatic at time s (Fig. 1C ). This 221 conceptual framework demonstrates that the distributions of τ i1 , τ g , and τ i2 (and therefore 222 the distributions of realized serial intervals) depend on the reference cohort, which is defined 223 by an event type (primary or secondary), temporal direction (forward or backward), and a 224 particular reference time.
To calculate realized serial-interval distributions, we begin by modeling T (p, s): the total 226 density of serial intervals that start (when infectors develop symptoms) at time p and end 227 (when infectees develop symptoms) at time s. The density of serial intervals between time 228 p and s, given that the infectors became infected at time α 1 ≤ p and the infectees became 229 infected at time α 2 ≤ s, depends on the amount of infection that occurs between time α 1 and 230 α 2 as well as the density of forward incubation periods between α 1 and p (realized incubation 231 periods of infectors) and between α 2 and s (realized incubation periods of infectees):
joint density of the forward incubation period p−α 1 and the forward generation interval α 2 −α 1 (of infectors)
where the case reproduction number R c (α 1 ) is defined as the average number of secondary 233 cases caused by a primary case infected at time α 1 over the course of their infection (Fraser, 234 2007). We describe the forward incubation periods and the forward generation intervals 235 using a joint probability distribution because onset of symptoms and transmission potential 236 jointly depend on the life history of a disease; for example, if an infected individual can only 237 7 . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review)
The copyright holder for this preprint this version posted June 5, 2020. . https://doi.org/10.1101/2020.06.04.20122713 doi: medRxiv preprint p + τ i1 p Infector (reference cohort) Infectee
Intrinsic serial interval, τ s = (τ g + τ i2 ) − τ i1
Intrinsic incubation period, τ i1
Intrinsic generation interval, τ g
A. Intrinsic serial interval
Backward incubation period, τ i2
Backward generation interval, τ g Forward incubation period, τ i1 C. Backward serial interval Figure 1 : Illustration of intrinsic, forward and backward serial intervals. (A) The intrinsic serial interval for a cohort of individuals infected at time p. In this case, τ i1 is drawn from the intrinsic incubation period distribution; τ g is drawn from the intrinsic generationinterval distribution; and τ i2 is drawn from the intrinsic incubation period distribution. (B) The forward serial interval for a cohort of infectors who became symptomatic at time p. In this case, τ i1 is drawn from the backward incubation period distribution; τ g is drawn from the forward generation-interval distribution; and τ i2 is drawn from the forward incubation period distribution. (C) The backward serial interval for a cohort of infectees who became symptomatic at time s. In this case, τ i1 is drawn from the forward incubation period distribution; τ g is drawn from the backward generation-interval distribution; and τ i2 is drawn from the backward incubation period distribution.
transmit the disease after symptom onset, the forward generation interval will necessarily be 238 longer than the forward incubation period.
The total amount of serial intervals can now be obtained by integrating over all possible 240 8 . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review)
The copyright holder for this preprint this version posted June 5, 2020. . https://doi.org/10.1101/2020.06.04.20122713 doi: medRxiv preprint infection times for the infector and the infectee:
Then, the forward serial-interval distribution f p (τ ) is given by the density of intervals of 242 length τ starting at p, relative to the total number of serial intervals starting at p:
Likewise, the backward serial-interval distribution b s (τ ) is given by the density of intervals 244 of length τ ending at s, relative to the total number of serial intervals ending at s:
This framework allows us to understand changes in the realized serial intervals for any 246 deterministic epidemic model. 
where R 0 is the basic reproduction number, and g(τ ) is the intrinsic generation-interval dis- 
:
which allows us to separate the joint probability distribution h p of the forward incubation 
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The copyright holder for this preprint this version posted June 5, 2020. .
We further assume that the forward incubation period distribution does not vary across cohorts over the course of an epidemic, as it represents the life history of a disease; we denote it as . Then, we have:
Finally, the case reproduction for this model is defined as follows:
The forward and backward serial-interval distributions are then calculated by substituting 259 these quantities into Eq. (7) and Eq. (8). We use this framework to illustrate how the realized 
Analogous to the intrinsic generation-interval distribution, forward serial-interval distribu-271 tions describe the renewal process of infection based on symptom onset dates. Therefore, we 272 expect the forward serial-interval distribution during the exponential growth phase -which 273 we refer to as the initial forward serial-interval distribution f 0 -to estimate the same value 274 of R 0 for a given r as the intrinsic generation-interval distribution (note, however, that the 275 forward serial interval is not necessarily positive):
where the initial forward serial-interval distribution is given by:
10 . CC-BY-NC-ND 4.0 International license It is made available under a 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 June 5, 2020. . https://doi.org/10.1101/2020.06.04.20122713 doi: medRxiv preprint where the normalization constant φ is determined by the requirement that ∞ −∞ f 0 (τ ) dτ = 1.
In Supplementary Materials, we provide a mathematical proof of this relationship. We 279 further compare this with the estimate of R 0 based on the intrinsic serial-interval distribution 280 q(τ ):
Here, the intrinsic serial-interval distribution q(τ ) does not depend on epidemic dynamics:
where the normalization constant φ q is determined by the requirement that ∞ −∞ q(τ ) dτ = 1.
Rather than numerically integrating over closed forms of g, f 0 , and q to estimate R 0 , we use 284 simulation-based approaches for simplicity (Supplementary Materials).
The initial forward serial-interval distribution depends on the exponential growth rate We parameterize our model based on parameter estimates for COVID-19 (Table 1) . We use 298 a bivariate lognormal distribution to model the joint probability distribution of the intrinsic 299 incubation periods and the intrinsic generation intervals while allowing for the possibility 300 that they might be correlated. For simplicity, we consider three values for the correlation 301 coefficients (on the log scale) of the bivariate lognormal distribution: ρ = −0.5, 0, 0.5.
3 Results
We use parameter estimates for COVID-19 to characterize the degree to which the realized are affected by the right censoring during an ongoing epidemic and provide a heuristic method 308 for addressing biases that can arise from using serial interval data to estimate R 0 . Finally, we 309 analyze the serial interval data from the COVID-19 epidemic in China under our framework.
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The copyright holder for this preprint this version posted June 5, 2020. . Table 1 : Parameter values used for simulations. The intrinsic generation-interval distribution is parameterized using a log-normal distribution with log mean µ G = 1.54 and log standard deviation σ G = 0.37. The intrinsic incubation period distribution is parameterized using a log-normal distribution with log mean µ I = 1.62 and log standard deviation σ I = 0.42. The joint probability distribution is modeled using a bivariate log-normal distribution with correlations (on the log scale) ρ = −0.5, 0, 0.5. The intrinsic generation-interval and incubation period distributions are chosen to match characteristic of COVID-19 to illustrate realistic magnitudes of time-varying/perspective effects in the current pandemic. we underestimate R 0 when we use the intrinsic serial-interval distribution.
The initial forward serial-interval distribution captures the exponential growth phase Table 1 ; we further assume 337 12 . CC-BY-NC-ND 4.0 International license It is made available under a 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 June 5, 2020. Figure 2 : Estimates of the reproduction number from the exponential growth rate based on serial-and generation-interval distributions. (A). The initial forward serial-interval distributions give a correct link between the exponential growth rate r and the reproduction number R 0 , for any correlation ρ between intrinsic incubation period and intrinsic generation interval of the underlying bivariate log-normal distribution. (B) The intrinsic serial-interval distributions give an incorrect link between r and R 0 . (C) The mean initial forward serial interval during the exponential growth phase increases with r. (D) The squared coefficient of variation of the initial forward serial intervals during the exponential growth phase decreases with r. Table 1 for parameter  values. 14 . CC-BY-NC-ND 4.0 International license It is made available under a 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 June 5, 2020. by assumption (Fig. 3B) . The mean forward generation interval decreases slightly when 348 incidence is high, which is when the susceptible population declines rapidly ( (2015)). In contrast, the mean forward serial interval 350 decreases over time (Fig. 3D) .
The forward serial-interval distributions depend on distributions of three intervals (Fig. 1) :
(i) the backward incubation period, (ii) the forward generation interval, and (iii) the for-353 ward incubation period. In these simulations, both forward incubation period (Fig. 3B) and because they are predominantly driven by the rate of change in incidence, which in turn 362 affects relative cohort sizes. When incidence is increasing, individuals are more likely to 363 have been infected recently and therefore, we are more likely to observe shorter intervals 364 (Eq. (3)). Similarly, when incidence decreases, we are more likely to observe longer intervals.
Neglecting these changes in the backward distributions will necessarily bias the inference of 366 the intrinsic distributions from the observed distributions. CC-BY-NC-ND 4.0 International license It is made available under a 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 June 5, 2020. 
Initial forward initial mean forward serial interval (and therefore R 0 ), likely by a large amount, because 386 the observed serial-interval distribution converges to the intrinsic serial-interval distribution; 387 in fact, we would even underestimate the intrinsic value slightly due to contraction of the 388 forward generation-interval distribution during the susceptible depletion phase (Fig. 3C) . 389 Here, we provide a heuristic way of assessing potential biases in the estimate of the 390 mean initial forward serial interval and therefore R 0 retrospectively. We can rearrange the 391 line list and group observed serial intervals based on the symptom onset date of infectors 392 16 . CC-BY-NC-ND 4.0 International license It is made available under a 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 June 5, 2020. . https://doi.org/10.1101/2020.06.04.20122713 doi: medRxiv preprint (Fig. 4C) . Then, we can compare how estimates of the mean serial interval as well as R 0 393 change as we incorporate more recent cohorts into the analysis; that is, we analyze observed 394 serial intervals from infectors who became symptomatic before time t and evaluate how 395 the estimates change as we increase t. This approach is analogous to averaging over a 396 set of forward intervals, just as using all information up to a certain time is analogous to 397 averaging over a set of backward intervals (Fig. 4D) . During the exponential growth phase, 398 the estimates of the mean serial interval and R 0 are consistent with the true value (see 'initial 399 forward' in Fig. 4B,D) ; adding more data allows us to make more precise inference during this 400 period. However, the cohort-averaged estimates decrease rapidly soon after the exponential to reduce the forward serial interval. Fig. 5B shows that the mean backward serial interval 420 increased over time, likely driven directly by the decrease in COVID-19 cases.
While the qualitative changes in the mean forward and backward serial interval are con-422 sistent with our earlier simulations (Fig. 3) , the initial mean forward serial interval (Fig. 5A ) 423 appears to be larger than what we calculated based on previously estimated incubation pe-424 riod and generation-interval distributions (Fig. 2C) . These may imply that the incubation 425 period and generation interval (Table 1) were underestimated, as neither study explicitly ac-426 counted for the fact that the observed intervals were drawn from the backward distributions 427 and were likely to have been censored. periods, respectively, using serial-interval data from infectors who developed symptoms by 435 day 7. These early cohort-averaged estimates of R 0 are unlikely to be affected by the right-436 censoring as we expect the degree of right-censoring to be low (Fig. 5A) . Therefore, the R 0 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 June 5, 2020. . Here, we assume that all individuals develop symptoms and that the entire transmission 494 process, including all relevant epidemiological delays, is known exactly. In practice, iden-495 19 . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.
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The copyright holder for this preprint this version posted June 5, 2020. We simulate the renewal equation model using a discrete-time approximation:
whereĝ is a discrete-time intrinsic generation-interval distribution that satisfies the following:
, m = 1, . . . , m max .
The continuous-time intrinsic generation-interval distribution is parameterized using a log-522 normal distribution (Table 1) . We define the intrinsic incubation period distribution in a 523 similar manner:
where its continuous-time analog is also based on a log-normal distribution. For simplic-525 ity, we assume that the forward incubation periods and intrinsic generation intervals are (Table 1) . Contactees are uniformly sampled from the total population. 539 21 . CC-BY-NC-ND 4.0 International license It is made available under a 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 June 5, 2020. . The intrinsic generation-interval distribution g(τ ) provides a link between r and R 0 via the 548 Euler-Lotka equation (Wallinga and Lipsitch, 2007) :
In this section, we prove that the initial forward serial-interval distribution f 0 (τ ) also esti-550 mates the same R 0 from r, except that integral extends to τ = −∞ rather than beginning 551 at τ = 0, because serial intervals can be negative:
Here, the initial forward serial-interval distribution f 0 (τ ) is defined as:
where h is the joint probability distribution describing the intrinsic generation-interval dis-554 tribution g and the intrinsic incubation period distribution (see Eq. (12) in the main text), 555 and the normalization constant φ is determined by the requirement that ∞ −∞ f 0 (τ ) dτ = 1.
In order to verify Eq. (25), we first rewrite the integral in Eq. (26) by substituting −α 1 for α 1 , and then changing the order of integration:
To further simplify the expression, we define z(α 2 ) as follows:
Substituting z(α 2 ) into Eq. (27) we obtain:
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The copyright holder for this preprint this version posted June 5, 2020. .
we can now express the initial forward serial-interval distribution f 0 as a convolution ofẑ 560 and :
Since the right hand side of Eq. (25) is also a Laplace transform of f 0 =ẑ * , we can 563 express it as the product of Laplace transforms ofẑ and :
In order to derive an expression for a Laplace transform ofẑ, we have to first derive an analytical expression for ∞ −∞ z(x) dx. By changing the order of integration, we have:
Since is a marginal probability distribution of h, it follows that:
Then, we have:
Substituting the expression into Eq. (32), we have:
Recall that g is also a marginal probability distribution of h:
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We can then substitute τ = α 1 + α 2 into Eq. (36) and apply change of variables to obtain:
Therefore, the initial forward serial-interval distribution and the intrinsic generation-interval 569 distribution give the same estimates of R 0 from r. We use a simulation-based approach to compare the estimates of R 0 based on the serial-573 and generation-interval distributions. To do so, we model the intrinsic generation-interval 574 distribution and the incubation period using a multivariate log-normal distribution with log 575 means µ G , µ I , log standard variances σ 2 G , σ 2 I , and log-scale correlation ρ; the multivariate log-576 normal distribution is parameterized based on parameter estimates for COVID-19 (Table 1) . 577 We construct forward serial intervals during the exponential growth period as follows:
where the backward incubation period X 1,i of an infector is simulated by drawing random log-579 normal samples Y i with log mean µ I and log variance σ 2 I and resampling Y i , each weighted by 580 the inverse of the exponential growth function exp(−rY i ); the intrinsic generation interval 581 conditional on the incubation period of the infector (G i |X 1,i ) is drawn from a log-normal 582 distribution with log mean µ G + σ G ρ(log(X 1,i ) − µ I )/σ I and log variance σ 2 G (1 − ρ 2 ); the 583 forward incubation period X 2,i of an infectee is drawn from a log-normal distribution with 584 log mean µ I and log variance σ 2 I . We then calculate the basic reproduction number R 0 using 585 the empirical estimator:
We compare this with an estimate of R 0 based on the intrinsic serial-interval distribution 587 which has the same mean as the intrinsic generation-interval distribution (Svensson, 2007 ; 
where 591 Q i = −Y i + (G i |Y i ) + X 2,i .
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The copyright holder for this preprint this version posted June 5, 2020. . https://doi.org/10.1101/2020.06.04.20122713 doi: medRxiv preprint 5.5 Applications: SEIR model 592 Consider a Susceptible-Exposed-Infectious-Recovered model:
where β is the transmission rate, 1/γ E is the mean latent period, and 1/γ I is the mean 593 infectious period. We further assume that the latent period is equivalent to incubation 594 period; in other words, infected individuals can only transmit after symptom onset. Then, 595 the generation interval will be always longer than the incubation period.
The joint probability distribution of the intrinsic incubation periods and intrinsic gener-597 ation intervals for this model can be written as:
Then, the intrinsic generation-interval distribution is given by:
On the other hand, the initial forward serial-interval distribution is given by:
Therefore, both the intrinsic generation intervals and the initial forward serial intervals are 599 identically distributed and have the same mean. 600 25 . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review)
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