# Best practices for estimating and reporting epidemiological delay distributions of infectious diseases using public health surveillance and healthcare data

Kelly Charniga, Sang Woo Park, Andrei R Akhmetzhanov, Anne Cori, Jonathan Dushoff, Sebastian Funk, Katelyn M Gostic, Natalie M Linton, Adrian Lison, Christopher E Overton, et al.

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## To cite this version:

Kelly Charniga, Sang Woo Park, Andrei R Akhmetzhanov, Anne Cori, Jonathan Dushoff, et al. Best practices for estimating and reporting epidemiological delay distributions of infectious diseases using public health surveillance and healthcare data. 2024. hal-04572940

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# Best practices for estimating and reporting epidemiological delay distributions of infectious diseases

Kelly Charniga, Sang Woo Park, Andrei R Akhmetzhanov, Anne Cori, Jonathan Dushoff, Sebastian Funk, Katelyn M Gostic, Natalie M Linton, Adrian Lison, Christopher E Overton, et al.

► To cite this version:

Kelly Charniga, Sang Woo Park, Andrei R Akhmetzhanov, Anne Cori, Jonathan Dushoff, et al. Best practices for estimating and reporting epidemiological delay distributions of infectious diseases. 2024. [hal-04572940v2](https://hal.science/hal-04572940v2)

HAL Id: hal-04572940

https://hal.science/hal-04572940v2

Preprint submitted on 10 Oct 2024

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

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Distributed under a Creative Commons Attribution 4.0 International License
Best practices for estimating and reporting epidemiological delay distributions of infectious diseases using public health surveillance and healthcare data

Kelly Charniga<fnref n="1" />, Sang Woo Park<fnref n="2" />, Andrei R. Akhmetzhanov<fnref n="3" />, Anne Cori<fnref n="4" />, Jonathan Dushoff<fnref n="5" />,<fnref n="6" />,<fnref n="7" />, Sebastian Funk<fnref n="8" />,<fnref n="9" />, Katelyn M. Gostic<fnref n="10" />, Natalie M. Linton<fnref n="11" />, Adrian Lison<fnref n="12" />, Christopher E. Overton<fnref n="13" />,<fnref n="14" />,<fnref n="15" />, Juliet R.C. Pulliam<fnref n="10" />, Thomas Ward<fnref n="14" />, Simon Cauchemez<fnref n="1" />, Sam Abbott<fnref n="8" />,<fnref n="9" />

Word count: 4,683

Contact information: Kelly Charniga, kelly.charniga@pasteur.fr

Keywords: epidemiological delays, natural history parameters, mathematical modeling, best practices, checklist

<footnote n="1">¹ Mathematical Modelling of Infectious Diseases Unit, Institut Pasteur, Université Paris Cité, CNRS UMR 2000, Paris, France</footnote>

<footnote n="2">² Department of Ecology and Evolutionary Biology, Princeton University, Princeton, New Jersey, USA</footnote>

<footnote n="3">³ College of Public Health, National Taiwan University, Taipei, Taiwan</footnote>

<footnote n="4">⁴ MRC Centre for Global Infectious Disease Analysis, School of Public Health, Imperial College London, London, UK</footnote>

<footnote n="5">⁵ Departments of Mathematics & Statistics and Biology, McMaster University, Hamilton, Ontario, Canada</footnote>

<footnote n="6">⁶ Department of Biology, McMaster University, Hamilton, Ontario, Canada</footnote>

<footnote n="7">⁷ M. G. DeGroote Institute for Infectious Disease Research, McMaster University, Hamilton, Ontario, Canada</footnote>

<footnote n="8">⁸ Department of Infectious Disease Epidemiology and Dynamics, London School of Hygiene & Tropical Medicine, London, UK</footnote>

<footnote n="9">⁹ Centre for Mathematical Modelling of Infectious Diseases, London School of Hygiene & Tropical Medicine, London, UK</footnote>

<footnote n="10">¹⁰ Center for Forecasting and Outbreak Analytics, U.S. Centers for Disease Control and Prevention, Atlanta, Georgia, USA</footnote>

<footnote n="11">¹¹ Graduate School of Medicine, Hokkaido University, Sapporo-shi, Hokkaido, Japan</footnote>

<footnote n="12">¹² Department of Biosystems Science and Engineering, ETH Zurich, Zurich, Switzerland</footnote>

<footnote n="13">¹³ Department of Mathematical Sciences, University of Liverpool, Liverpool, UK</footnote>

<footnote n="14">¹⁴ All Hazards Intelligence, Infectious Disease Modelling Team, Data Analytics and Surveillance, UK Health Security Agency, UK</footnote>

<footnote n="15">¹⁵ Department of Mathematics, University of Manchester, Manchester, UK</footnote>
1 Best practices for estimating and reporting epidemiological delay distributions of infectious diseases

2

3

4

5 Kelly Charniga¹, Sang Woo Park², Andrei R. Akhmetzhanov³, Anne Cori⁴, Jonathan Dushoff⁵,⁶,⁷,

6 Sebastian Funk⁸,⁹, Katelyn M. Gostic¹⁰, Natalie M. Linton¹¹, Adrian Lison¹², Christopher E.

7 Overton¹³,¹⁴,¹⁵, Juliet R.C. Pulliam¹⁰, Thomas Ward¹⁴, Simon Cauchemez¹, Sam Abbott⁸,⁹

8

9

10 ¹ Mathematical Modelling of Infectious Diseases Unit, Institut Pasteur, Université Paris Cité,

11 CNRS UMR 2000, Paris, France

12 ²Department of Ecology and Evolutionary Biology, Princeton University, Princeton, New Jersey,

13 USA

14 ³College of Public Health, National Taiwan University, Taipei, Taiwan

15 ⁴MRC Centre for Global Infectious Disease Analysis, School of Public Health, Imperial College

16 London, London, UK

17 ⁵Departments of Mathematics & Statistics and Biology, McMaster University, Hamilton, Ontario,

18 Canada

19 ⁶Department of Biology, McMaster University, Hamilton, Ontario, Canada

20 ⁷M. G. DeGroote Institute for Infectious Disease Research, McMaster University, Hamilton,

21 Ontario, Canada

22 ⁸Department of Infectious Disease Epidemiology and Dynamics, London School of Hygiene &

23 Tropical Medicine, London, UK

24 ⁹Centre for Mathematical Modelling of Infectious Diseases, London School of Hygiene &

25 Tropical Medicine, London, UK

26 ¹⁰Center for Forecasting and Outbreak Analytics, U.S. Centers for Disease Control and

27 Prevention, Atlanta, Georgia, USA

28 ¹¹Graduate School of Medicine, Hokkaido University, Sapporo-shi, Hokkaido, Japan

29 ¹²Department of Biosystems Science and Engineering, ETH Zurich, Zurich, Switzerland

30 ¹³Department of Mathematical Sciences, University of Liverpool, Liverpool, UK

31 ¹⁴All Hazards Intelligence, Infectious Disease Modelling Team, Data Analytics and Surveillance,

32 UK Health Security Agency, UK

33 ¹⁵Department of Mathematics, University of Manchester, Manchester, UK

34

35

36

37 Word count: 5,273

38

39 Contact information: Kelly Charniga, kelly.charniga@pasteur.fr

40

41

42

43 Keywords: epidemiological delays, natural history parameters, mathematical modeling, best

44 practices, checklist
**Abstract**

Epidemiological delays, such as incubation periods, serial intervals, and hospital lengths of stay, are among key quantities in infectious disease epidemiology that inform public health policy and clinical practice. This information is used to inform mathematical and statistical models, which in turn can inform control strategies. There are three main challenges that make delay distributions difficult to estimate. First, the data are commonly censored (e.g., symptom onset may only be reported by date instead of the exact time of day). Second, delays are often right truncated when being estimated in real time (not all events that have occurred have been observed yet). Third, during a rapidly growing or declining outbreak, overrepresentation or underrepresentation, respectively, of recently infected cases in the data can lead to bias in estimates. Studies that estimate delays rarely address all these factors and sometimes report several estimates using different combinations of adjustments, which can lead to conflicting answers and confusion about which estimates are most accurate. In this work, we formulate a checklist of best practices for estimating and reporting epidemiological delays with a focus on the incubation period and serial interval. We also propose strategies for handling common biases and identify areas where more work is needed. Our recommendations can help improve the robustness and utility of reported estimates and provide guidance for the evaluation of estimates for downstream use in transmission models or other analyses.
45 Abstract
46
47 Epidemiological delays are key quantities that inform public health policy and clinical practice.
48 They are used as inputs for mathematical and statistical models, which in turn can guide control
49 strategies. In recent work, we found that censoring, right truncation, and dynamical bias were
50 rarely addressed correctly when estimating delays and that these biases were large enough to
51 have knock-on impacts across a large number of use cases. Here, we formulate a checklist of
52 best practices for estimating and reporting epidemiological delays. We also provide a flowchart
53 to guide practitioners based on their data. Our examples are focused on the incubation period
54 and serial interval due to their importance in outbreak response and modeling, but our
55 recommendations are applicable to other delays. The recommendations, which are based on
56 the literature and our experience estimating epidemiological delay distributions during outbreak
57 responses, can help improve the robustness and utility of reported estimates and provide
58 guidance for the evaluation of estimates for downstream use in transmission models or other
59 analyses.
60

61
62
63
64
65
66
Introduction

Epidemiological delay distributions are key quantities for public health policy [1], clinical
practice [2], and infectious disease modeling [3–7]. An epidemiological delay is the time
between two epidemiological events. Examples of delays in infectious disease epidemiology
include the incubation period (the time between infection and symptom onset), the serial interval
(the time between symptom onset in a given infected person and someone they infect), and the
generation interval (the time between infection in a given person and someone they infect).
Other commonly used epidemiological delays include hospital lengths of stay and delays from
symptom onset to hospitalization, hospitalization to death, and symptom onset to specimen
date. In the aftermath of the COVID-19 pandemic, there is considerable need for and interest in
the estimation of epidemiological parameters, including delays, with at least two R packages
[8,9], a World Health Organization working group [10], and meta-analyses of priority pathogens
[11–13] working to collect and make these data accessible. In this perspective, we present best
practices for the estimation and reporting of epidemiological delays, illustrated by examples for
the incubation period and serial interval of directly transmitted (person-to-person) infectious
diseases.

The incubation period may vary according to age, vaccination status, underlying
conditions [14], and transmission route or intensity of exposure [15,16], among other factors.
The serial interval depends on the incubation period, contact patterns in the population, control
measures, and the dynamics of infectiousness within a host. While the incubation period must
be positive, the serial interval can be positive or negative [17]. The right tail of the incubation
period distribution informs the length of quarantine, while the left tail indicates the earliest time
symptoms might develop after infection [2]. By comparing the incubation period and serial
interval, we can learn about a pathogen's tendency for pre-symptomatic vs. symptomatic
transmission [18], which can inform the controllability of an epidemic [19]. At the individual level,
pre-symptomatic transmission occurs precisely when the serial interval is shorter than the
incubation period of the infectee. In addition, both the incubation period and serial interval are
key inputs in mathematical and statistical models, such as those used for
nowcasting/forecasting [3,20] or for scenario-based modeling [21,22], which can inform control
strategies. The serial interval is often used as a proxy for the generation interval in these
applications.

Methods for estimating epidemiological delays have been improving, especially during
the COVID-19 pandemic, and recent research has highlighted the importance of appropriately
adjusting for three statistical issues inherent in the data collection process: right truncation,
censoring, and dynamical bias [23–27]. Indeed, use of delay parameters that are not adjusted
for these biases may lead to the propagation of bias into downstream modeling [23] and
therefore an incorrect understanding of an epidemic (e.g., over- or under-estimating the risk to
the host population) [25].

We provide details about the data needed to estimate delays and how they should be
prepared for analysis. Then, we discuss biases that can affect the estimation of epidemiological
delay distributions, followed by strategies that can be used to reduce the impact of these biases.
Technical details about these biases and how to adjust for them can be found in Park et al. [23].
67 **Introduction**
68

69 In recent years, an increasing number of real-time data streams have become available
70 for a variety of computational biology applications. For example, NEON [1] is a large-scale
71 observation platform for environmental data in the United States, and EarthRanger [2] is used
72 for wildlife conservation in several countries. These new data streams may improve the ability to
73 perform real-time analyses, such as monitoring biodiversity [3,4] and insect populations [5].
74 However, delays inherent in these systems (e.g., larval development times for mosquitoes,
75 ecological lags [6]) may pose challenges. The estimation of epidemiological delay distributions,
76 which we discuss in this paper, is a special case of using real-time data to understand the time
77 course of biological processes. Methods for estimating these delays could be broadly applicable
78 to other areas of computational biology.

79 Epidemiological delay distributions are key quantities for public health policy [7], clinical
80 practice [8], and infectious disease modeling [9–13]. An epidemiological delay is the time
81 between two epidemiological events, a primary event and a secondary event. There are
82 numerous examples of delays in infectious disease epidemiology. Three of the most important
83 delays include the incubation period (the time between infection and symptom onset), the serial
84 interval (the time between symptom onset in a given infected person and someone they infect),
85 and the generation interval (the time between infection in a given person and someone they
86 infect). Other commonly used epidemiological delays include hospital lengths of stay and delays
87 from symptom onset to hospitalization, hospitalization to death, and symptom onset to specimen
88 date, among others. Delays vary in length (e.g., between infected individuals) due to biological
89 and epidemiological factors [14–16], such that samples of delays can be characterized by a
90 distribution. These distributions are typically described by one or more parameters. Pair-
91 dependent delays are expected to be more heterogeneous than single-individual delays
92 because they involve more than one person.
Finally, we introduce a reporting checklist to aid researchers and reviewers based on our experience estimating epidemiological delay distributions during outbreak responses.

## Data

Epidemiological delays have start and end times, marked by a primary and secondary event. These events can be observed (e.g., symptom onset time) or unobserved (e.g., usually infection time). Incubation period data include the times of probable exposure and perceived symptom onset for each case. Infection time can be inferred from exposure time because exposure is a necessary, but not sufficient, condition for infection. Data that can be used to estimate the serial interval are those in which symptom onset has been observed for primary and secondary cases. Examples of study designs or public health activities that generate such data include contact tracing [24,28], prospective cohort studies [29–31], household studies [32], or other types of intensive cohort monitoring [33]. Data from passive surveillance, which involves healthcare providers reporting cases to public health agencies, can also be used to estimate delay distributions; however, exposure information may be unreliable because key information may be missing, incomplete, or abstracted from other variables [34].

## Biases in delay data

Three main biases can affect the estimation of epidemiological delay distributions, 1. censoring, 2. right truncation bias, and 3. dynamical (or epidemic-phase) bias (Table 1).

Censoring is knowing that an event occurred but not precisely when. Data can be right censored (the event is known to have occurred after a certain time), left censored (the event is known to have occurred before a certain time), or interval censored (the event is known to have occurred within a certain time interval). In epidemiological delay data, censoring can affect either primary or secondary events (single interval censoring) or both (double interval censoring) [35]. Epidemiological data are almost always doubly interval-censored due to the time scales of reporting. For example, when reporting occurs daily with a cutoff at midnight, a patient could experience the event of interest (e.g., symptom onset) at any time between 12:00 am and 11:59 pm on a particular day. Moreover, some events are prone to longer censoring intervals than others (e.g., exposure intervals may be longer than one day for cases with multiple possible exposures). Not or incorrectly accounting for censoring of event intervals can lead to biased estimates of a delay [23].

Right truncation is defined as the inability to observe intervals (e.g. incubation periods) greater than some threshold (e.g. greater than the number of days elapsed since infection). It typically applies to real-time settings, when recently occurred events with longer intervals may not have been observed yet, leading to an overrepresentation of shorter intervals when estimating the incubation period or serial interval distribution. However, right truncation should not be confused with right censoring. The latter occurs when we observe the primary event of a case or future case but cannot observe it long enough to witness its secondary event [36], which could be due to, for instance, a study ending prematurely. As a result, we only know that the secondary event did not occur during the observation period and therefore have a right-censored interval for a data point. In contrast, right truncation means that certain intervals are
Here, we focus on the incubation period and serial interval because they are key inputs in mathematical and statistical models, such as those used for nowcasting/forecasting [9,17] or for scenario-based modeling [18,19], which can guide control strategies. While the incubation period must be positive, the serial interval can be positive or negative [20]. The right tail of the incubation period distribution informs the length of quarantine, while the left tail indicates the earliest time symptoms might develop after infection [8]. By comparing the incubation period and serial interval, we can learn about a pathogen's tendency for pre-symptomatic versus symptomatic transmission [21], which can inform the controllability of an epidemic [22]. At the individual level, pre-symptomatic transmission occurs precisely when the generation interval is shorter than the incubation period of the *infector* (or, equivalently, when the serial interval is shorter than the incubation period of the *infectee*). The serial interval is often used as a proxy for the generation interval.

Methods for estimating epidemiological delays have been improving, especially during the COVID-19 pandemic, and recent research has highlighted the importance of appropriately adjusting for three statistical issues inherent in the data collection process: censoring, right truncation, and dynamical bias [23–27] (Table 1, Figures 1-2).

Not adjusting for bias can lead to incorrect estimates of delays, which can have direct implications for public health practice. For example, Overton et al. found that the mean incubation period for COVID-19 in early 2020 (corresponding to the ancestral strain of SARS-CoV-2) was 3.49 days without adjusting for right truncation compared to 4.69 days when adjusted [25]. Overton et al. further showed that the unadjusted estimate would suggest a quarantine of 10 days would capture 99% of cases, compared to a 14-day quarantine for the adjusted estimate [25]. If the unadjusted estimate from this study had been used to inform the length of quarantine, more still-infectious individuals would have gone on to infect others.

Similarly, Park et al. found that ignoring right truncation for a fast-growing epidemic with
completely missing from our data as observing primary events depends on identifying
secondary events first. Right truncation is common in data where case ascertainment depends
on the secondary event, e.g. we rarely observe an individual's incubation period until after
symptoms develop. Not accounting for right truncation can lead to underestimating the mean
delay [23]. Although right truncation is mainly a problem for real-time analyses, retrospective
data can be right-truncated if surveillance ended prematurely.

Dynamical bias is another type of common sampling bias which is related to right truncation. During the increasing phase of an epidemic, patients who were infected recently are overrepresented in the recent data, leading to underestimation of delay intervals. Conversely, when the epidemic is decreasing, patients with short delays are underrepresented in the recent data, leading to the overestimation of delay intervals. Dynamical bias is especially problematic during periods of exponential growth and decay of cases when cases are exponentially more and less likely, respectively, to be infected recently rather than further back in time.

Measuring epidemiological delays

We aim to estimate the true underlying distribution for each epidemiological delay which characterizes the time between the primary and secondary event. In general, we assume that this distribution does not change over the course of an epidemic (although this may not always be the case [37]). Cases can enter a dataset due to observation of either their primary or secondary event. Regardless of how data were collected, we can organize our data into cohorts using either a forward or backward approach (Figure 1). For the forward approach, we start from primary events that occurred during the same period and prospectively determine when the secondary events occurred—the resulting distribution of the delays is the forward distribution. In contrast, for the backward approach, we start with secondary events that occurred during the same period and retrospectively determine when the primary events occurred—the resulting distribution of the delays is the backward distribution.

Data observed in real-time can be subject to either right truncation or right censoring.
Right truncation causes the observed forward distribution to be shorter than the true underlying
distribution and has the largest effect when the epidemic is growing because recently infected
individuals are overrepresented. In a declining epidemic, right truncation will have a smaller
impact on the forward distribution because the proportion of recently infected individuals with
unobserved secondary events is lower. Excluding right-censored data from the analysis is
equivalent to right truncating the data and leads to underestimation of the delay. Backward
distributions are not susceptible to right truncation but can have a delay distribution that is
shorter or longer than the forward distribution and the true underlying distribution depending on
the phase of the epidemic (i.e. dynamical bias). Both right truncation and dynamical biases are
minimal if data from the entire epidemic are available and included in a delay estimate.

Figure 2 shows the impact of different biases on the forward and backward distributions. We recommend always analyzing delay distributions as forward distributions and accounting for potential biases (i.e., censoring and right truncation) as this approach is generally more robust than working with backward distributions and correcting for dynamical bias.

Adjusting for common biases
118 relatively long delays could result in underestimation of the mean delay distribution by up to
119 50% [23].
120 Not adjusting for bias can also lead to incorrect estimates of other parameters of interest
121 that rely on accurate estimates of delays, such as the reproduction number. The time-varying
122 reproduction number, $R_t$, is the average number of secondary cases caused by a single infected
123 individual in a population. Gostic et al. showed how mis-specifying the mean, variance, or form
124 of the generation interval led to biased estimates of $R_t$ when using three empirical methods on
125 synthetic data [28]. The bias was greatest early in the epidemic. Underestimation of $R_t$ during
126 this phase could lead to an insufficient public health response. Similarly, if changes in the
127 generation interval (or serial interval) over the course of the epidemic are not accounted for, $R_t$
128 may not be accurate [29].

129 In the aftermath of the COVID-19 pandemic, there is considerable need for and interest
130 in the estimation and reporting of epidemiological parameters, including delays, with at least two
131 R packages [30,31], a World Health Organization working group [32], and meta-analyses of
132 priority pathogens [33–35] working to collect and make these data accessible. In this
133 perspective, we present best practices for the estimation and reporting of epidemiological
134 delays, illustrated by examples for the incubation period and serial interval of directly transmitted
135 (person-to-person) infectious diseases. Our recommendations are based on our experience
136 estimating and using delays across multiple outbreaks and recent methodological work [23].

137 To make these best practices easier to follow, we developed two checklists (Tables 2-3)
138 and a flowchart (Figure 3), which can be used to understand which biases need to be adjusted
139 for based on available data. To provide context for our recommendations, we additionally
140 provide details about the data needed to estimate delays and how they should be prepared for
141 analysis. Then, we discuss biases that can affect the estimation of epidemiological delay
142 distributions, followed by strategies that can be used to reduce the impact of these biases.
143 Technical details about these biases and how to adjust for them can be found in Park et al. [23]
This section is a synthesis of Park et al. [23] in which further details can be found. In general, adjusting for double interval censoring involves estimating the conditional probabilities of the primary and secondary events occurring between their observed lower and upper bounds. Interval censoring should always be adjusted for, and the adjustment method is the same irrespective of the epidemic phase. Right censoring can be handled using methods for survival analysis, such as the Kaplan-Meier approach [36]. Adjusting for right truncation involves normalizing the probability of observing a given delay from the untruncated forward distribution by the probability of observing any delay before the final observation time, and adjusting for dynamical bias involves incorporating the epidemic trajectory (e.g., the growth or decay rate of the epidemic) into the analysis [23, 26, 37, 39].

Although there are several available methods and tools for estimating epidemiological delay distributions, most either have not been validated or do not correct for all potential biases. One example is the *coarseDataTools* R package developed by Reich et al. [35]. This tool has been validated and can correct for double interval censoring, but it does not adjust for right truncation or dynamical bias. In contrast, *epidist*, an R package developed by some of the authors of this study, contains methods which can adjust for all three potential biases. In [23], *epidist* was used to evaluate multiple methods and found that the approximate latent variable censoring and truncation method emerged as the best performer for both real-time and retrospective analyses for most real-world use cases. This method corresponds to the double interval censoring and right truncation adjusted model developed by Ward et al. [24] which we recommend.

Ward et al.'s approach estimates the probability of observing a secondary event conditional on observing the primary event by a given final observation date. Estimated event times for each case are included in the model as unobserved, or latent, variables, and uniform prior distributions are used for both the primary and secondary event times, which can accommodate censoring intervals of arbitrary length. However, when censoring intervals are long, the event time distribution within the censoring interval will deviate from the uniform approximation (as its shape depends on underlying epidemic dynamics) and should be taken into account.

While we recommend Ward et al.'s double interval censoring and right truncation adjusted model for correcting common biases [24], this approach has important limitations; Park et al. found that this method was not able to estimate the mean or standard deviation well in epidemic simulations characterized by very rapid exponential growth and long delays. Details about other possible approaches can be found in Park et al. We suggest avoiding approaches that adjust for right truncation and dynamical biases simultaneously because they lead to overestimation of the mean delay by overcompensating for intervals that have not yet been observed.

## Additional modeling recommendations

Beyond correctly adjusting for biases, there are several common issues with reported epidemiological delays that may lead to biased conclusions when used in practice or impact their ability to be used at all. Historically, the incubation period and serial interval were often
along with extensive simulation and case studies. Our paper goes beyond the work of Park et al. by offering practical guidelines, a suggested workflow, and checklists, for a broader technical audience, such as modelers at public health agencies.

## 147 Data to estimate delays

Primary and secondary events can be observed (e.g., symptom onset time) or unobserved (e.g., usually infection time). Data to estimate the incubation period include the times of probable exposure (from which infection time can be inferred) and perceived symptom onset for each case. Data to estimate the serial interval are those in which symptom onset has been observed for primary and secondary cases. Examples of study designs or public health activities that generate such data include contact tracing [24,36], prospective cohort studies [37–39], household studies [40], or other types of intensive cohort monitoring [41]. Data from passive surveillance, which involves healthcare providers reporting cases to public health agencies, can also be used to estimate delay distributions; however, key information about exposures (e.g., dates, settings, and types of contact) may be missing, incomplete, or abstracted from other variables [42,43].

## 159 Biases in delay data

Three main biases can affect the estimation of epidemiological delay distributions, 1. censoring, 2. right truncation bias, and 3. dynamical (or epidemic-phase) bias (Table 1). All of them affect both single-individual delays, such as the incubation period, and pair-dependent delays, such as the serial interval.

Censoring is knowing that an event occurred but not precisely when. Data can be right censored (the event is known to have occurred after a certain time), left censored (the event is known to have occurred before a certain time), or interval censored (the event is known to have occurred within a certain time interval). In epidemiological delay data, censoring can affect either primary or secondary events (single interval censoring) or both (double interval censoring) [44]. Epidemiological data are almost always doubly interval-censored due to discretization of
reported using only the mean (and sometimes the range) [40–43]. However, models can be
fitted to delay data to adjust for some of the biases we have described and better characterize
the tail of the distribution. Assuming a modeling approach is taken, we summarize our
recommendations for estimating and reporting for epidemiological delay distributions in Table 2
and give more details in the following two sections.

We recommend fitting a parametric distribution to summarize the empirical delay
distribution. Multiple probability distributions should be fitted to delay data [44] and compared
using appropriate model comparison criteria (e.g., widely applicable information criterion [WAIC]
or leave-one-out information criterion [LOOIC] for Bayesian models). Common distributions for
epidemiological delays in the literature include the gamma, lognormal, and Weibull distributions
[45]. For delays that can have negative values, distributions that can accept negative values,
such as the skew-normal or skew-logistic distributions [46], may be used, or less ideally, the
delay data may be shifted to allow for fitting of distributions that only allow positive numbers
[47]. Mixture distributions may be appropriate for some delays and should also be considered
[48–51].

It is also important to visualize the fitted distributions to check that they fit the data [28]. When doing so, we recommend visualizing the estimated distribution in conjunction with the modeled observation process (e.g., double interval censoring and right truncation). In other words, estimate the latent (continuous) distribution. Then simulate elements of the observation process, such as double interval censoring (e.g. for date-level censoring this will transform continuous delay times into an integer number of days elapsed) and right truncation (this will change the shape of the observed distribution, relative to the latent distribution), from the latent distribution [52]. Not accounting for the observation process after estimating the latent distribution makes visual assessment of the fit difficult, because the observation data and the latent distribution may differ in shape and data type.

Care should be taken when converting the parameters of fitted probability distributions to the summary statistics of interest. For example, the gamma distribution may use either a scale or a rate parameter, in addition to its shape parameter [53], while the standard lognormal parameters, log mean and log sd do not correspond to the log of the mean and the log of the standard deviation of the lognormal [54]. Some R packages contain functions to perform parameter conversion, such as EpiNow2 (lognormal) [55], epitrix (gamma) [56], mixR (gamma, lognormal, and Weibull) [57], and epiparameter (gamma, lognormal, Weibull, negative binomial, and geometric) [8].

We recommend stratifying delay estimates whenever there are hypothesized differences
across groups as delays such as the incubation period and the serial interval may vary by route
of exposure [15], viral species [58] or clade, disease severity, or other factors. Ideally, this
stratification should be done jointly in a statistically robust framework [52,59,60]. Wider
application of joint modeling approaches could be achieved with more availability of easy-to-use
tools [52].

The serial interval and generation interval are two transmission intervals commonly used
in infectious disease modeling. For diseases with non-specific initial symptoms, consider
estimating delays according to more than one definition of symptom onset (e.g., onset of any
symptoms vs rash onset in the case of mpox or measles) [28,61]. Comparing the serial interval
and/or generation interval to other transmission intervals (e.g., diagnosis intervals, lab
170 the end points of the intervals being measured. For example, when reporting occurs daily with a
171 cutoff at midnight, a patient could experience the event of interest (e.g., symptom onset) at any
172 time between 12:00 am and 11:59 pm on a particular day. Double interval censoring is shown in
173 Figure 1 by the brackets around each event (circles and squares). Events occur anywhere on
174 the x axis, which represents continuous time, but the reporting of events only occurs at discrete
175 time points, or observation times. Some events are prone to longer censoring intervals than
176 others (e.g., exposure intervals may be longer than one day for cases with multiple possible
177 exposures). Not or incorrectly accounting for censoring of event intervals can lead to biased
178 estimates of a delay [23].

179 Right truncation is defined as the inability to observe intervals (e.g. incubation periods)
180 greater than a threshold (e.g. greater than the number of days elapsed since infection). It
181 typically applies to real-time settings, when events with longer intervals may not have occurred
182 yet, leading to an overrepresentation of shorter intervals when estimating delays. Right
183 truncation is common in data where case ascertainment depends on the secondary event, e.g.
184 we rarely observe an individual's incubation period until after symptoms develop. Not
185 accounting for right truncation can lead to underestimating the mean delay [23]. Although right
186 truncation is mainly a problem for real-time analyses, retrospective data can be right-truncated if
187 surveillance ended prematurely.

188 Right truncation should not be confused with right censoring. The latter occurs when we
189 observe the primary event of a case or future case but cannot observe it long enough to witness
190 its secondary event [45], which could be due to, for instance, a study ending prematurely. As a
191 result, we only know that the secondary event did not occur during the observation period and
192 therefore have a right-censored interval for a data point. Right censoring is shown in Figure 1A.
193 The intervals on the top and bottom of the panel are included in the analysis, but we do not
194 know when the secondary events will occur because they happen after the observation time. In
195 contrast, right truncation means that certain delays are completely missing from our data as
confirmation intervals, or reporting intervals) can provide useful information about the time scales of transmission as long as the interval that is being studied is clearly defined [62].

When Markov chain Monte Carlo methods are used to estimate delay distributions, it is important to visualize posterior predictions against data and check model diagnostics, such as R-hat values, divergent transitions, and effective sample sizes, and report them [63–65]. Convergence issues may indicate that the model is misspecified, making the results unreliable.

## Reporting items

We recommend reporting an estimate of variability (e.g., standard deviation or dispersion) along with central tendency (e.g., mean or median) for all estimated delay distributions. These quantities are often used as inputs in infectious disease models and can inform both clinical practice and public health policy [2]. The parameter estimates and quantiles of fitted probability distributions should also be reported as they are often used in modeling.

All summary statistics should always be accompanied by credible intervals or confidence intervals for Bayesian and frequentist analyses, respectively (usually 90% or 95% with the width of the reported interval also being reported). High uncertainty in parameter estimates can have substantial impacts on downstream modeling [66,67] and can indicate that more data need to be collected.

Estimates of delays should be accompanied by contextual information to aid in interpretation. For example, we recommend reporting the study sample size; the epidemic curve; which, if any, control measures are in place; and summary statistics on age, sex, geographic location, vaccination status, and possible exposure route(s). The epidemic curve can indicate at which stage of the epidemic the analysis took place and whether the outbreak is now over.

Code and data should be uploaded to repositories, such as GitHub (https://github.com/) or Zenodo (https://zenodo.org/), to ensure reproducibility of the analysis and facilitate re-use of the code. Apart from allowing others to reproduce, validate and potentially improve analyses, providing data along with estimates of delay distributions also ensures that the estimates can be integrated in future pooled estimation efforts as methods continue to be improved. These data should ideally be provided in linelist format, alongside information on censoring and stratifying variables. Importantly, the data should be anonymized to protect patient privacy according to local health data laws and regulations.

If data cannot be shared, we recommend at minimum providing samples of the posterior distribution in a permanent online repository to facilitate future re-analyses (as in [24]). This step is particularly important for distributions with correlated parameters to avoid inflation of uncertainty in downstream use (using parameterizations that have less correlations between parameters, such as those with location and scale, can also help).

## Reporting items specific to the incubation period and serial interval

In addition to the checklist for reporting epidemiological delay distributions, we recommend additional considerations specific to the incubation period and serial interval (Table 3).
196 observing primary events depends on identifying secondary events first. Right truncation is
197 shown in Figure 1B. Here, the intervals at the top and bottom of the panel are not included in
198 the analysis; we are unaware of these data points because their secondary events occur after
199 the observation time.

200 Dynamical bias is another type of common sampling bias which is analogous to right truncation. During the increasing phase of an epidemic, patients with short delays are overrepresented in the recent data, leading to underestimation of delay intervals. Conversely, when the epidemic is decreasing, patients with long delays are overrepresented in the recent data, leading to the overestimation of delay intervals. Dynamical bias is especially problematic during periods of exponential growth and decay of cases when cases are exponentially more and less likely, respectively, to be infected recently rather than further back in time.

207 Measuring epidemiological delays

208 We aim to estimate the true underlying distribution for each epidemiological delay which
209 characterizes the time between the primary and secondary event. In general, we assume that
210 this distribution does not change over the course of an epidemic (although this may not always
211 be the case [29]). Cases can enter a dataset due to observation of either their primary or
212 secondary event (shown by the arrows in Figure 2). Regardless of how data were collected, we
213 can organize our data into cohorts using either a forward or backward approach. For the forward
214 approach, we start from primary events that occurred during the same period and prospectively
215 determine when the secondary events occurred—the resulting distribution of the delays is the
216 forward distribution (Figure 2A and 2C). In contrast, for the backward approach, we start with
217 secondary events that occurred during the same period and retrospectively determine when the
218 primary events occurred—the resulting distribution of the delays is the backward distribution
219 (Figure 2B) [23].

220 Data observed in real-time can be subject to either right truncation or right censoring.
221 Right truncation causes the observed forward distribution to be shorter than the true underlying
For the incubation period, a case may have had multiple possible exposures prior to symptom onset, especially when community transmission of a pathogen is high. If a case reports multiple exposures, we recommend defining an exposure window that includes all possible exposure dates [1], using disjointed exposure windows where appropriate. Other methods that take this uncertainty into account could be used (such as [68] who used a Bayesian framework to infer the incubation period of the infector). We caution against restricting the analysis to cases with a high degree of certainty about their exposure periods as this can introduce biases [69].

For the serial interval, we only use case pairs where we are fairly confident transmission has occurred [47] (usually based on exposure information collected from patient interviews, see [28,70]). Although this approach could bias the serial interval towards specific lengths of intervals, it is usually preferable to using mis-specified case pairs. In terms of the direction of transmission, a number of approaches can be taken to order the case pairs. Where there is strong evidence that pre-symptomatic transmission does not occur for the disease of interest, reported negative intervals are presumed to be erroneous, and can be removed [28] or reversed [24]. Some studies assume the direction of transmission between epidemiologically linked cases based on the date order of symptom onset [24]; where negative serial intervals are possible, this should be avoided. It is also possible to use genomic data to order the pairs [71,72]. The ideal approach where negative intervals are possible is to use information about pair ordering and fit a distribution that allows for negative values. When no such information is available, a method that does not rely on knowing the order of case pairs would be ideal as this would enable the use of more available data and avoid biases from certain types or durations of exposure being easier or more difficult to link epidemiologically. Although such methods have been developed, they assume a fully sampled population [73,74]. It is important to be transparent about the approach taken. Uncertainty in the source of infection (such as the potential for multiple possible infectors) should also be considered.

## Other considerations

*How to best use new data.* Delays should be (re-)estimated when possible, especially if current estimates are poor or lacking. New estimates can be compared to those from previous epidemics or from a different phase of the same epidemic. One could consider any new dataset on its own. Alternatively, using mixed effects models to partially pool information across different outbreaks is likely a better use of available data. When choosing an approach, it is important to consider whether the primary modes of transmission or pathogen properties are different than in the past [45,75].

*Choice of prior distributions.* When using Bayesian methods to estimate epidemiological delay distributions, it is important to think about the choice of prior distributions. Some R packages that estimate epidemiological delays, including `coarseDataTools` [76] and `epidist` [52], have default prior distributions. Users of these tools should carefully consider the appropriateness of default prior distributions for their analyses and should explore the impact of different prior distributions on their results. We recommend against using uninformed prior distributions, especially uniform prior distributions [77], for the parameters of epidemiological delay
distribution and has the largest effect when the epidemic is growing because relatively many
recently infected individuals with long delays are not observed. In a declining epidemic, right
truncation will have a smaller impact on the forward distribution because the proportion of
recently infected individuals with unobserved secondary events is lower [23]. Excluding right-
censored data from the analysis is equivalent to right truncating the data and leads to
underestimation of the delay. Backward distributions are not susceptible to right truncation but
can have a delay distribution that is shorter or longer than the forward distribution and the true
underlying distribution depending on the phase of the epidemic (i.e. dynamical bias). Both right
truncation and dynamical biases are minimal if data from the entire epidemic are available and
included in a delay estimate [23].

Figure 1 shows the impact of different biases on the forward and backward distributions.
We recommend always analyzing delay distributions as forward distributions and accounting for
potential biases (i.e., censoring and right truncation) as this approach does not require
additional information on incidence trajectories, which are used to adjust for dynamical bias in
the backward distribution [23].

## Adjusting for common biases

Figure 3 shows a decision tree for assessing which biases need to be addressed
depending on the approaches taken for data collection and processing. For each of three
possible scenarios, it includes an explanation about the potential impact of the biases as well as
the methods needed to adjust for them.

In general, adjusting for double interval censoring involves estimating the conditional
probabilities of the primary and secondary events occurring between their observed lower and
upper bounds. Interval censoring should always be adjusted for, and the adjustment method is
the same irrespective of the epidemic phase [23]. Right censoring can be handled using
methods for survival analysis, such as the Kaplan-Meier approach [45]. Adjusting for right
truncation involves normalizing the probability of observing a given delay from the untruncated
distributions. When the delay distribution is already well reported in the literature, it is sensible to use this knowledge to inform prior distributions; however, the methods should be clearly communicated and accompanied by estimates generated from weakly informed prior distributions as sensitivity analyses.

***Meta-analyses.*** To reduce uncertainty from small sample sizes, some researchers have combined estimates of epidemiological delays from different studies through meta-analysis [12,13] or pooled analysis (re-analyzing published individual-level data) [2,78]. The latter method is preferred but may not always be possible. If performing a meta-analysis, we recommend performing sensitivity analyses when some estimates from the literature have not been corrected for bias [79]. Published estimates can also be adjusted for bias post-hoc by using the relationship between the backward and forward distributions, as in Park et al. [79]. Some authors have designed custom quality assessment scales to assess bias in studies included in meta-analyses of epidemiological parameters [12,47].

***Time-varying delays.*** While some delay distributions are expected to remain constant during an epidemic wave, others can change over time in response to interventions [37] and changes in reporting, among other factors. Specifically, we can study time-varying delays by analyzing changes in forward, rather than backward, delay distributions across cohorts. For example, Ali et al. found that using time-varying estimates of the serial interval of COVID-19 in early 2020 resulted in more accurate estimates of the time-varying reproduction number in mainland China compared to using fixed serial interval distributions [37]. Figure 3 shows a decision tree which can help determine which biases need to be adjusted for depending on the approaches taken for data collection and processing, including whether one is interested in time-varying delays.

Discussion

Epidemiological delay distributions are key parameters for preparedness and response to epidemics and pandemics. Their importance has been highlighted during the COVID-19 pandemic [37], the global mpox outbreak in 2022 [1], and other epidemics over the last two decades [7]. We have focused on the incubation period and serial interval, for which estimates are often made at the beginning of an epidemic when contact tracing data are available to support early characterization of the pathogen. While the estimates are most useful for real-time response during this time, they are also the most susceptible to bias. In addition to adjusting for bias, estimates need to be clearly and fully reported to maximize utility and make the most of data that are both costly and difficult to collect [80,81]. Our recommendations can assist with this.

Adjusting for bias when estimating delay distributions is one of the most important recommendations we highlight. Not adjusting for bias can lead to incorrect estimates of delays, which can have direct implications for public health practice. For example, Overton et al. found that the mean incubation period for COVID-19 in early 2020 (corresponding to the ancestral strain of SARS-CoV-2) was 3.49 days without adjusting for right truncation compared to 4.69 days when adjusted [25]. The unadjusted estimate would suggest a quarantine of 10 days would capture 99% of cases, compared to a 14-day quarantine for the adjusted estimate [25]. If
248 forward distribution by the probability of observing any delay before the final observation time,
249 and adjusting for dynamical bias involves incorporating the epidemic trajectory (e.g., the growth
250 or decay rate of the epidemic) into the analysis [23, 26, 45, 47].

251 There are several available methods and tools for estimating epidemiological delay
252 distributions; however, most of these approaches have not been validated or do not correct for
253 all potential biases [23]. One example is the *coarseDataTools* R package developed by Reich et
254 al. [48]. This tool has been validated and can correct for double interval censoring, but it does
255 not adjust for right truncation or dynamical bias. In contrast, *epidist* [49], an R package
256 developed by some of the authors of this study, contains methods which can adjust for all three
257 potential biases. In Park et al. [23], a simulation study was used to evaluate multiple methods
258 and found that the approximate latent variable censoring and truncation method emerged as the
259 best performer for both real-time and retrospective analyses for most real-world use cases. This
260 method corresponds to the double interval censoring and right truncation adjusted model
261 developed by Ward et al. [24] which we recommend.

262 Ward et al.'s approach estimates the probability of observing a secondary event
263 conditional on observing the primary event by a given final observation time [24]. Estimated
264 event times for each case are included in the model as unobserved, or latent, variables, and
265 uniform prior distributions are used for both the primary and secondary event times, which can
266 accommodate censoring intervals of arbitrary length. However, when censoring intervals are
267 long, the event time distribution within the censoring interval will deviate from the uniform
268 approximation (as its shape depends on underlying epidemic dynamics) and should be taken
269 into account [24]. This model has important limitations. For example, Park et al. [23] found that
270 this method was not able to estimate the mean or standard deviation as well in epidemic
271 simulations characterized by very rapid exponential growth and long delays because primary
272 event times are poorly approximated by a uniform distribution.
the unadjusted estimate from this study had been used to inform the length of quarantine, more still-infectious individuals would have gone on to infect others. Similarly, Park et al. found that ignoring right truncation for a fast-growing epidemic with relatively long delays could result in underestimation of the mean delay distribution by up to 50% [23].

Not adjusting for bias can also lead to incorrect estimates of other parameters of interest that rely on accurate estimates of delays, such as the reproduction number. The basic reproduction number, $R_0$, is the average number of secondary cases caused by a single infected individual in a large, completely susceptible population. If $R_0$ is underestimated, new epidemics may be erroneously perceived as less severe, leading to an insufficient initial public health response.

A limitation of current methods for correcting common biases is that they do not fully account for time-varying changes in delay distributions [23]. Future work on delay distributions or nowcasting (which demonstrates how to model time-varying delays using time-to-event modeling) [82–84] could extend current methods or develop new methods to account for these changes.

To our knowledge, the *epidist* R package [52], developed by some of this paper's authors, is the only software tool that correctly handles interval censoring, right truncation, and dynamical bias. However, it does not currently implement all the best practices we have recommended. A major, though soon to be addressed, limitation is that it can only model lognormal distributions. In addition, while it can handle joint modeling (i.e., across strata) and time-varying delays, these features are not documented and require validation. Its implementation of our recommended method, by Ward et al., does not support custom primary-event prior distributions, so it cannot account for the growth rate in the primary-event window. The tool also scales poorly with increasing amounts of data. Further limitations are discussed in Park et al. [23]. Future development work is planned on the package to address these issues, and community contributions are also welcome.

Many of the best practices outlined in this paper also apply to other epidemiological delays. However, there are issues which we did not cover. For example, we did not focus on methods for estimating delay distributions for vector-borne diseases, as these require additional considerations (e.g., accounting for vector biology) [7,85]. Also, we did not aim to provide a systematic review, and the examples presented in this work were selected to illustrate specific points.

In conclusion, we have provided recommendations for generating and evaluating epidemiological delay distributions. We gave examples of good practice for the incubation period and serial interval from various infectious disease outbreaks over the last few decades; though few examples in the literature incorporate all the best practices outlined in this paper. We hope that our recommendations will provide clarity and structured guidance about what should be reported and how to adjust for biases in delay data. We also hope this checklist will be adopted by the infectious disease modeling community to understand the limitations of existing estimates and improve future estimates.

## Funding statement
Certain practices should be avoided if using an alternative approach to adjust for biases in delays. For example, we suggest avoiding approaches that adjust for right truncation and dynamical biases simultaneously because they lead to overestimation of the mean delay by overcompensating for intervals that have not yet been observed [23]. We also do not recommend using a midpoint imputation rule (i.e., use the midpoint of the interval as the "observed" value and construct downstream inferences based on that imputation) on interval-censored data as such imputation approaches may introduce bias [23,50].

## Additional modeling recommendations

Beyond correctly adjusting for biases, there are several common issues with reported epidemiological delays that may lead to biased conclusions when used in practice or impact their ability to be used at all. Historically, the incubation period and serial interval were often reported using only the mean (and sometimes the range) [51–54]. However, models can be fitted to delay data to adjust for some of the biases we have described and better characterize the tail of the distribution. Assuming a modeling approach is taken, we summarize our recommendations for estimating and reporting for epidemiological delay distributions in Table 2 and give more details in the sections “Reporting epidemiological delay distributions” and “Reporting the incubation period and serial interval.” Table 2 contains details about each recommendation, examples of diseases for which it has been implemented, and possible solutions. It is divided into two sections, estimation and reporting. Whenever estimating delays, we recommend going through the table to make sure all the steps are taken.

*Fit multiple probability distributions.* We recommend fitting multiple probability distributions to summarize the empirical delay distribution [55]. This approach has greater utility for users of the estimates compared to non-parametric approaches. Use appropriate model comparison criteria (e.g., widely applicable information criterion [WAIC] or leave-one-out information criterion [LOOIC] for Bayesian models) to suggest the best-fitting model alongside visual checks.

Common distributions for epidemiological delays in the literature include the gamma, lognormal,
AC is funded by the National Institute for Health and Care Research (NIHR) Health Protection Research Unit in Modelling and Health Economics, a partnership between the UK Health Security Agency, Imperial College London and LSHTM (grant code NIHR200908); and acknowledges funding from the MRC Centre for Global Infectious Disease Analysis (reference MR/X020258/1), funded by the UK Medical Research Council (MRC). This UK funded award is carried out in the frame of the Global Health EDCTP3 Joint Undertaking. SC acknowledges funding support from the Laboratoire d'Excellence Integrative Biology of Emerging Infectious Diseases program (grant ANR-10-LABX-62-IBEID) and the INCEPTION project (PIA/ANR16-CONV-0005). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

## Disclaimer

The findings and conclusions in this report are those of the authors and do not necessarily represent the official position of the CDC, U.S. Department of Health and Human Services, NIHR, UK Health Security Agency, or the UK Department of Health and Social Care.

## Author contributions

Conceptualization: KC, SWP, SA

Data curation: NA

Formal analysis: NA

Funding acquisition: NA

Investigation: KC, SWP, SA

Methodology: NA

Project administration: NA

Resources: NA

Software: NA

Supervision: SA

Validation: NA

Visualization: KC, SWP, SA, AC

Writing - original draft preparation: KC, SWP

Writing - review & editing: All authors

## Data availability statement

No data were used in this study.
and Weibull distributions [56]. For delays that can have negative values, distributions that accept negative values, such as the skew-normal or skew-logistic distributions [57], may be used, or less ideally, the delay data may be shifted to allow for fitting of distributions that only allow positive numbers [58]. Mixture distributions may be appropriate for some delays, such as those with bimodal distributions [59–62]. Fitting parametric distributions may not be appropriate for all delay distributions. Semi- or non-parametric approaches, such as hazard models, may be considered [63]. Non-parametric estimates can also be used to assess the relative fit of parametric models [27].

*Visualize the distributions.* It is also important to visualize the fitted distributions to check that they fit the data [36]. When doing so, we recommend visualizing the estimated distribution in conjunction with the modeled observation process (e.g., double interval censoring and right truncation). In other words, estimate the latent (continuous) distribution. Then, from the latent distribution, simulate elements of the observation process, such as double interval censoring (e.g. for date-level censoring this will transform continuous delay times into an integer number of days elapsed) and right truncation (this will change the shape of the observed distribution, relative to the latent distribution) [49]. Not accounting for the observation process after estimating the latent distribution makes visual assessment of the fit difficult, because the observation data and the latent distribution may differ in shape and data type. For example, Sender et al. illustrate how good visualizations can help make intuitive comparisons across different distributions [64].

*Correctly convert parameters.* Care should be taken when converting the parameters of fitted probability distributions to the summary statistics of interest. For example, the gamma distribution may use either a scale or a rate parameter, in addition to its shape parameter [65], while the standard lognormal parameters, log mean and log sd do *not* correspond to the log of the mean and the log of the standard deviation of the lognormal [66]. Some R packages contain functions to perform parameter conversion, such as EpiNow2 (lognormal) [67], epitrix (gamma)
Tables

<table><caption>Table 1. Adjusting for biases in epidemiological delay distributions. Examples of diseases for each bias were selected based on convenience (either papers written by this study's authors or those encountered during the course of their work). Note that not all methods in this table are discussed in the text, and more details can be found in Park et al. [23].</caption><thead><tr><th>Bias</th><th>Interval censoring</th><th>Right truncation</th><th>Dynamical bias</th></tr></thead><tbody><tr><td>Details</td><td>The exact timing of the primary event or secondary event (single interval censoring) or both events (double interval censoring) is unknown (e.g., except for experimental studies, exposure is usually reported as a date or range of dates, rather than a time of day).</td><td>Right truncation is a type of sampling bias. It arises because only cases whose secondary event has occurred can be observed. In an ongoing epidemic, right truncation biases the incubation period and serial interval toward shorter intervals because individuals with longer incubation periods may not have developed symptoms or have been reported yet.</td><td>Dynamical bias is another type of sampling bias that can be present when case ascertainment is based on the secondary event. It is related to epidemic dynamics: during a growth phase, cases that developed symptoms recently are overrepresented in the observed data, while during a declining phase, these cases are underrepresented. This means that the backward distribution is not representative of the forward distribution during these periods.</td></tr><tr><td>Impact</td><td>Not accounting for interval censoring can lead to biased estimates of a delay's standard deviation. Incorrectly accounting for it can also bias the mean.</td><td>Not accounting for right truncation can lead to underestimation of the mean delay [23].</td><td>Not adjusting for dynamical bias when estimating the forward distribution from the backward distribution can lead to under- or over-estimation of delay intervals depending on whether the epidemic is in a growth or declining phase, respectively.</td></tr><tr><td>Diseases for which this bias has been considered in analyses</td><td>Incubation period: mpox [28]; Zika [86]; COVID-19 [87]; 6 vector-borne diseases [78]<br/>Serial interval: mpox [28]</td><td>Incubation period: COVID-19 [25,87]<br/>Serial interval: mpox [24]</td><td>Serial interval: mpox [24]</td></tr></tbody></table>
[68], mixR (gamma, lognormal, and Weibull) [69], and epiparameter (gamma, lognormal, Weibull, negative binomial, and geometric) [30].

*Add subgroups or stratify estimates.* If sample size allows, we recommend stratifying delay estimates whenever there are hypothesized differences across groups as delays, such as the incubation period and the serial interval, may vary by route of exposure [15], viral species [70] or clade, disease severity, sex [71], age [71], vaccination status, or other factors. Ideally, this stratification should be done jointly in a statistically robust framework [49,63,72]. Wider application of joint modeling approaches could be achieved with more availability of easy-to-use tools [49].

*Check model diagnostics.* When Bayesian methods are used, visualize the posterior distribution against data and check model diagnostics, such as trace plots, the potential scale reduction statistic (R-hat), divergent transitions, and effective sample sizes, and report them [73–75].

Convergence issues may indicate that the model is mis-specified, making the results unreliable.
For more advice about using Bayesian methods, see [73,74,76].

## Reporting epidemiological delay distributions

We recommend reporting an estimate of variability (e.g., standard deviation or dispersion) along with central tendency (e.g., mean or median) for all estimated delay distributions (sometimes more than one distribution fits the data similarly well as in [36]) alongside the quantiles and underlying parameters of the fitted distributions. These quantities are often used as inputs in infectious disease models and can inform both clinical practice and public health policy [8]. If possible, the probability density function should be specified to avoid ambiguity about the parameters. For Bayesian analyses, samples of the posterior distribution should be made available as summarizing estimates may obfuscate valuable information about the correlations between parameters and the shape of the posterior distributions.

All summary statistics should always be accompanied by credible intervals or confidence intervals for Bayesian and frequentist analyses, respectively (usually 90% or 95% with the width
<table><tr><td>Possible solutions</td><td>Use methods, such as Reich et al. 2009 [35], that adjust for double interval censoring; however, this method does not adjust for right truncation. Alternatively, use Ward et al.'s double interval censoring and right truncation adjusted model, which adjusts for interval censoring and right truncation simultaneously [23,24].</td><td>Use an approximate latent variable method, such as Ward et al.'s double interval censoring and right truncation adjusted model [24] or similar alternatives [23].</td><td>If both primary and secondary event dates are known and the incidence of primary events is changing exponentially at a constant rate, it is possible to use the approach of Verity et al., Britton et al., and Park et al. [26,38,39] to adjust for dynamical bias; however, uncertainty in both growth rate estimates and observed delays need to be taken into account carefully with this approach and the assumption of constant growth rates may not be met in practice. Park et al. present a version of this method that allows for a time-varying growth rate, but it requires untruncated incidence data or assumptions to be made about the recent growth rate [23]. For most settings, considering the forward distribution and accounting for right truncation is recommended.</td></tr><tr><td>Practices to avoid</td><td>Do not adjust for single censoring if the data are doubly interval-censored as this will result in a biased mean [23]. Note that even if both primary and secondary events are reported to have occurred on a single date, the data should be considered doubly interval-censored.</td><td>Do not adjust for right truncation and dynamical bias at the same time early in an outbreak as doing this can lead to overadjustment of the downward bias and therefore to overestimation of the delay [23].</td><td>Avoid adjusting for right truncation and dynamical bias at the same time [23]. When analyzing the forward distribution, adjust for right truncation; when analyzing the forward distribution via the backward distribution, adjust for dynamical bias. We recommend the former when possible.</td></tr></table>
of the reported interval also being reported). High uncertainty in parameter estimates can have substantial impacts on downstream modeling [77,78] and can indicate that more data need to be collected.

Estimates of delays should be accompanied by contextual information to aid in interpretation. For example, we recommend reporting the study sample size; the epidemic curve; which, if any, control measures are in place; and summary statistics on age, sex, geographic location, vaccination status, and possible exposure route(s). Control measures and summary statistics can be used to assess generalizability of the estimates (see earlier advice on stratification of estimates). The epidemic curve can indicate at which stage of the epidemic the analysis took place and whether the outbreak is now over (i.e., whether certain biases need to be adjusted for).

Code and data should be uploaded to repositories, such as GitHub (https://github.com/) or Zenodo (https://zenodo.org/), to ensure reproducibility of the analysis and facilitate re-use of the code. Apart from allowing others to reproduce, validate and potentially improve analyses, providing data along with estimates of delay distributions also ensures that the estimates can be integrated in future pooled estimation efforts as methods continue to be improved. These data should ideally be provided in linelist format with all necessary information required for estimation (e.g., the left and right boundaries of the possible infection and symptom onset times for the incubation period [44]). Importantly, the data should be anonymized/de-identified to protect patient privacy according to local health data laws and regulations. If data cannot be shared, we recommend at minimum providing samples of the posterior distribution in a permanent online repository to facilitate future re-analyses (as in [24]).

## Reporting the incubation period and serial interval

In addition to the checklist for reporting epidemiological delay distributions, we recommend additional considerations specific to the incubation period and serial interval in Table 3. Like Table 2, this checklist contains details about each recommendation, examples
<table><caption>Table 2. Checklist for reporting/reported epidemiological delay distributions. Examples of diseases for each checklist item were selected based on convenience (either papers written by this study's authors or those encountered during the course of their work).</caption><thead><tr><th>Checklist item</th><th>Details</th><th>Diseases for which this item has been implemented</th><th>Possible solutions</th></tr></thead><tbody><tr><td colspan="4" style="text-align: center; font-weight: bold;">Estimation</td></tr><tr><td>Adjust for biases</td><td>See Table 1</td><td>See Table 1</td><td>Adjust for censoring (always), right truncation (when needed), and dynamical bias (when needed). Clearly state that all these adjustments have been made, and report both right-truncation-adjusted and right-truncation-unadjusted estimates. Consider using the approach of Reich et al. [35] or Ward et al's double interval censoring corrected model [23,24] to obtain estimates that are not adjusted for right truncation.</td></tr><tr><td>Compare multiple probability distributions</td><td>Estimated delays may depend on the fitted probability distribution, so it is important to use the distribution that best represents the data.</td><td>Incubation period: mpox [45]; COVID-19 [87,88]; dengue [85]; HIV [89]; malaria [49]<br/>Serial interval: MERS [68]</td><td>Fit more than one probability distribution to the data [44] and use appropriate model selection criteria to compare them. Visualize the fit of distributions to the data.</td></tr><tr><td>Correctly convert parameters of probability distributions to summary statistics</td><td>Incorrectly converting parameters leads to wrong estimates of delays.</td><td>Incubation period: COVID-19 [87]</td><td>If writing equations, double check them or use a software package with built-in functions for parameter conversion, such as EpiNow2 (lognormal) [55], epitrix (gamma) [56], mixR (gamma, lognormal, and Weibull) [57], and epiparameter (gamma, lognormal, Weibull, negative binomial, and geometric) [8].</td></tr></tbody></table>
377 from real outbreaks, and possible solutions. It should be used each time an incubation period or
378 serial interval is estimated.

379 For the incubation period, a case may have had multiple possible exposures prior to
380 symptom onset, especially when community transmission of a pathogen is high. If a case
381 reports multiple exposures, we recommend defining an exposure window that includes all
382 possible exposure dates [7], using disjointed exposure windows where appropriate. Other
383 methods that take this uncertainty into account could be used (such as [79] who used a
384 Bayesian framework to infer the incubation period of the infector). We caution against restricting
385 the analysis to cases with a high degree of certainty about their exposure periods as this can
386 introduce biases [80].

387 For the serial interval, we only use case pairs where we are fairly confident transmission
388 has occurred [58] (usually based on exposure information collected from patient interviews, see
389 [36,81]). Although this approach could bias the serial interval towards specific lengths of
390 intervals, it is usually preferable than to using mis-specified case pairs. Bias from including mis-
391 specified case pairs in the analysis is likely larger than that from removing pairs between which
392 transmission likely did not occur; however, future research should formally assess this
393 convention with simulations. In terms of the direction of transmission, a number of approaches
394 can be taken to order the case pairs. Where there is strong evidence that pre-symptomatic
395 transmission does not occur for the disease of interest, reported negative intervals are
396 presumed to be erroneous, and can be removed [36] or reversed [24]. Some studies assume
397 the direction of transmission between epidemiologically linked cases based on the date order of
398 symptom onset [24]; where negative serial intervals are possible, this should be avoided. It is
399 also possible to use genomic data to order the pairs [82,83]. The ideal approach where negative
400 intervals are possible is to use information about pair ordering and fit a distribution that allows
401 for negative values (see "Additional modeling recommendations"). When no such information is
402 available, a method that does not rely on knowing the order of case pairs would be ideal as this
<table><tr><td>Add subgroups to your model or stratify the estimates whenever appropriate</td><td>Estimated delays may vary based on a variety of factors.</td><td>Incubation period: Ebola (virus species) [58]; West Nile virus disease (transplant/transfusion vs not) [78]; dengue (serotype) [85]; HIV (sex and age category) [89]</td><td>If sample size allows and a difference across groups is hypothesized, add subgroups to your model or stratify the estimates by exposure type, genetic variant/clade, or other factors (e.g., sex, age, vaccination status, etc.). The former approach, also known as joint modeling, is preferred.</td></tr><tr><td>Consider estimating other transmission intervals</td><td>Comparing other indicators of infections can provide useful information about the time scales of transmission.</td><td>Incubation period: mpox (symptom onset definition) [28]; COVID-19 (symptom onset definition) [88]<br/>Serial interval: mpox (symptom onset definition) [28]; measles (symptom onset definition) [61]</td><td>Depending on available data, other useful delays may be estimated, such as diagnosis-based, lab-confirmation-based, and report-based transmission intervals. Consider joint modeling approaches for estimation and clearly report the time points used for primary and secondary events.</td></tr><tr><td>Check model diagnostics</td><td>If model requirements are not met, the estimated delay distributions may not be reliable.</td><td>Incubation period: dengue [85]</td><td>If using Markov chain Monte Carlo methods, make sure all models converge. Visually inspect Markov chain Monte Carlo traces. Check R-hat values, divergent transitions, effective sample sizes, and any other diagnostics and report when appropriate.</td></tr><tr><td colspan="4" style="text-align: center; font-weight: bold;">Reporting</td></tr><tr><td>Report measures of central tendency and variability</td><td>These may be used as inputs for infectious disease models and can inform both clinical practice and public health</td><td>Incubation period: mpox [45], COVID-19 [87,88]; 6 vector-borne diseases [78]; Zika</td><td>Report multiple summary statistics and clearly state which is which. Report at least the mean or median as well as standard deviation, variance, or dispersion.</td></tr></table>
403 would enable the use of more available data and avoid biases from certain types or durations of
404 exposure being easier or more difficult to link epidemiologically. Although such methods have
405 been developed, they assume a fully sampled population [84,85]. It is important to be
406 transparent about the approach taken. Uncertainty in the source of infection (such as the
407 potential for multiple possible infectors) should also be considered.

408 **Other considerations**

409 *How to best use new data.* Delays should be (re-)estimated when possible, especially if current
410 estimates are poor or lacking. New estimates can be compared to those from previous
411 epidemics or from a different phase of the same epidemic. One could consider any new dataset
412 on its own. Alternatively, using mixed effects models to partially pool information across different
413 outbreaks is likely a better use of available data. When choosing an approach, it is important to
414 consider whether the primary modes of transmission or pathogen properties are different than in
415 the past [56,86].

416 *Choice of prior distributions.* When using Bayesian methods to estimate epidemiological delay
417 distributions, it is important to think about the choice of prior distributions. Some R packages
418 that estimate epidemiological delays have default prior distributions [48,49]. Users of these tools
419 should carefully consider the appropriateness of default prior distributions for their analyses and
420 should explore the impact of different prior distributions on their results. We recommend against
421 using uninformative prior distributions, especially uniform prior distributions [87], for the
422 parameters of epidemiological delay distributions. When the delay distribution is already well
423 reported in the literature, this knowledge can be used to inform prior distributions; however, the
424 methods should be clearly communicated and accompanied by estimates generated from
425 weakly informative prior distributions as sensitivity analyses.

426 *Meta-analyses.* To reduce uncertainty from small sample sizes, some researchers have
427 combined estimates of epidemiological delays from different studies through meta-analysis
428 [34,35] or pooled analysis (re-analyzing published individual-level data) [8,88]. The latter
<table><tr><td></td><td>policy [2].</td><td>[86]; 9 respiratory diseases [2]<br/>Serial interval: MERS [68]; mpox [28]</td><td></td></tr><tr><td>Report quantiles of the probability distribution</td><td>The left tail of the incubation period distribution indicates the earliest time symptoms could develop following infection, while the right tail is often of interest for control strategies, such as monitoring people who have been exposed [25,88]. The right tail of the serial interval can also inform the length of quarantine.</td><td>Incubation period: mpox [45]; COVID-19 [87,88]; 6 vector-borne diseases [78]; Zika [86]; 9 respiratory diseases [2]<br/>Serial interval: mpox [24]</td><td>Report key quantiles (e.g., 2.5, 5, 25, 50, 75, 95, 97.5, 99) of the distribution in a table.</td></tr><tr><td>Report the parameters for the fitted probability distributions</td><td>These inputs may be used for mathematical modeling and are key for correctly defining the distribution.</td><td>Serial interval: mpox [28,90]</td><td>E.g. for the gamma distribution, report shape and scale; for the lognormal distribution, report logmean and log standard deviation. If possible, the probability density function should be specified to avoid ambiguity about the parameters.</td></tr><tr><td>Report uncertainty in the estimates</td><td>Communicating uncertainty is a key aspect of outbreak analysis and modeling [67]. Also, high uncertainty in parameter estimates can affect downstream modeling [66].</td><td>Incubation period: mpox [1,45,90]; COVID-19 [87,88]; 6 vector-borne diseases [78]; dengue [85]; Zika [2,7]; MERS [68,91]; 9 respiratory diseases [2]<br/>Serial interval: mpox [24,28]; COVID-19 [17]</td><td>Report 90% or 95% credible intervals or confidence intervals for all estimates (central tendency, variability, quantiles, and parameters for the fitted probability distributions). Ideally, provide joint posterior samples for Bayesian analyses as these are important for characterizing covariance in the posterior distribution. Make sure to report how you have defined your intervals (i.e. report that they are 95% credible intervals).</td></tr></table>
method is preferred but may not always be possible. If performing a meta-analysis, we
recommend performing sensitivity analyses when some estimates from the literature have not
been corrected for bias [89]. Published estimates can also be adjusted for bias post-hoc by
using the relationship between the backward and forward distributions, as in Park et al. [89].
Some authors have designed custom quality assessment scales to assess bias in studies
included in meta-analyses of epidemiological parameters [33,34].

*Time-varying delays.* While some delay distributions are expected to remain stationary during
an epidemic wave, others can change over time in response to interventions [29] and changes
in reporting, among other factors. For example, Ali et al. found that the serial interval of COVID-
19 in mainland China shortened after non-pharmaceutical interventions were implemented in
early 2020 [29]. We can study time-varying delays by analyzing changes in forward delay
distributions across cohorts. When doing so, we still need to account for both censoring and
right truncation. We do not recommend inferring time-varying delays from backward delay
distributions because dynamical bias also causes changes in this distribution even when the
forward distribution remains stationary.

444

Discussion

Epidemiological delay distributions are key parameters for preparedness and response
to epidemics and pandemics. Their importance has been highlighted during the COVID-19
pandemic [29], the global mpox (formerly known as monkeypox) outbreak in 2022 [7], and other
epidemics over the last two decades [13]. We have focused on the incubation period and serial
interval, for which estimates are often made at the beginning of an epidemic when contact
tracing data are available to support early characterization of the pathogen. While the estimates
are most useful for real-time response during this time, they are also the most susceptible to
bias. Indeed, adjusting for bias when estimating delay distributions is one of the most important
recommendations we highlight. as not doing so may lead to the propagation of bias into
downstream modeling [23] and therefore an incorrect understanding of an epidemic (e.g., over-
<table><tr><td>Report characteristics about study sample</td><td>Characteristics about the study sample can provide epidemiological context for the estimates which can help with interpretation.</td><td>Incubation period: MERS [68]; 9 respiratory diseases [2] Serial interval: MERS [68]</td><td>Report the sample size of the study, demographic characteristics of patients (e.g., age, sex, geographic location, vaccination status), and route of exposure(s) (if known).</td></tr><tr><td>Report the epidemic curve and which, if any, control measures are in place</td><td>The epidemic curve can provide context about the epidemic phase (increasing, decreasing, or stable) and whether right truncation bias or dynamical bias needs to be considered.</td><td>Incubation period: mpox [90] Serial interval: COVID-19 [38]; MERS [91]</td><td>Include a figure of the epidemic curve or provide a reference to the curve on a permanent website (doi). Ideally, the underlying data for the curve would be made available to download. Alternatively, provide an estimate of the growth rate for the study period.</td></tr><tr><td>Provide anonymized data and documented code</td><td>This step improves the reproducibility of the study, and the code can be reused by other teams during future epidemics. It can also facilitate meta-analyses and joint analyses of multiple datasets.</td><td>Incubation period: COVID-19 [87,88] Serial interval: mpox [70]</td><td>Deidentified linelist-level data should be provided with relevant stratifying variables. For small epidemics, some authors have reported data relative to an unspecified reference date to protect patient identities [45]. However, the data should still ideally be linked to the epidemic trajectory to address dynamical bias issues. An alternative approach could be to widen the censoring intervals.</td></tr></table>
or under-estimating the risk to the host population) [25]. In addition to adjusting for bias, estimates need to be clearly and fully reported to maximize utility and make the most of data that are both costly and difficult to collect [90,91]. Our recommendations can assist with this.

A limitation of current methods for correcting common biases is that they do not fully account for time-varying changes in delay distributions [23]. Future work on delay distributions or nowcasting (which demonstrates how to model time-varying delays using time-to-event modeling) [92–94] should extend current methods or develop new methods to account for these changes.

Many of the best practices outlined in this paper also apply to other epidemiological delays. However, there are issues which we did not cover. For example, we did not focus on methods for estimating delay distributions for vector-borne diseases, such as dengue and Yellow fever, as these require additional considerations (e.g., accounting for vector biology) [13,95]. Also, we did not aim to provide a systematic review; rather, we provided insights based on our experiences. The examples presented in this work were selected to illustrate specific points.

In conclusion, we have provided recommendations in the form of two checklists for generating and evaluating epidemiological delay distributions. We gave examples of good practice for the incubation period and serial interval from various infectious disease outbreaks over the last few decades; though few examples in the literature incorporate all the best practices outlined in this paper. We hope that our recommendations will provide clarity and structured guidance about what should be reported and how to adjust for biases in delay data. We also hope our flowchart and checklists will be adopted by the infectious disease modeling community to understand the limitations of existing estimates and improve future estimates.

## Funding statement
<table><caption>Table 3. Additional checklist items for reporting the incubation period and serial interval.</caption><thead><tr><td>Delay</td><td>Checklist item</td><td>Details</td><td>Diseases for which this item has been implemented</td><td>Possible solutions</td></tr></thead><tbody><tr><td>Incubation period</td><td>Investigate the potential for multiple possible exposures</td><td>Sometimes, a case may have had multiple opportunities to be exposed before symptom onset.</td><td>Mpox [92]; MERS [68]</td><td>For these cases, use an exposure window that includes all possible exposure dates, such as travel to a high-risk area. Use a disjointed exposure window where appropriate (however, this approach may cause some issues for the sampler with Ward et al.'s double interval censoring and right truncation adjusted model [24]). The method developed by Cowling et al. 2015 [68] for MERS could be used as well.</td></tr><tr><td rowspan="2">Serial interval</td><td>Check for negative serial intervals</td><td>Negative serial intervals can occur when symptom onset in the infectee occurs before symptom onset in the infector [17].</td><td>Mpox [24,28]</td><td>Assuming the data are correct, including negative serial intervals, keep the ordering of the pairs and fit a distribution that allows for negative values (such as normal). If there are negative serial intervals but there is strong evidence that pre-symptomatic transmission does not occur for the disease of interest, consider removing [28] or reversing the order [24] of those case pairs. There are also methods that do not depend on knowing the order of case pairs [73,74].</td></tr><tr><td>Investigate the potential for multiple possible infectors</td><td>Sometimes, a case could have been exposed to more than one infected person prior to symptom onset.</td><td>Mpox [28,70]; MERS [68]</td><td>Restrict the analysis to only cases with a high degree of certainty that the secondary case was infected by the primary case. Then, do a sensitivity analysis with all cases and compare the results.</td></tr></tbody></table>
481 AC is funded by the National Institute for Health and Care Research (NIHR) Health Protection
482 Research Unit in Modelling and Health Economics, a partnership between the UK Health
483 Security Agency, Imperial College London and LSHTM (grant code NIHR200908); and
484 acknowledges funding from the MRC Centre for Global Infectious Disease Analysis (reference
485 MR/X020258/1), funded by the UK Medical Research Council (MRC). This UK funded award is
486 carried out in the frame of the Global Health EDCTP3 Joint Undertaking. SC acknowledges
487 funding support from the Laboratoire d'Excellence Integrative Biology of Emerging Infectious
488 Diseases program (grant ANR-10-LABX-62-IBEID) and the INCEPTION project (PIA/ANR16-
489 CONV-0005). The funders had no role in study design, data collection and analysis, decision to
490 publish, or preparation of the manuscript.

491

### Disclaimer

493

494 The findings and conclusions in this report are those of the authors and do not necessarily
495 represent the official position of the CDC, U.S. Department of Health and Human Services,
496 NIHR, UK Health Security Agency, or the UK Department of Health and Social Care.

497

### Author contributions

499

500 Conceptualization: KC, SWP, SA

501 Data curation: NA

502 Formal analysis: NA

503 Funding acquisition: NA

504 Investigation: KC, SWP, SA

505 Methodology: NA

506 Project administration: NA

507 Resources: NA

508 Software: NA

509 Supervision: SA

510 Validation: NA

511 Visualization: KC, SWP, SA, AC

512 Writing - original draft preparation: KC, SWP

513 Writing - review & editing: All authors

514

### Data availability statement

516

517 No data were used in this study.
## Figures

Figure 1. Forward and backward approaches for cohorting and analyzing data to estimate epidemiological delay distributions. The y-axis represents unique observations of delays. The yellow circles represent the primary events, while the green squares represent the secondary events. The black horizontal lines represent the delay between primary and secondary events, and the vertical dotted lines show the cohorts. The arrows represent how cases enter the dataset: arrows pointing toward the right indicate that the case's primary event was observed first, while arrows pointing toward the left indicate that the case's secondary event was observed first. Note that the case ascertainment method does not impact the direction we can cohort the data. For forward cohorts (A and C), all primary events that occurred during the same period are selected and prospectively followed until the secondary event occurs. For backward cohorts (B), all secondary events that occurred during the same period are selected; the timing of the primary events is identified retrospectively.

A

Forward cohort/forward distribution

<figure><img src="image_3.png" /></figure>

B

Backward cohort/backward distribution

<figure><img src="image_4.png" /></figure>

C

Forward cohort/forward distribution

<figure><img src="image_5.png" /></figure>
Tables

<table><caption>Table 1. Adjusting for biases in epidemiological delay distributions. Examples of diseases for each bias were selected based on convenience (either papers written by this study's authors or those encountered during the course of their work). Note that not all methods in this table are discussed in the text, and more details can be found in Park et al. [23].</caption><thead><tr><td>Bias</td><td>Interval censoring</td><td>Right truncation</td><td>Dynamical bias</td></tr></thead><tbody><tr><td>Details</td><td>The exact timing of the primary event or secondary event (single interval censoring) or both events (double interval censoring) is unknown (e.g., except for experimental studies, exposure is usually reported as a date or range of dates, rather than a time of day).</td><td>Right truncation is a type of sampling bias. It arises because only cases whose secondary event has occurred can be observed. In an ongoing epidemic, right truncation biases the incubation period and serial interval toward shorter intervals because individuals with longer incubation periods may not have developed symptoms or have been reported yet.</td><td>Dynamical bias is another type of sampling bias that can be present when case ascertainment is based on the secondary event. It is related to epidemic dynamics: during a growth phase, cases that developed symptoms recently are overrepresented in the observed data, while during a declining phase, these cases are underrepresented. This means that the backward distribution is not representative of the forward distribution during these periods.</td></tr><tr><td>Impact</td><td>Not accounting for interval censoring can lead to biased estimates of a delay's standard deviation. Incorrectly accounting for it can also bias the mean.</td><td>Not accounting for right truncation can lead to underestimation of the mean delay [23].</td><td>Not adjusting for dynamical bias when estimating the forward distribution from the backward distribution can lead to under- or over-estimation of delay intervals depending on whether the epidemic is in a growth or declining phase, respectively.</td></tr><tr><td>Diseases for which this bias has been considered in analyses</td><td>Incubation period: mpox [36]; Zika [96]; COVID-19 [97]; 6 vector-borne diseases [88]<br/>Serial interval: mpox [36]</td><td>Incubation period: COVID-19 [25,97]<br/>Serial interval: mpox [24]</td><td>Incubation period: COVID-19 [89]<br/>Serial interval: mpox [24]</td></tr></tbody></table>
Figure 2. Common biases involved in the estimation of epidemiological delay distributions. The y axis in each panel represents unique observations of delays. The circles represent primary events, while the squares represent secondary events. The horizontal lines represent the delay between events, while the vertical lines represent the time at which the data are observed. The arrows represent how cases enter the dataset: arrows pointing toward the right indicate that the case's primary event was observed first, while arrows pointing toward the left indicate that the case's secondary event was observed first. The brackets "[ ]" represent interval censoring of the primary and secondary events. Delays and events in gray are unobserved. (A) and (B) demonstrate the same scenario, but in (A), observation of the delay is based on the primary event, and there is right censoring, while in (B), observation is based on the secondary event, and there is right truncation. (C) demonstrates an example of a backward distribution in a growing epidemic, when the majority of delays that make up the distribution will be short; hence, the backward distribution will be shorter than the equivalent forward distribution. (D) demonstrates the reverse in a declining epidemic. Both (C) and (D) show the impact of dynamical bias.
<table><tr><td>Possible solutions</td><td>Use methods, such as Reich et al. 2009 [44], that adjust for double interval censoring; however, this method does not adjust for right truncation. Alternatively, use Ward et al.'s double interval censoring and right truncation adjusted model, which adjusts for interval censoring and right truncation simultaneously [23,24].</td><td>Use an approximate latent variable method, such as Ward et al's double interval censoring and right truncation adjusted model [24] or similar alternatives [23].</td><td>If both primary and secondary event dates are known and the incidence of primary events is changing exponentially at a constant rate, it is possible to use the approach of Verity et al., Britton et al., and Park et al. [26,46,47] to adjust for dynamical bias; however, uncertainty in both growth rate estimates and observed delays need to be taken into account carefully with this approach and the assumption of constant growth rates may not be met in practice. Park et al. present a version of this method that allows for a time-varying growth rate, but it requires untruncated incidence data or assumptions to be made about the recent growth rate [23]. For most settings, considering the forward distribution and accounting for right truncation is recommended.</td></tr><tr><td>Practices to avoid</td><td>Do not adjust for single censoring if the data are doubly interval-censored as this will result in a biased mean [23]. Note that even if both primary and secondary events are reported to have occurred on a single date, the data should be considered doubly interval-censored. Do not use a midpoint imputation rule on interval-censored data as this may introduce bias [50].</td><td>Do not adjust for right truncation and dynamical bias at the same time early in an outbreak as doing this can lead to overadjustment of the downward bias and therefore to overestimation of the delay [23].</td><td>Avoid adjusting for right truncation and dynamical bias at the same time [23]. When analyzing the forward distribution, adjust for right truncation; when analyzing the forward distribution via the backward distribution, adjust for dynamical bias. We recommend the former when possible.</td></tr></table>
A

Forward cohort/forward distribution

<figure><img src="image_6.png" /></figure>

B

Forward cohort/forward distribution

<figure><img src="image_7.png" /></figure>

C

Backward cohort/backward distribution

Growing epidemic

<figure><img src="image_8.png" /></figure>

D

Backward cohort/backward distribution

Declining epidemic

<figure><img src="image_9.png" /></figure>
<figure><img src="image_10.png" /><figcaption>Figure 3. Decision tree for bias adjustment when estimating delay distributions, assuming that double interval censoring is always adjusted for and that the forward distribution is being modeled directly (i.e not via the backwards distribution and dynamical correction) as we recommend regardless of data collection approach. If you have an estimate of the backwards distribution from the literature, see the section on other considerations for advice.</figcaption></figure>
<table><caption>Table 2. Checklist for reporting/reported epidemiological delay distributions. Examples of diseases for each checklist item were selected based on convenience (either papers written by this study's authors or those encountered during the course of their work).</caption><thead><tr><th>Checklist item</th><th>Details</th><th>Diseases for which this item has been implemented</th><th>Possible solutions</th></tr></thead><tbody><tr><td colspan="4" style="text-align: center; font-weight: bold;">Estimation</td></tr><tr><td>Adjust for biases</td><td>See Table 1</td><td>See Table 1</td><td>Adjust for censoring (always), right truncation (when needed), and dynamical bias (when needed). Clearly state that all these adjustments have been made, and report both right-truncation-adjusted and right-truncation-unadjusted estimates. Consider using the approach of Reich et al. [44] or Ward et al's double interval censoring corrected model [23,24] to obtain estimates that are not adjusted for right truncation.</td></tr><tr><td>Compare multiple probability distributions</td><td>Estimated delays may depend on the fitted probability distribution, so it is important to use the distribution that best represents the data.</td><td>Incubation period: mpox [56]; COVID-19 [97,98]; dengue [95]; HIV [71]; malaria [60]<br/>Serial interval: MERS [79]</td><td>Fit more than one probability distribution to the data [55] and use appropriate model selection criteria to compare them. Visualize the fit of distributions to the data.</td></tr><tr><td>Correctly convert parameters of probability distributions to summary statistics</td><td>Incorrectly converting parameters leads to wrong estimates of delays.</td><td>Incubation period: COVID-19 [97]</td><td>If writing equations, double check them or use a software package with built-in functions for parameter conversion, such as EpiNow2 (lognormal) [67], epitrix (gamma) [68], mixR (gamma, lognormal, and Weibull) [69], and epiparameter (gamma, lognormal, Weibull, negative binomial, and geometric) [30].</td></tr></tbody></table>
References

1. Charniga K, Masters NB, Slayton RB, Gosdin L, Minhaj FS, Philpott D, et al. Estimating the incubation period of monkeypox virus during the 2022 multi-national outbreak. medRxiv. 2022; 2022.06.22.22276713. doi:10.1101/2022.06.22.22276713

2. Lessler J, Reich NG, Brookmeyer R, Perl TM, Nelson KE, Cummings DA. Incubation periods of acute respiratory viral infections: a systematic review. Lancet Infect Dis. 2009;9: 291–300. doi:10.1016/S1473-3099(09)70069-6

3. Overton CE, Abbott S, Christie R, Cumming F, Day J, Jones O, et al. Nowcasting the 2022 mpox outbreak in England. Lau EHY, editor. PLOS Comput Biol. 2023;19: e1011463. doi:10.1371/journal.pcbi.1011463

4. Lavezzo E, Franchin E, Ciavarella C, Cuomo-Dannenburg G, Barzon L, Del Vecchio C, et al. Suppression of a SARS-CoV-2 outbreak in the Italian municipality of Vo'. Nature. 2020;584: 425–429. doi:10.1038/s41586-020-2488-1

5. WHO Ebola Response Team. Ebola Virus Disease in West Africa — The First 9 Months of the Epidemic and Forward Projections. N Engl J Med. 2014;371: 1481–1495. doi:10.1056/NEJMoa1411100

6. Fraser C, Donnelly CA, Cauchemez S, Hanage WP, Van Kerkhove MD, Hollingsworth TD, et al. Pandemic Potential of a Strain of Influenza A (H1N1): Early Findings. Science. 2009;324: 1557–1561. doi:10.1126/science.1176062

7. Ferguson NM, Cucunubá ZM, Dorigatti I, Nedjati-Gilani GL, Donnelly CA, Basáñez M-G, et al. Countering the Zika epidemic in Latin America. Science. 2016;353: 353–354. doi:10.1126/science.aag0219

8. Lambert JW, Kucharski A. epiparameter. 2023. Available: https://github.com/epiverse-trace/epiparameter

9. Nash R, Morgenstern C, Cuomo-Dannenburg G, McCabe R, McCain K. epireview. [cited 28 Dec 2023]. Available: https://mrc-ide.github.io/epireview/

10. WorldHealthOrganization/collaboratory-epiparameter-community. World Health Organization; 2023. Available: https://github.com/WorldHealthOrganization/collaboratory-epiparameter-community

11. McAloon C, Collins Á, Hunt K, Barber A, Byrne AW, Butler F, et al. Incubation period of COVID-19: a rapid systematic review and meta-analysis of observational research. BMJ Open. 2020;10: e039652. doi:10.1136/bmjopen-2020-039652

12. Cuomo-Dannenburg G, McCain K, McCabe R, Unwin HJT, Doohan P, Nash RK, et al. Marburg virus disease outbreaks, mathematical models, and disease parameters: a systematic review. Lancet Infect Dis. 2023; S1473309923005157. doi:10.1016/S1473-3099(23)00515-7
<table><tr><td>Add subgroups to your model or stratify the estimates whenever appropriate</td><td>Estimated delays may vary based on a variety of factors.</td><td>Incubation period: Ebola (virus species) [70]; West Nile virus disease (transplant/transfusion versus not) [88]; dengue (serotype) [95]; HIV (sex and age category) [71]</td><td>If sample size allows and a difference across groups is hypothesized, add subgroups to your model or stratify the estimates by exposure type, genetic variant/clade, or other factors (e.g., sex, age, vaccination status, etc.). The former approach, also known as joint modeling, is preferred.</td></tr><tr><td>Check model diagnostics</td><td>If model requirements are not met, the estimated delay distributions may not be reliable.</td><td>Incubation period: dengue [95]</td><td>If using Bayesian methods, make sure all models converge. Visually inspect trace plots. Check potential scale reduction statistic (R-hat), divergent transitions, effective sample sizes, and any other diagnostics and report when appropriate. Examples of R packages to perform these checks include coda [99], rstan [75], and bayesplot [100].</td></tr></table>

## Reporting

<table><tr><td>Report measures of central tendency and variability</td><td>These may be used as inputs for infectious disease models and can inform both clinical practice and public health policy [8].</td><td>Incubation period: mpox [56], COVID-19 [97,98]; 6 vector-borne diseases [88]; Zika [96]; 9 respiratory diseases [8]<br/>Serial interval: MERS [79]; mpox [36]</td><td>Report multiple summary statistics and clearly state which is which. Report at least the mean or median as well as standard deviation, variance, or dispersion.</td></tr><tr><td>Report quantiles of the probability distribution</td><td>The left tail of the incubation period distribution indicates the earliest time symptoms could develop following</td><td>Incubation period: mpox [56]; COVID-19 [97,98]; 6 vector-borne diseases [88]; Zika</td><td>Report key quantiles (e.g., 2.5, 5, 25, 50, 75, 95, 97.5, 99) of the distribution in a table.</td></tr></table>
13. Doohan P, Jorgensen D, Naidoo TM, McCain K, Hicks JT, McCabe R, et al. Lassa fever outbreaks, mathematical models, and disease parameters: a systematic review and meta-analysis. medRxiv; 2024. p. 2024.03.23.24304596. doi:10.1101/2024.03.23.24304596

14. Ward T, Glaser A, Overton CE, Carpenter B, Gent N, Seale AC. Replacement dynamics and the pathogenesis of the Alpha, Delta and Omicron variants of SARS-CoV-2. Epidemiol Infect. 2023;151: e32. doi:10.1017/S0950268822001935

15. Reynolds MG, Yorita KL, Kuehnert MJ, Davidson WB, Huhn GD, Holman RC, et al. Clinical Manifestations of Human Monkeypox Influenced by Route of Infection. J Infect Dis. 2006;194: 773-780. doi:10.1086/505880

16. Galmiche S, Cortier T, Charmet T, Schaeffer L, Chény O, Von Platen C, et al. SARS-CoV-2 incubation period across variants of concern, individual factors, and circumstances of infection in France: a case series analysis from the ComCor study. Lancet Microbe. 2023;4: e409-e417. doi:10.1016/S2666-5247(23)00005-8

17. Nishiura H, Linton NM, Akhmetzhanov AR. Serial interval of novel coronavirus (COVID-19) infections. Int J Infect Dis. 2020;93: 284-286. doi:10.1016/j.ijid.2020.02.060

18. Liu Y, Centre for Mathematical Modelling of Infectious Diseases nCoV Working Group, Funk S, Flasche S. The contribution of pre-symptomatic infection to the transmission dynamics of COVID-2019 [version 1; peer review: 3 approved]. Wellcome Open Res. 2020;5: 58. doi:10.12688/wellcomeopenres.15788.1

19. Fraser C, Riley S, Anderson RM, Ferguson NM. Factors that make an infectious disease outbreak controllable. Proc Natl Acad Sci. 2004;101: 6146-6151. doi:10.1073/pnas.0307506101

20. Cramer E, Ray E, Lopez V, Bracher J, Brennen A, Rivadeneira A, et al. Evaluation of individual and ensemble probabilistic forecasts of COVID-19 mortality in the United States. PNAS. 2022;119: e2113561119.

21. Borchering RK, Healy JM, Cadwell BL, Johansson MA, Slayton RB, Wallace M, et al. Public health impact of the U.S. Scenario Modeling Hub. Epidemics. 2023;44: 100705. doi:10.1016/j.epidem.2023.100705

22. Clay PA, Pollock ED, Saldarriaga EM, Pathela P, Macaraig M, Zucker JR, et al. Modeling the impact of prioritizing first or second vaccine doses during the 2022 mpox outbreak. Infectious Diseases (except HIV/AIDS); 2023 Oct. doi:10.1101/2023.10.13.23297005

23. Park SW, Akhmetzhanov AR, Charniga K, Cori A, Davies NG, Dushoff J, et al. Estimating epidemiological delay distributions for infectious diseases. Epidemiology; 2024 Jan. doi:10.1101/2024.01.12.24301247

24. Ward T, Christie R, Paton RS, Cumming F, Overton CE. Transmission dynamics of monkeypox in the United Kingdom: contact tracing study. BMJ. 2022; e073153. doi:10.1136/bmj-2022-073153
<table><tr><td></td><td>infection, while the right tail is often of interest for control strategies, such as monitoring people who have been exposed [25,98]. The right tail of the serial interval can also inform the length of quarantine.</td><td>[96]; 9 respiratory diseases [8]<br/>Serial interval: mpox [24]</td><td></td></tr><tr><td>Report the parameters for all fitted probability distributions</td><td>These inputs may be used for mathematical modeling and are key for correctly defining the distribution.</td><td>Serial interval: mpox [36,101]</td><td>E.g. for the gamma distribution, report shape and scale; for the lognormal distribution, report logmean and log standard deviation. If possible, the probability density function should be specified to avoid ambiguity about the parameters.</td></tr><tr><td>Report uncertainty in the estimates</td><td>Communicating uncertainty is a key aspect of outbreak analysis and modeling [78]. Also, high uncertainty in parameter estimates can affect downstream modeling [77].</td><td>Incubation period: mpox [7,56,101]; COVID-19 [97,98]; 6 vector-borne diseases [88]; dengue [95]; Zika [8,13]; MERS [79,102]; 9 respiratory diseases [8]<br/>Serial interval: mpox [24,36]; COVID-19 [20]</td><td>Report 90% or 95% credible intervals or confidence intervals for all estimates (central tendency, variability, quantiles, and parameters for the fitted probability distributions). Ideally, provide joint posterior samples for Bayesian analyses as these are important for characterizing covariance in the posterior distribution. Make sure to report how intervals were defined (i.e. report that they are 95% credible intervals).</td></tr><tr><td>Report characteristics about study sample</td><td>Characteristics about the study sample can provide epidemiological context for the estimates which can help with interpretation.</td><td>Incubation period: MERS [79]; 9 respiratory diseases [8]; mpox [56]<br/>Serial interval: MERS [79]</td><td>Report the sample size of the study, demographic characteristics of patients (e.g., age, sex, geographic location, vaccination status), and route of exposure(s) (if known).</td></tr></table>
25. Overton CE, Stage HB, Ahmad S, Curran-Sebastian J, Dark P, Das R, et al. Using statistics and mathematical modelling to understand infectious disease outbreaks: COVID-19 as an example. Infect Dis Model. 2020;5: 409-441. doi:10.1016/j.idm.2020.06.008

26. Britton T, Scalia Tomba G. Estimation in emerging epidemics: biases and remedies. J R Soc Interface. 2019;16: 20180670. doi:10.1098/rsif.2018.0670

27. Seaman SR, Presanis A, Jackson C. Estimating a time-to-event distribution from right-truncated data in an epidemic: A review of methods. Stat Methods Med Res. 2022;31: 1641-1655. doi:10.1177/09622802211023955

28. Madewell ZJ, Charniga K, Masters NB, Asher J, Fahrenwald L, Still W, et al. Serial Interval and Incubation Period Estimates of Monkeypox Virus Infection in 12 Jurisdictions, United States, May-August 2022. Emerg Infect Dis. 2023;29: 818-821. doi:10.3201/eid2904.221622

29. Humphrey JH, Marinda E, Mutasa K, Moulton LH, Iliff PJ, Ntozini R, et al. Mother to child transmission of HIV among Zimbabwean women who seroconverted postnatally: prospective cohort study. BMJ. 2010;341: c6580-c6580. doi:10.1136/bmj.c6580

30. Antonucci G, Girardi E, Raviglione M, Ippolito G, Almi P, Angarano G, et al. Risk Factors for Tuberculosis in HIV-Infected Persons: A Prospective Cohort Study. JAMA. 1995;274: 143-148. doi:10.1001/jama.1995.03530020061033

31. Munywoki PK, Koech DC, Agoti CN, Lewa C, Cane PA, Medley GF, et al. The Source of Respiratory Syncytial Virus Infection In Infants: A Household Cohort Study In Rural Kenya. J Infect Dis. 2014;209: 1685-1692. doi:10.1093/infdis/jit828

32. Levy JW, Cowling BJ, Simmerman JM, Olsen SJ, Fang VJ, Suntarattiwong P, et al. The Serial Intervals of Seasonal and Pandemic Influenza Viruses in Households in Bangkok, Thailand. Am J Epidemiol. 2013;177: 1443-1451. doi:10.1093/aje/kws402

33. Russell TW, Hellewell J, Jarvis CI, Van Zandvoort K, Abbott S, Ratnayake R, et al. Estimating the infection and case fatality ratio for coronavirus disease (COVID-19) using age-adjusted data from the outbreak on the Diamond Princess cruise ship, February 2020. Eurosurveillance. 2020;25. doi:10.2807/1560-7917.ES.2020.25.12.2000256

34. Centers for Disease Control and Prevention. Introduction to Public Health. In: Public Health 101 Series. 2014 [cited 9 Jan 2024]. Available: https://www.cdc.gov/training/publichealth101/surveillance.html

35. Reich NG, Lessler J, Cummings DAT, Brookmeyer R. Estimating incubation period distributions with coarse data. Stat Med. 2009;28: 2769-2784. doi:10.1002/sim.3659

36. Dey T, Lipsitz SR, Cooper Z, Trinh Q-D, Krzywinski M, Altman N. Survival analysis—time-to-event data and censoring. Nat Methods. 2022;19: 906-908. doi:10.1038/s41592-022-01563-7

37. Ali ST, Wang L, Lau EHY, Xu X-K, Du Z, Wu Y, et al. Serial interval of SARS-CoV-2 was shortened over time by nonpharmaceutical interventions. Science. 2020;369: 1106-1109. doi:10.1126/science.abc9004
<table><tr><td>Report the epidemic curve and which, if any, control measures are in place</td><td>The epidemic curve can provide context about the epidemic phase (increasing, decreasing, or stable) and whether right truncation bias or dynamical bias needs to be considered.</td><td>Incubation period: mpox [101]<br/>Serial interval: COVID-19 [46]; MERS [102]</td><td>Include a figure of the epidemic curve or provide a reference to the curve on a permanent website (doi). Ideally, the underlying data for the curve would be made available to download. Alternatively, provide an estimate of the growth rate for the study period.</td></tr><tr><td>Provide anonymized/de-identified data and documented code</td><td>This step improves the reproducibility of the study, and the code can be reused by other teams during future epidemics. It can also facilitate meta-analyses and joint analyses of multiple datasets.</td><td>Incubation period: COVID-19 [97,98]<br/>Serial interval: mpox [81]</td><td>Anonymized/de-identified linelist-level data should be provided with relevant stratifying variables. For small epidemics, some authors have reported data relative to an unspecified reference date to protect patient identities [56]. However, the data should still ideally be linked to the epidemic trajectory to address dynamical bias issues. An alternative approach could be to widen the censoring intervals.</td></tr></table>
38. Park SW, Sun K, Champredon D, Li M, Bolker BM, Earn DJD, et al. Forward-looking serial intervals correctly link epidemic growth to reproduction numbers. Proc Natl Acad Sci. 2021;118: e2011548118. doi:10.1073/pnas.2011548118

39. Verity R, Okell LC, Dorigatti I, Winskill P, Whittaker C, Imai N, et al. Estimates of the severity of coronavirus disease 2019: a model-based analysis. Lancet Infect Dis. 2020;20: 669–677. doi:10.1016/S1473-3099(20)30243-7

40. Nolen LD, Osadebe L, Katomba J, Likofata J, Mukadi D, Monroe B, et al. Extended Human-to-Human Transmission during a Monkeypox Outbreak in the Democratic Republic of the Congo. Emerg Infect Dis. 2016;22: 1014–1021. doi:10.3201/eid2206.150579

41. Mertens PE, Patton R, Baum JJ, Monath TP. Clinical presentation of lassa fever cases during the hospital epidemic at Zorzor, Liberia, March-April 1972. Am J Trop Med Hyg. 1973;22: 780–784. doi:10.4269/ajtmh.1973.22.780

42. Martini G. Marburg virus disease. Postgrad Med J. 1973;49: 542–546.

43. WHO International Study Team. Ebola haemorrhagic fever in Sudan, 1976. Bull World Health Organ. 1978;56: 247–270.

44. Nishiura H. Early efforts in modeling the incubation period of infectious diseases with an acute course of illness. Emerg Themes Epidemiol. 2007;4: 2. doi:10.1186/1742-7622-4-2

45. Miura F, Van Ewijk CE, Backer JA, Xiridou M, Franz E, Op De Coul E, et al. Estimated incubation period for monkeypox cases confirmed in the Netherlands, May 2022. Eurosurveillance. 2022;27. doi:10.2807/1560-7917.ES.2022.27.24.2200448

46. Ferretti L, Ledda A, Wymant C, Zhao L, Ledda V, Abeler-Dörner L, et al. The timing of COVID-19 transmission. Epidemiology; 2020 Sep. doi:10.1101/2020.09.04.20188516

47. McAloon CG, Wall P, Griffin J, Casey M, Barber A, Codd M, et al. Estimation of the serial interval and proportion of pre-symptomatic transmission events of COVID-19 in Ireland using contact tracing data. BMC Public Health. 2021;21: 805. doi:10.1186/s12889-021-10868-9

48. Akhmetzhanov AR, Cheng H-Y, Dushoff J. Incubation-period estimates of Omicron (BA.1) variant from Taiwan, December 2021–January 2022, and its comparison to other SARS-CoV-2 variants: a statistical modeling, systematic search and meta-analysis. Infectious Diseases (except HIV/AIDS); 2023 Jul. doi:10.1101/2023.07.20.23292983

49. Lover AA, Zhao X, Gao Z, Coker RJ, Cook AR. The distribution of incubation and relapse times in experimental human infections with the malaria parasite Plasmodium vivax. BMC Infect Dis. 2014;14: 539. doi:10.1186/1471-2334-14-539

50. Klinkenberg D, Nishiura H. The correlation between infectivity and incubation period of measles, estimated from households with two cases. J Theor Biol. 2011;284: 52–60. doi:10.1016/j.jtbi.2011.06.015
<table><caption>Table 3. Additional checklist items for reporting the incubation period and serial interval.</caption><thead><tr><td>Delay</td><td>Checklist item</td><td>Details</td><td>Diseases for which this item has been implemented</td><td>Possible solutions</td></tr></thead><tbody><tr><td>Incubation period</td><td>Investigate the potential for multiple possible exposures</td><td>Sometimes, a case may have had multiple opportunities to be exposed before symptom onset.</td><td>Mpox [103]; MERS [79]</td><td>For these cases, use an exposure window that includes all possible exposure dates, such as travel to a high-risk area. Use a disjointed exposure window where appropriate (however, this approach may cause some issues for the sampler with Ward et al.'s double interval censoring and right truncation adjusted model [24]). The method developed by Cowling et al. 2015 [79] for MERS could be used as well.</td></tr><tr><td rowspan="2">Serial interval</td><td>Check for negative serial intervals</td><td>Negative serial intervals can occur when symptom onset in the infectee occurs before symptom onset in the infector [20].</td><td>Mpox [24,36]</td><td>Assuming the data are correct, including negative serial intervals, keep the ordering of the pairs and fit a distribution that allows for negative values (such as normal). If there are negative serial intervals but there is strong evidence that pre-symptomatic transmission does not occur for the disease of interest, consider removing [36] or reversing the order [24] of those case pairs. There are also methods that do not depend on knowing the order of case pairs [84,85].</td></tr><tr><td>Investigate the potential for multiple possible infectors</td><td>Sometimes, a case could have been exposed to more than one infected person prior to symptom onset.</td><td>Mpox [36,81]; MERS [79]</td><td>Restrict the analysis to only cases with a high degree of certainty that the secondary case was infected by the primary case. Then, do a sensitivity analysis with all cases and compare the results.</td></tr></tbody></table>
51. Hoffmann T, Alsing J. Faecal shedding models for SARS-CoV-2 RNA among hospitalised patients and implications for wastewater-based epidemiology. J R Stat Soc Ser C Appl Stat. 2023;72: 330–345. doi:10.1093/jrsssc/qlad011

52. Park SW, Abbott S. Estimating delay distributions accounting for common biases. 2023. Available: https://github.com/parksw3/epidist-paper

53. Gamma distribution. Wikipedia. 2024. Available: https://en.wikipedia.org/w/index.php?title=Gamma_distribution&oldid=1210997995

54. Log-normal distribution. Wikipedia. 2024. Available: https://en.wikipedia.org/w/index.php?title=Log-normal_distribution&oldid=1210518171

55. Abbott S, Hellewell J, Sherratt K, Gostic K, Hickson J, Badr HS, et al. EpiNow2: Estimate Real-Time Case Counts and Time-Varying Epidemiological Parameters. 2023. Available: https://cran.r-project.org/web/packages/EpiNow2/index.html

56. Jombart T, Cori A, Kamvar ZN, Schumacher D, Finger F, Whittaker C. epitrix: Small Helpers and Tricks for Epidemics Analysis. 2023. Available: https://cran.r-project.org/web/packages/epitrix/index.html

57. Yu Y. mixR: Finite Mixture Modeling for Raw and Binned Data. 2021. Available: https://cran.r-project.org/web/packages/mixR/index.html

58. Van Kerkhove MD, Bento AI, Mills HL, Ferguson NM, Donnelly CA. A review of epidemiological parameters from Ebola outbreaks to inform early public health decision-making. Sci Data. 2015;2: 150019. doi:10.1038/sdata.2015.19

59. Abbott S, Lison A, Funk S, Choi-Hannah, Gruson H, Pearson CAB, et al. epinowcast/epinowcast: Epinowcast 0.2.2. Zenodo; 2023. doi:10.5281/ZENODO.5637165

60. Lison A, Abbott S, Huisman J, Stadler T. Generative Bayesian modeling to nowcast the effective reproduction number from line list data with missing symptom onset dates. 2023 [cited 21 Feb 2024]. doi:10.48550/ARXIV.2308.13262

61. Hope Simpson RE. INFECTIOUSNESS OF COMMUNICABLE DISEASES IN THE HOUSEHOLD. The Lancet. 1952;260: 549–554. doi:10.1016/S0140-6736(52)91357-3

62. Abbott S, Sherratt K, Gerstung M, Funk S. Estimation of the test to test distribution as a proxy for generation interval distribution for the Omicron variant in England. medRxiv; 2022. p. 2022.01.08.22268920. doi:10.1101/2022.01.08.22268920

63. Hamra G, MacLehose R, Richardson D. Markov Chain Monte Carlo: an introduction for epidemiologists. Int J Epidemiol. 2013;42: 627–634. doi:10.1093/ije/dyt043

64. Lambert B. A student's guide to Bayesian statistics. Los Angeles London New Delhi Singapore Washington DC Melbourne: SAGE; 2018.

65. RStan: the R interface to Stan. [cited 10 Apr 2024]. Available: https://mc-stan.org/rstan/articles/rstan.html
Figures

Figure 1. Common biases involved in the estimation of epidemiological delay distributions. The y axis in each panel represents distinct observations of delays. The circles represent primary events, while the squares represent secondary events. The horizontal lines represent the delay between events, while the vertical lines represent the time at which the data are observed. The arrows represent how cases enter the dataset: arrows pointing toward the right indicate that the primary event was observed first, while arrows pointing toward the left indicate that the secondary event was observed first. The brackets "[ ]" represent interval censoring of the primary and secondary events. Note how primary and secondary events occur in continuous time, while the reporting of events (brackets) always occur at discrete intervals. Delays and events in gray are unobserved. (A) and (B) demonstrate the same scenario, but the direction of the arrows is different. In (A), observation of the delay is based on the primary event, and there is right censoring, while in (B), observation is based on the secondary event, and there is right truncation. (C) demonstrates an example of a backward distribution in a growing epidemic, when the majority of delays that make up the distribution will be short; hence, the backward distribution will be shorter than the equivalent forward distribution. (D) demonstrates the reverse in a declining epidemic. Both (C) and (D) show the impact of dynamical bias.
66. Elderd BD, Dukic VM, Dwyer G. Uncertainty in predictions of disease spread and public health responses to bioterrorism and emerging diseases. Proc Natl Acad Sci. 2006;103: 15693-15697. doi:10.1073/pnas.0600816103

67. Pollett S, Johansson MA, Reich NG, Brett-Major D, Del Valle SY, Venkatramanan S, et al. Recommended reporting items for epidemic forecasting and prediction research: The EPIFORGE 2020 guidelines. PLOS Med. 2021;18: e1003793. doi:10.1371/journal.pmed.1003793

68. Cowling BJ, Park M, Fang VJ, Wu P, Leung GM, Wu JT. Preliminary epidemiological assessment of MERS-CoV outbreak in South Korea, May to June 2015. Eurosurveillance. 2015;20. doi:10.2807/1560-7917.ES2015.20.25.21163

69. Cheng H-Y, Akhmetzhanov AR, Dushoff J. SARS-CoV-2 Incubation Period during Omicron BA.5-Dominant Period, Japan. Emerg Infect Dis. 2024;30. doi:10.3201/eid3001.230208

70. Miura F, Backer JA, Van Rijckevorsel G, Bavalia R, Raven S, Petrignani M, et al. Time Scales of Human Mpox Transmission in The Netherlands. J Infect Dis. 2023; jiad091. doi:10.1093/infdis/jiad091

71. Campbell F, Didelot X, Fitzjohn R, Ferguson N, Cori A, Jombart T. outbreaker2: a modular platform for outbreak reconstruction. BMC Bioinformatics. 2018;19: 363. doi:10.1186/s12859-018-2330-z

72. Hall M, Woolhouse M, Rambaut A. Epidemic Reconstruction in a Phylogenetics Framework: Transmission Trees as Partitions of the Node Set. PLOS Comput Biol. 2015;11: e1004613. doi:10.1371/journal.pcbi.1004613

73. Forsberg White L, Pagano M. A likelihood-based method for real-time estimation of the serial interval and reproductive number of an epidemic. Stat Med. 2008;27: 2999-3016. doi:10.1002/sim.3136

74. Wu KM, Riley S. Estimation of the Basic Reproductive Number and Mean Serial Interval of a Novel Pathogen in a Small, Well-Observed Discrete Population. PLOS ONE. 2016;11: e0148061. doi:10.1371/journal.pone.0148061

75. Madewell ZJ, Yang Y, Longini IM, Halloran ME, Vespignani A, Dean NE. Rapid review and meta-analysis of serial intervals for SARS-CoV-2 Delta and Omicron variants. BMC Infect Dis. 2023;23: 429. doi:10.1186/s12879-023-08407-5

76. Reich NG, Lessler J, Azman A, Kamvar ZN. coarseDataTools: Analysis of Coarsely Observed Data. 2021. Available: https://cran.r-project.org/web/packages/coarseDataTools/index.html

77. Stan Development Team. Prior Choice Recommendations. In: GitHub [Internet]. [cited 16 Apr 2024]. Available: https://github.com/stan-dev/stan/wiki/Prior-Choice-Recommendations

78. Rudolph KE, Lessler J, Moloney RM, Kmush B, Cummings DAT. Incubation Periods of Mosquito-Borne Viral Infections: A Systematic Review. Am J Trop Med Hyg. 2014;90: 882-891. doi:10.4269/ajtmh.13-0403
<caption>A</caption>

Forward cohort/forward distribution

<figure><img src="image_2.png" /><figcaption>B</figcaption></figure>

Forward cohort/forward distribution

<figure><img src="image_3.png" /><figcaption>C</figcaption></figure>

Backward cohort/backward distribution
Growing epidemic

<figure><img src="image_4.png" /><figcaption>D</figcaption></figure>

Backward cohort/backward distribution
Declining epidemic

<figure><img src="image_5.png" /></figure>

Figure 2. Forward and backward approaches for organizing events into cohorts and analyzing data to estimate epidemiological delay distributions. The y-axis represents distinct observations of delays. The yellow circles represent the primary events, while the green squares represent the secondary events. The black horizontal lines represent the delay between primary and secondary events, and the vertical dotted lines show the cohorts. The arrows represent how cases enter the dataset: arrows pointing toward the right indicate that the primary event was observed first, while arrows pointing toward the left indicate that the secondary event was observed first. Note that the case ascertainment method does not impact the direction we can cohort the data. For forward cohorts (A and C), all primary events that occurred during the same period are selected and prospectively followed until the secondary event occurs. For backward cohorts (B), all secondary events that occurred during the same period are selected; the timing of the primary events is identified retrospectively. For simplicity, interval censoring has been omitted from this figure; events appear to occur at discrete intervals when in reality, they occur in continuous time.
79. Park SW, Sun K, Abbott S, Sender R, Bar-on YM, Weitz JS, et al. Inferring the differences in incubation-period and generation-interval distributions of the Delta and Omicron variants of SARS-CoV-2. Proc Natl Acad Sci. 2023;120: e2221887120. doi:10.1073/pnas.2221887120

80. Fields VL, Kracalik IT, Carthel C, Lopez A, Schwartz A, Lewis NM, et al. Coronavirus Disease Contact Tracing Outcomes and Cost, Salt Lake County, Utah, USA, March-May 2020. Emerg Infect Dis. 2021;27: 2999-3008. doi:10.3201/eid2712.210505

81. Wu Zeng, Hadia Samaha, Michel Yao, Steve Ahuka-Mundeke, Thomas Wilkinson, Thibaut Jombart, et al. The cost of public health interventions to respond to the 10th Ebola outbreak in the Democratic Republic of the Congo. BMJ Glob Health. 2023;8: e012660. doi:10.1136/bmjgh-2023-012660

82. Höhle M, An Der Heiden M. Bayesian nowcasting during the STEC O104:H4 outbreak in Germany, 2011. Biometrics. 2014;70: 993-1002. doi:10.1111/biom.12194

83. Van De Kassteele J, Eilers PHC, Wallinga J. Nowcasting the Number of New Symptomatic Cases During Infectious Disease Outbreaks Using Constrained P-spline Smoothing. Epidemiology. 2019;30: 737-745. doi:10.1097/EDE.0000000000001050

84. Bastos LS, Economou T, Gomes MFC, Villela DAM, Coelho FC, Cruz OG, et al. A modelling approach for correcting reporting delays in disease surveillance data. Stat Med. 2019;38: 4363-4377. doi:10.1002/sim.8303

85. Chan M, Johansson MA. The Incubation Periods of Dengue Viruses. Vasilakis N, editor. PLoS ONE. 2012;7: e50972. doi:10.1371/journal.pone.0050972

86. Lessler J, Ott CT, Carcelen AC, Konikoff JM, Williamson J, Bi Q, et al. Times to key events in Zika virus infection and implications for blood donation: a systematic review. Bull World Health Organ. 2016;94: 841-849. doi:10.2471/BLT.16.174540

87. Linton N, Kobayashi T, Yang Y, Hayashi K, Akhmetzhanov A, Jung S, et al. Incubation Period and Other Epidemiological Characteristics of 2019 Novel Coronavirus Infections with Right Truncation: A Statistical Analysis of Publicly Available Case Data. J Clin Med. 2020;9: 538. doi:10.3390/jcm9020538

88. Lauer SA, Grantz KH, Bi Q, Jones FK, Zheng Q, Meredith HR, et al. The Incubation Period of Coronavirus Disease 2019 (COVID-19) From Publicly Reported Confirmed Cases: Estimation and Application. Ann Intern Med. 2020;172: 577-582. doi:10.7326/M20-0504

89. Medley GF, Anderson RM, Cox DR, Billard L. Incubation period of AIDS in patients infected via blood transfusion. Nature. 1987;328: 719-721. doi:10.1038/328719a0

90. Guzzetta G, Mammone A, Ferraro F, Caraglia A, Rapiti A, Marziano V, et al. Early Estimates of Monkeypox Incubation Period, Generation Time, and Reproduction Number, Italy, May-June 2022. Emerg Infect Dis. 2022;28: 2078-2081. doi:10.3201/eid2810.221126
A

Forward cohort/forward distribution

<figure><img src="image_6.png" /></figure>

B

Backward cohort/backward distribution

<figure><img src="image_7.png" /></figure>

C

Forward cohort/forward distribution

<figure><img src="image_8.png" /></figure>
91. Assiri A, McGeer A, Perl TM, Price CS, Al Rabeeah AA, Cummings DAT, et al. Hospital Outbreak of Middle East Respiratory Syndrome Coronavirus. N Engl J Med. 2013;369: 407-416. doi:10.1056/NEJMoa1306742

92. McFarland SE, Marcus U, Hemmers L, Miura F, Iñigo Martínez J, Martínez FM, et al. Estimated incubation period distributions of mpox using cases from two international European festivals and outbreaks in a club in Berlin, May to June 2022. Eurosurveillance. 2023;28. doi:10.2807/1560-7917.ES.2023.28.27.2200806
<figure><img src="image_9.png" /><figcaption>Figure 3. Flowchart for bias adjustment when estimating delay distributions, assuming that double interval censoring is always adjusted for and that the forward distribution is being modeled directly (i.e., not via the backwards distribution and dynamical bias correction) as we recommend regardless of data collection approach. If you have an estimate of the backwards distribution from the literature, see the section on "Other considerations" for advice.</figcaption></figure>
References

1. Home | NSF NEON | Open Data to Understand our Ecosystems. [cited 23 Aug 2024].
Available: https://www.neonscience.org/

2. EarthRanger: Protecting Wildlife With Real-Time Data. [cited 23 Aug 2024]. Available:
https://www.earthranger.com/

3. Paz A, Silva TS, Carnaval AC. A framework for near-real time monitoring of diversity patterns based on indirect remote sensing, with an application in the Brazilian Atlantic rainforest. PeerJ. 2022;10: e13534. doi:10.7717/peerj.13534

4. Panigrahi S, Maski P, Thondiyath A. Real-time biodiversity analysis using deep-learning algorithms on mobile robotic platforms. PeerJ Comput Sci. 2023;9: e1502. doi:10.7717/peerj-cs.1502

5. Venverloo T, Duarte F. Towards real-time monitoring of insect species populations. Sci Rep. 2024;14: 18727. doi:10.1038/s41598-024-68502-8

6. Cornford R, Spooner F, McRae L, Purvis A, Freeman R. Ongoing over-exploitation and delayed responses to environmental change highlight the urgency for action to promote vertebrate recoveries by 2030. Proc R Soc B Biol Sci. 2023;290: 20230464. doi:10.1098/rspb.2023.0464

7. Charniga K, Masters NB, Slayton RB, Gosdin L, Minhaj FS, Philpott D, et al. Estimating the incubation period of monkeypox virus during the 2022 multi-national outbreak. medRxiv. 2022; 2022.06.22.22276713. doi:10.1101/2022.06.22.22276713

8. Lessler J, Reich NG, Brookmeyer R, Perl TM, Nelson KE, Cummings DA. Incubation periods of acute respiratory viral infections: a systematic review. Lancet Infect Dis. 2009;9: 291-300. doi:10.1016/S1473-3099(09)70069-6

9. Overton CE, Abbott S, Christie R, Cumming F, Day J, Jones O, et al. Nowcasting the 2022 mpox outbreak in England. Lau EHY, editor. PLOS Comput Biol. 2023;19: e1011463. doi:10.1371/journal.pcbi.1011463

10. Lavezzo E, Franchin E, Ciavarella C, Cuomo-Dannenburg G, Barzon L, Del Vecchio C, et al. Suppression of a SARS-CoV-2 outbreak in the Italian municipality of Vo'. Nature. 2020;584: 425-429. doi:10.1038/s41586-020-2488-1

11. WHO Ebola Response Team. Ebola Virus Disease in West Africa — The First 9 Months of the Epidemic and Forward Projections. N Engl J Med. 2014;371: 1481–1495. doi:10.1056/NEJMoa1411100

12. Fraser C, Donnelly CA, Cauchemez S, Hanage WP, Van Kerkhove MD, Hollingsworth TD, et al. Pandemic Potential of a Strain of Influenza A (H1N1): Early Findings. Science. 2009;324: 1557–1561. doi:10.1126/science.1176062
13. Ferguson NM, Cucunubá ZM, Dorigatti I, Nedjati-Gilani GL, Donnelly CA, Basáñez M-G, et al. Countering the Zika epidemic in Latin America. Science. 2016;353: 353–354.
doi:10.1126/science.aag0219

14. Ward T, Glaser A, Overton CE, Carpenter B, Gent N, Seale AC. Replacement dynamics and the pathogenesis of the Alpha, Delta and Omicron variants of SARS-CoV-2. Epidemiol Infect. 2023;151: e32. doi:10.1017/S0950268822001935

15. Reynolds MG, Yorita KL, Kuehnert MJ, Davidson WB, Huhn GD, Holman RC, et al. Clinical Manifestations of Human Monkeypox Influenced by Route of Infection. J Infect Dis. 2006;194: 773–780. doi:10.1086/505880

16. Galmiche S, Cortier T, Charmet T, Schaeffer L, Chény O, Von Platen C, et al. SARS-CoV-2 incubation period across variants of concern, individual factors, and circumstances of infection in France: a case series analysis from the ComCor study. Lancet Microbe. 2023;4: e409–e417. doi:10.1016/S2666-5247(23)00005-8

17. Cramer E, Ray E, Lopez V, Bracher J, Brennen A, Rivadeneira A, et al. Evaluation of individual and ensemble probabilistic forecasts of COVID-19 mortality in the United States. PNAS. 2022;119: e2113561119.

18. Borchering RK, Healy JM, Cadwell BL, Johansson MA, Slayton RB, Wallace M, et al. Public health impact of the U.S. Scenario Modeling Hub. Epidemics. 2023;44: 100705.
doi:10.1016/j.epidem.2023.100705

19. Clay PA, Pollock ED, Saldarriaga EM, Pathela P, Macaraig M, Zucker JR, et al. Modeling the impact of prioritizing first or second vaccine doses during the 2022 mpox outbreak. Infectious Diseases (except HIV/AIDS); 2023 Oct. doi:10.1101/2023.10.13.23297005

20. Nishiura H, Linton NM, Akhmetzhanov AR. Serial interval of novel coronavirus (COVID-19) infections. Int J Infect Dis. 2020;93: 284–286. doi:10.1016/j.ijid.2020.02.060

21. Liu Y, Centre for Mathematical Modelling of Infectious Diseases nCoV Working Group, Funk S, Flasche S. The contribution of pre-symptomatic infection to the transmission dynamics of COVID-2019 [version 1; peer review: 3 approved]. Wellcome Open Res. 2020;5: 58. doi:10.12688/wellcomeopenres.15788.1

22. Fraser C, Riley S, Anderson RM, Ferguson NM. Factors that make an infectious disease outbreak controllable. Proc Natl Acad Sci. 2004;101: 6146–6151.
doi:10.1073/pnas.0307506101

23. Park SW, Akhmetzhanov AR, Charniga K, Cori A, Davies NG, Dushoff J, et al. Estimating epidemiological delay distributions for infectious diseases. medRxiv; 2024.
doi:10.1101/2024.01.12.24301247

24. Ward T, Christie R, Paton RS, Cumming F, Overton CE. Transmission dynamics of monkeypox in the United Kingdom: contact tracing study. BMJ. 2022; e073153.
doi:10.1136/bmj-2022-073153
25. Overton CE, Stage HB, Ahmad S, Curran-Sebastian J, Dark P, Das R, et al. Using statistics and mathematical modelling to understand infectious disease outbreaks: COVID-19 as an example. *Infect Dis Model.* 2020;5: 409–441. doi:10.1016/j.idm.2020.06.008

26. Britton T, Scalia Tomba G. Estimation in emerging epidemics: biases and remedies. *J R Soc Interface.* 2019;16: 20180670. doi:10.1098/rsif.2018.0670

27. Seaman SR, Presanis A, Jackson C. Estimating a time-to-event distribution from right-truncated data in an epidemic: A review of methods. *Stat Methods Med Res.* 2022;31: 1641–1655. doi:10.1177/09622802211023955

28. Gostic KM, McGough L, Baskerville EB, Abbott S, Joshi K, Tedijanto C, et al. Practical considerations for measuring the effective reproductive number, Rt. *Pitzer VE*, editor. *PLOS Comput Biol.* 2020;16: e1008409. doi:10.1371/journal.pcbi.1008409

29. Ali ST, Wang L, Lau EHY, Xu X-K, Du Z, Wu Y, et al. Serial interval of SARS-CoV-2 was shortened over time by nonpharmaceutical interventions. *Science.* 2020;369: 1106–1109. doi:10.1126/science.abc9004

30. Lambert JW, Kucharski A. *epiparameter*. 2023. Available: https://github.com/epiverse-trace/epiparameter

31. Nash R, Morgenstern C, Cuomo-Dannenburg G, McCabe R, McCain K. *epireview*. [cited 28 Dec 2023]. Available: https://mrc-ide.github.io/epireview/

32. WorldHealthOrganization/collaboratory-epiparameter-community. World Health Organization; 2023. Available: https://github.com/WorldHealthOrganization/collaboratory-epiparameter-community

33. McAloon C, Collins Á, Hunt K, Barber A, Byrne AW, Butler F, et al. Incubation period of COVID-19: a rapid systematic review and meta-analysis of observational research. *BMJ Open*. 2020;10: e039652. doi:10.1136/bmjopen-2020-039652

34. Cuomo-Dannenburg G, McCain K, McCabe R, Unwin HJT, Doohan P, Nash RK, et al. Marburg virus disease outbreaks, mathematical models, and disease parameters: a systematic review. *Lancet Infect Dis.* 2023; S1473309923005157. doi:10.1016/S1473-3099(23)00515-7

35. Doohan P, Jorgensen D, Naidoo TM, McCain K, Hicks JT, McCabe R, et al. Lassa fever outbreaks, mathematical models, and disease parameters: a systematic review and meta-analysis. *medRxiv*; 2024. p. 2024.03.23.24304596. doi:10.1101/2024.03.23.24304596

36. Madewell ZJ, Charniga K, Masters NB, Asher J, Fahrenwald L, Still W, et al. Serial Interval and Incubation Period Estimates of Monkeypox Virus Infection in 12 Jurisdictions, United States, May–August 2022. *Emerg Infect Dis.* 2023;29: 818–821. doi:10.3201/eid2904.221622

37. Humphrey JH, Marinda E, Mutasa K, Moulton LH, Iliff PJ, Ntozini R, et al. Mother to child transmission of HIV among Zimbabwean women who seroconverted postnatally: prospective cohort study. *BMJ*. 2010;341: c6580–c6580. doi:10.1136/bmj.c6580
38. Antonucci G, Girardi E, Raviglione M, Ippolito G, Almi P, Angarano G, et al. Risk Factors for Tuberculosis in HIV-Infected Persons: A Prospective Cohort Study. JAMA. 1995;274: 143-148. doi:10.1001/jama.1995.03530020061033

39. Munywoki PK, Koech DC, Agoti CN, Lewa C, Cane PA, Medley GF, et al. The Source of Respiratory Syncytial Virus Infection In Infants: A Household Cohort Study In Rural Kenya. J Infect Dis. 2014;209: 1685-1692. doi:10.1093/infdis/jit828

40. Levy JW, Cowling BJ, Simmerman JM, Olsen SJ, Fang VJ, Suntarattiwong P, et al. The Serial Intervals of Seasonal and Pandemic Influenza Viruses in Households in Bangkok, Thailand. Am J Epidemiol. 2013;177: 1443-1451. doi:10.1093/aje/kws402

41. Russell TW, Hellewell J, Jarvis CI, Van Zandvoort K, Abbott S, Ratnayake R, et al. Estimating the infection and case fatality ratio for coronavirus disease (COVID-19) using age-adjusted data from the outbreak on the Diamond Princess cruise ship, February 2020. Eurosurveillance. 2020;25. doi:10.2807/1560-7917.ES.2020.25.12.2000256

42. Centers for Disease Control and Prevention. Introduction to Public Health. In: Public Health 101 Series. 2014 [cited 9 Jan 2024]. Available: https://www.cdc.gov/training/publichealth101/surveillance.html

43. Sharpe JD, Charniga K, Byrd KM, Stefanos R, Lewis L, Watson J, et al. Possible Exposures Among Mpox Patients Without Reported Male-to-Male Sexual Contact — Six U.S. Jurisdictions, November 1–December 14, 2022. MMWR Morb Mortal Wkly Rep. 2023;72: 944–948. doi:10.15585/mmwr.mm7235a2

44. Reich NG, Lessler J, Cummings DAT, Brookmeyer R. Estimating incubation period distributions with coarse data. Stat Med. 2009;28: 2769-2784. doi:10.1002/sim.3659

45. Dey T, Lipsitz SR, Cooper Z, Trinh Q-D, Krzywinski M, Altman N. Survival analysis—time-to-event data and censoring. Nat Methods. 2022;19: 906–908. doi:10.1038/s41592-022-01563-7

46. Park SW, Sun K, Champredon D, Li M, Bolker BM, Earn DJD, et al. Forward-looking serial intervals correctly link epidemic growth to reproduction numbers. Proc Natl Acad Sci. 2021;118: e2011548118. doi:10.1073/pnas.2011548118

47. Verity R, Okell LC, Dorigatti I, Winskill P, Whittaker C, Imai N, et al. Estimates of the severity of coronavirus disease 2019: a model-based analysis. Lancet Infect Dis. 2020;20: 669-677. doi:10.1016/S1473-3099(20)30243-7

48. Reich NG, Lessler J, Azman A, Kamvar ZN. coarseDataTools: Analysis of Coarsely Observed Data. 2021. Available: https://cran.r-project.org/web/packages/coarseDataTools/index.html

49. Park SW, Abbott S. Estimating delay distributions accounting for common biases. 2023. Available: https://github.com/parksw3/epidist-paper

50. Leung K-M, Elashoff RM, Afifi AA. CENSORING ISSUES IN SURVIVAL ANALYSIS. Annu Rev Public Health. 1997;18: 83-104. doi:10.1146/annurev.publhealth.18.1.83
51. Nolen LD, Osadebe L, Katomba J, Likofata J, Mukadi D, Monroe B, et al. Extended Human-to-Human Transmission during a Monkeypox Outbreak in the Democratic Republic of the Congo. Emerg Infect Dis. 2016;22: 1014–1021. doi:10.3201/eid2206.150579

52. Mertens PE, Patton R, Baum JJ, Monath TP. Clinical presentation of lassa fever cases during the hospital epidemic at Zorzor, Liberia, March-April 1972. Am J Trop Med Hyg. 1973;22: 780–784. doi:10.4269/ajtmh.1973.22.780

53. Martini G. Marburg virus disease. Postgrad Med J. 1973;49: 542–546.

54. WHO International Study Team. Ebola haemorrhagic fever in Sudan, 1976. Bull World Health Organ. 1978;56: 247–270.

55. Nishiura H. Early efforts in modeling the incubation period of infectious diseases with an acute course of illness. Emerg Themes Epidemiol. 2007;4: 2. doi:10.1186/1742-7622-4-2

56. Miura F, Van Ewijk CE, Backer JA, Xiridou M, Franz E, Op De Coul E, et al. Estimated incubation period for monkeypox cases confirmed in the Netherlands, May 2022. Eurosurveillance. 2022;27. doi:10.2807/1560-7917.ES.2022.27.24.2200448

57. Ferretti L, Ledda A, Wymant C, Zhao L, Ledda V, Abeler-Dörner L, et al. The timing of COVID-19 transmission. Epidemiology; 2020 Sep. doi:10.1101/2020.09.04.20188516

58. McAloon CG, Wall P, Griffin J, Casey M, Barber A, Codd M, et al. Estimation of the serial interval and proportion of pre-symptomatic transmission events of COVID-19 in Ireland using contact tracing data. BMC Public Health. 2021;21: 805. doi:10.1186/s12889-021-10868-9

59. Akhmetzhanov AR, Cheng H-Y, Dushoff J. Incubation-period estimates of Omicron (BA.1) variant from Taiwan, December 2021–January 2022, and its comparison to other SARS-CoV-2 variants: a statistical modeling, systematic search and meta-analysis. Infectious Diseases (except HIV/AIDS); 2023 Jul. doi:10.1101/2023.07.20.23292983

60. Lover AA, Zhao X, Gao Z, Coker RJ, Cook AR. The distribution of incubation and relapse times in experimental human infections with the malaria parasite Plasmodium vivax. BMC Infect Dis. 2014;14: 539. doi:10.1186/1471-2334-14-539

61. Klinkenberg D, Nishiura H. The correlation between infectivity and incubation period of measles, estimated from households with two cases. J Theor Biol. 2011;284: 52–60. doi:10.1016/j.jtbi.2011.06.015

62. Hoffmann T, Alsing J. Faecal shedding models for SARS-CoV-2 RNA among hospitalised patients and implications for wastewater-based epidemiology. J R Stat Soc Ser C Appl Stat. 2023;72: 330–345. doi:10.1093/jrsssc/qlad011

63. Abbott S, Lison A, Funk S, Choi-Hannah, Gruson H, Pearson CAB, et al. epinowcast/epinowcast: Epinowcast 0.2.2. Zenodo; 2023. doi:10.5281/ZENODO.5637165

64. Sender R, Bar-On Y, Park SW, Noor E, Dushoff J, Milo R. The unmitigated profile of COVID-19 infectiousness. eLife. 2022;11: e79134. doi:10.7554/eLife.79134
65. Gamma distribution. Wikipedia. 2024. Available:
https://en.wikipedia.org/w/index.php?title=Gamma_distribution&oldid=1210997995

66. Log-normal distribution. Wikipedia. 2024. Available:
https://en.wikipedia.org/w/index.php?title=Log-normal_distribution&oldid=1210518171

67. Abbott S, Hellewell J, Sherratt K, Gostic K, Hickson J, Badr HS, et al. EpiNow2: Estimate Real-Time Case Counts and Time-Varying Epidemiological Parameters. 2023. Available:
https://cran.r-project.org/web/packages/EpiNow2/index.html

68. Jombart T, Cori A, Kamvar ZN, Schumacher D, Finger F, Whittaker C. epitrix: Small Helpers and Tricks for Epidemics Analysis. 2023. Available: https://cran.r-project.org/web/packages/epitrix/index.html

69. Yu Y. mixR: Finite Mixture Modeling for Raw and Binned Data. 2021. Available:
https://cran.r-project.org/web/packages/mixR/index.html

70. Van Kerkhove MD, Bento AI, Mills HL, Ferguson NM, Donnelly CA. A review of epidemiological parameters from Ebola outbreaks to inform early public health decision-making. Sci Data. 2015;2: 150019. doi:10.1038/sdata.2015.19

71. Medley GF, Anderson RM, Cox DR, Billard L. Incubation period of AIDS in patients infected via blood transfusion. Nature. 1987;328: 719-721. doi:10.1038/328719a0

72. Lison A, Abbott S, Huisman J, Stadler T. Generative Bayesian modeling to nowcast the effective reproduction number from line list data with missing symptom onset dates. 2023 [cited 21 Feb 2024]. doi:10.48550/ARXIV.2308.13262

73. Hamra G, MacLehose R, Richardson D. Markov Chain Monte Carlo: an introduction for epidemiologists. Int J Epidemiol. 2013;42: 627-634. doi:10.1093/ije/dyt043

74. Lambert B. A student's guide to Bayesian statistics. Los Angeles London New Delhi Singapore Washington DC Melbourne: SAGE; 2018.

75. RStan: the R interface to Stan. [cited 10 Apr 2024]. Available: https://mc-stan.org/rstan/articles/rstan.html

76. Stan Development Team. Stan Modeling Language Users Guide and Reference Manual Version 2.34. 2022. Available: https://mc-stan.org

77. Elderd BD, Dukic VM, Dwyer G. Uncertainty in predictions of disease spread and public health responses to bioterrorism and emerging diseases. Proc Natl Acad Sci. 2006;103: 15693-15697. doi:10.1073/pnas.0600816103

78. Pollett S, Johansson MA, Reich NG, Brett-Major D, Del Valle SY, Venkatramanan S, et al. Recommended reporting items for epidemic forecasting and prediction research: The EPIFORGE 2020 guidelines. PLOS Med. 2021;18: e1003793. doi:10.1371/journal.pmed.1003793
79. Cowling BJ, Park M, Fang VJ, Wu P, Leung GM, Wu JT. Preliminary epidemiological assessment of MERS-CoV outbreak in South Korea, May to June 2015. Eurosurveillance. 2015;20. doi:10.2807/1560-7917.ES2015.20.25.21163

80. Cheng H-Y, Akhmetzhanov AR, Dushoff J. SARS-CoV-2 Incubation Period during Omicron BA.5-Dominant Period, Japan. Emerg Infect Dis. 2024;30. doi:10.3201/eid3001.230208

81. Miura F, Backer JA, Van Rijckevorsel G, Bavalia R, Raven S, Petrignani M, et al. Time Scales of Human Mpox Transmission in The Netherlands. J Infect Dis. 2023; jiad091. doi:10.1093/infdis/jiad091

82. Campbell F, Didelot X, Fitzjohn R, Ferguson N, Cori A, Jombart T. outbreaker2: a modular platform for outbreak reconstruction. BMC Bioinformatics. 2018;19: 363. doi:10.1186/s12859-018-2330-z

83. Hall M, Woolhouse M, Rambaut A. Epidemic Reconstruction in a Phylogenetics Framework: Transmission Trees as Partitions of the Node Set. PLOS Comput Biol. 2015;11: e1004613. doi:10.1371/journal.pcbi.1004613

84. Forsberg White L, Pagano M. A likelihood-based method for real-time estimation of the serial interval and reproductive number of an epidemic. Stat Med. 2008;27: 2999-3016. doi:10.1002/sim.3136

85. Wu KM, Riley S. Estimation of the Basic Reproductive Number and Mean Serial Interval of a Novel Pathogen in a Small, Well-Observed Discrete Population. PLOS ONE. 2016;11: e0148061. doi:10.1371/journal.pone.0148061

86. Madewell ZJ, Yang Y, Longini IM, Halloran ME, Vespignani A, Dean NE. Rapid review and meta-analysis of serial intervals for SARS-CoV-2 Delta and Omicron variants. BMC Infect Dis. 2023;23: 429. doi:10.1186/s12879-023-08407-5

87. Stan Development Team. Prior Choice Recommendations. In: GitHub [Internet]. [cited 16 Apr 2024]. Available: https://github.com/stan-dev/stan/wiki/Prior-Choice-Recommendations

88. Rudolph KE, Lessler J, Moloney RM, Kmush B, Cummings DAT. Incubation Periods of Mosquito-Borne Viral Infections: A Systematic Review. Am J Trop Med Hyg. 2014;90: 882-891. doi:10.4269/ajtmh.13-0403

89. Park SW, Sun K, Abbott S, Sender R, Bar-on YM, Weitz JS, et al. Inferring the differences in incubation-period and generation-interval distributions of the Delta and Omicron variants of SARS-CoV-2. Proc Natl Acad Sci. 2023;120: e2221887120. doi:10.1073/pnas.2221887120

90. Fields VL, Kracalik IT, Carthel C, Lopez A, Schwartz A, Lewis NM, et al. Coronavirus Disease Contact Tracing Outcomes and Cost, Salt Lake County, Utah, USA, March-May 2020. Emerg Infect Dis. 2021;27: 2999-3008. doi:10.3201/eid2712.210505

91. Wu Zeng, Hadia Samaha, Michel Yao, Steve Ahuka-Mundeke, Thomas Wilkinson, Thibaut Jombart, et al. The cost of public health interventions to respond to the 10th Ebola outbreak in the Democratic Republic of the Congo. BMJ Glob Health. 2023;8: e012660. doi:10.1136/bmjgh-2023-012660
92. Höhle M, An Der Heiden M. Bayesian nowcasting during the STEC O104:H4 outbreak in Germany, 2011. Biometrics. 2014;70: 993–1002. doi:10.1111/biom.12194

93. Van De Kassteele J, Eilers PHC, Wallinga J. Nowcasting the Number of New Symptomatic Cases During Infectious Disease Outbreaks Using Constrained P-spline Smoothing. Epidemiology. 2019;30: 737–745. doi:10.1097/EDE.00000000000001050

94. Bastos LS, Economou T, Gomes MFC, Villela DAM, Coelho FC, Cruz OG, et al. A modelling approach for correcting reporting delays in disease surveillance data. Stat Med. 2019;38: 4363–4377. doi:10.1002/sim.8303

95. Chan M, Johansson MA. The Incubation Periods of Dengue Viruses. Vasilakis N, editor. PLoS ONE. 2012;7: e50972. doi:10.1371/journal.pone.0050972

96. Lessler J, Ott CT, Carcelen AC, Konikoff JM, Williamson J, Bi Q, et al. Times to key events in Zika virus infection and implications for blood donation: a systematic review. Bull World Health Organ. 2016;94: 841–849. doi:10.2471/BLT.16.174540

97. Linton N, Kobayashi T, Yang Y, Hayashi K, Akhmetzhanov A, Jung S, et al. Incubation Period and Other Epidemiological Characteristics of 2019 Novel Coronavirus Infections with Right Truncation: A Statistical Analysis of Publicly Available Case Data. J Clin Med. 2020;9: 538. doi:10.3390/jcm9020538

98. Lauer SA, Grantz KH, Bi Q, Jones FK, Zheng Q, Meredith HR, et al. The Incubation Period of Coronavirus Disease 2019 (COVID-19) From Publicly Reported Confirmed Cases: Estimation and Application. Ann Intern Med. 2020;172: 577–582. doi:10.7326/M20-0504

99. Plummer M, Best N, Cowles K, Vines K. CODA: Convergence Diagnosis and Output Analysis for MCMC. R News. 6: 7–11.

100. Gabry J, Mahr T. bayesplot: Plotting for Bayesian Models. Available: bayesplot: Plotting for Bayesian Models

101. Guzzetta G, Mammone A, Ferraro F, Caraglia A, Rapiti A, Marziano V, et al. Early Estimates of Monkeypox Incubation Period, Generation Time, and Reproduction Number, Italy, May–June 2022. Emerg Infect Dis. 2022;28: 2078–2081. doi:10.3201/eid2810.221126

102. Assiri A, McGeer A, Perl TM, Price CS, Al Rabeeah AA, Cummings DAT, et al. Hospital Outbreak of Middle East Respiratory Syndrome Coronavirus. N Engl J Med. 2013;369: 407–416. doi:10.1056/NEJMoa1306742

103. McFarland SE, Marcus U, Hemmers L, Miura F, Iñigo Martínez J, Martínez FM, et al. Estimated incubation period distributions of mpox using cases from two international European festivals and outbreaks in a club in Berlin, May to June 2022. Eurosurveillance. 2023;28. doi:10.2807/1560-7917.ES.2023.28.27.2200806