key: cord-1004051-raav221g
authors: Deng, Yuhao; You, Chong; Liu, Yukun; Qin, Jing; Zhou, Xiao‐Hua
title: Estimation of incubation period and generation time based on observed length‐biased epidemic cohort with censoring for COVID‐19 outbreak in China
date: 2020-07-06
journal: Biometrics
DOI: 10.1111/biom.13325
sha: ef15dd59ac037de73a5f3b7f046c27ec92b53949
doc_id: 1004051
cord_uid: raav221g

The incubation period and generation time are key characteristics in the analysis of infectious diseases. The commonly used contact‐tracing based estimation of incubation distribution is highly influenced by the individuals' judgment on the possible date of exposure, and might lead to significant errors. On the other hand, interval censoring based methods are able to utilize a much larger set of traveling data but may encounter biased sampling problems. The distribution of generation time is usually approximated by observed serial intervals. However, it may result in a biased estimation of generation time, especially when the disease is infectious during incubation. In this paper, the theory from renewal process is partially adopted by considering the incubation period as the inter‐arrival time, and the duration between departure from Wuhan and onset of symptoms as the mixture of forward time and inter‐arrival time with censored intervals. In addition, a consistent estimator for the distribution of generation time based on incubation period and serial interval is proposed for incubation‐infectious diseases. A real case application to the current outbreak of COVID‐19 is implemented. We find that the incubation period has a median of 8.50 days (95% CI: 7.22, 9.15). The basic reproduction number in the early phase of COVID‐19 outbreak based on the proposed generation time estimation is estimated to be 2.96 (95% CI: 2.15, 3.86). This article is protected by copyright. All rights reserved

In epidemiology, incubation period is the time between the infection of an individual by a pathogen and the manifestation of symptoms, while generation time is defined as the time between the infection of a primary case and its secondary cases (Fine, 2003; Svensson, 2007) .

Both are vital clinical characteristics that depict an epidemic and are essential for policy making. For example, a good understanding of incubation period offers an optimal length of quarantine, and a good understanding of generation time is essential in estimating the transmission potential of a disease measured by the basic reproduction number R 0 (Farewell et al., 2005; Wallinga and Lipsitch, 2007; Nishiura, 2010) .

In most of literature, such as Li et al. (2020) and Guan et al. (2020) , the distribution of incubation period is either described through a parametric model, for example, log-normal and Weibull, or, its empirical distribution based on the observed incubation period from contact-tracing data. However, contact-tracing data is usually difficult to obtain, and can be highly influenced by the individuals' judgment on the possible date of exposure rather than the actual date of exposure which, in turn, might not be accurately monitored and determined leading to significant errors (Cowling et al., 2007) .

An alternative approach to study incubation period is to take advantage of the mechanism of truncation or censoring. Lui et al. (1988) ; Struthers and Farewell (1989) ; De Gruttola and Lagakos (1989) ; Kuo et al. (1991) estimated incubation distribution of contagious diseases using external truncation or censoring information. Kuk and Ma (2005) studied the incubation period of SARS by deconvolution, but the proposed method was only feasible for the disease which is non-infectious during incubation period, which is not the case for . It also assumed that the ability of infectiousness is uniform during the infectious period, which is a strong assumption. In the studies of Lessler et al. (2009); Reich et al. (2009), double censoring was used to characterize the problem caused by daily reports rather than Biometrics, 000 0000 continuous observed symptoms onset time. Nishiura and Inaba (2011) used 72 confirmed imported cases who traveled to Japan from Hawaii during the early phase of the 2009 H1N1 pandemic to estimate the incubation by addressing censoring and infection-age.

For COVID-19, Backer et al. (2020) and Linton et al. (2020) used confirmed cases detected outside Wuhan to estimate the distribution of the incubation by interval-censoring likelihood.

In their studies, for each selected case, a censored interval for incubation period was obtained by travel histories and dates of symptoms onset, and the distribution of incubation was then estimated by fitting censored intervals into Weibull, Gamma and log-normal. However, such estimations may lead to biased estimations of incubation period due to the biased sampling issues. Qin et al. (2020) adopted the theory from renewal process and carefully selected the studying cohort to overcome the biased sampling problems but fitted a continuous parametric model with discrete observations, while the discreteness of data is in fact a sort of interval censoring caused by daily reports.

To the best of our knowledge, generation time is usually directly estimated by the time difference between symptoms onset of successive cases in a chain of transmission rather than the actual time of infection, that is, the serial interval. This is because it is challenging to obtain both the corresponding infection dates of the primary case and its secondary cases in a chain of transmission, while the dates of symptoms onsets are relatively easier to obtain.

However, the distribution of serial interval may be biased for estimating generation time, especially when the disease is infectious during incubation, in which the variance could be over-estimated (Britton and Scalia Tomba, 2019). As a result, the subsequent quantities estimated based on the generation time is biased. For example, the basic reproduction number, indicating the spreading ability of an infectious disease, would be under-estimated.

Note that COVID-19 is incubation-infectious, hence the estimation of generation time simply based on observed serial intervals is not consistent.

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To overcome the issues aforementioned, in this paper we estimate the distribution of incubation period using the well-studied renewal process where there exists a censoring event within the incubation period. ; Vardi ( , 1989 ) discussed nonparametric maximum likelihood estimation based on length-biased sampling and renewal process with incomplete renewal data, and further the multiplicative censoring problem. A brief review can be found in Qin (2017) . Issues related to the length-biased sampling and interval censoring sampling are both taken into consideration in the estimation of incubation distribution in this study. We have shown that under mild assumptions, parameters in the incubation distribution are identifiable and enjoy desirable asymptotic properties. Furthermore, a consistent estimator for the distribution of generation time is also proposed based on incubation period and observed serial interval for incubation-infectious and incubation-noninfectious diseases respectively. Our approaches increase available sample size and utilize censored information in the early phase of an epidemic outbreak.

The rest of this paper is organized as follows. Section 2 describes the motivation data.

In Section 3, we propose algorithms to estimate the distribution of incubation period and

show that under mild assumptions the model parameters are identifiable and enjoy desirable asymptotic properties. In Section 4, we propose algorithms to estimate the distribution of generation time. Simulation studies are performed in Section 5, and the analyzed results to the current outbreak of COVID-19 in China are shown in Section 6. Further discussion is given in Section 7.

The COVID-19 outbreak in Wuhan, China has attracted world-wide attention Huang et al., 2020; Tu et al., 2020) . Publicly available data were collected from provincial and municipal health commissions in China and ministry of health in other countries and areas. The following details were collected on each confirmed case: case ID, In the collected data, 645 chains of transmission were found in the collected data, and n = 198 of them have their dates of symptoms onset available which can be used to calculate serial intervals . These 198 observed serial intervals, {s j , j = 1, . . . , n}, range from −13 to 21 days, with a mean of 4.6 days and quartiles of 1, 4 and 7 days. The same subset of the data used in Qin et al. (2020) is considered in this study for the estimation of the incubation period. This subset includes the confirmed cases who left Wuhan between January 19 and 23, 2020, and excludes cases who developed symptoms before leaving Wuhan.

There is a total of m = 1, 211 cases which meet such criteria in the collected data. These 1,211 observed durations between departure from Wuhan and symptoms onset outside Hubei Province, {t j , j = 1, . . . , m}, ranges from 0 to 22 days with a mean of 5.4 days and quartiles of 2, 5 and 8 days. It is worth noting that Bi et al. (2020) reported that 191 travelers developed symptoms 4.9 days on average after arriving in Shenzhen (Guangdong Province, China).

It is arguable that people who left Wuhan might have higher chance to be infected on the day of departure since it is easier to be exposed to the human-to-human transmitted virus in a crowded environment. Hence in our dataset, there might be two types of individuals: (1) those who got infected during their stay in Wuhan and developed symptoms outside Hubei Province, and (2) those who got infected at the time of leaving Wuhan, for example, at the airport, railway station or on the way from Wuhan to their destinations. Thus, the observed durations between departure from Wuhan and symptoms onset are from a mixture of two distributions: the time between departure from Wuhan and symptoms onset (forward time)

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Estimation of Incubation Period and Generation Time 5 and the complete incubation period. Note that the selected cohort is length-biased since the ones with shorter incubation periods who got infected were less likely to be captured as they had higher chance to develop symptoms before leaving Wuhan. The length-biased issue cannot be tested easily in the data but is naturally arised from the data collection process, since only those who developed symptoms after departure from Wuhan can be collected.

In this section, the distribution of incubation is estimated through theory of renewal process and interval censoring with a mixture distribution. Here we have to assume that the distribution of incubation period is the same between the Wuhan residents who had a schedule to leave Wuhan and the general population. Furthermore, given an individual who got infected in Wuhan and developed symptoms outside Wuhan, it is reasonable to assume that the event of departing from Wuhan is independent of the event of infection and manifestation of symptoms. Hence, we can consider the incubation period as a continuous random variable, I, as the sum of forward and backward times, and the duration between departure from

Wuhan and onset of symptoms as the forward time V in renewal process (see Figure 1 as an illustration). Suppose that I and V are continuous and let f I (·) be the probability density function (pdf) of incubation period, and h(·) be the probability density function of forward time. According to Qin (2017) and Qin et al. (2020) , we have

where S(·) is the survival function and E(I) is the expectation of I.

Note that I is not observable in our dataset but V is observable with observations of {t j }, j = 1, . . . , m. From Equation (1), we can see that the forward time V should have a monotonically decreasing density. However, the observed density of {t j } does not seem to be monotone (see Figure 3 ). A possible explanation towards it would be that {t j } are not observations of V only but mixture of V and I. As aforementioned, due to the nature of a human to human infectious disease, it is easier to get infected at the airport/train station or on the flight/train/bus, namely, the infection occurs at the departure. In such case, the duration between departure from Wuhan and onset of symptoms is no longer the forward time, but the complete incubation period. Taking such possibility into account, let π be the (unknown) probability of getting infected at the departure time from Wuhan, and 1−π be the probability of getting infected before departure. Therefore, the duration between departure from Wuhan and symptoms onset follows a mixture distribution with density

where θ is the model parameter in f I (·) and h(·).

[ Figure 1 about here.]

Accounting for the error caused by daily reports, we can simply let t + j = t j + 0.5 and t − j = t j − 0.5. The estimates of θ and π can be estimated by directly maximizing the likelihood function with interval censoring, that is,

where F I and H are the cumulative distribution functions (cdf) of I and V respectively. We denote the maximum likelihood estimate (MLE) of (θ , π) by ( θ , π) = arg sup θ,π (θ, π),

where (θ, π) = log L(θ, π; t 1 , . . . , t m ). In the Web Appendix B, we will provide an alternative interpretation for the likelihood function.

In general, it is difficult to derive asymptotic properties of the estimator for interval censoring cases (see Lehmann and Romano, 2006; Gentleman and Geyer, 1994) . However, the asymptotic properties can be proved under our particular setting, in which we have identical interval lengths for all observations, namely t + j − t − j = 1 for j = 1, . . . , m. Let (t − j , t + j ) for j = 1, . . . , m be independently and identically distributed (iid) observations from the mixture model (2). Define a pseudo-pdf for the mixed model (2) as

Define two likelihood ratio functions

Let (θ 0 , π 0 ) be the true parameter value. For notational simplicity, let g(t; ϕ) denote the density in (4) with ϕ = (θ , π) , that is, g(t; ϕ) = Q p (t; θ, π). In addition, let q θ denote the dimension of θ, ∇ ϕ = ∂/∂ϕ and ∇ ϕϕ = ∂ 2 /(∂ϕ∂ϕ ). The upcoming expectations are taken with respect to the true density g(t; ϕ 0 ) where ϕ 0 = (θ 0 , π 0 ) . To establish the asymptotic result, we make the following regularity condition.

The non-singularity of U in Condition 1(b) excludes the cases where at least one of θ and π is not identifiable. Theorem 1 below shows the asymptotic properties of the estimator ( θ , π)

if the true parameter value is an interior point in the parameter space while Theorem 2

shows the case if π 0 is at the boundary.

Theorem 1: Suppose that g(t; ϕ) and ϕ 0 satisfy Condition 1, and that (θ 0 , π 0 ) is an

Theorem 2: Suppose that g(t; ϕ) and ϕ 0 satisfy Condition 1, and that θ 0 is an interior point in the parameter space of θ and π 0 = 1. As m → ∞,

The proof of Theorem 1 and 2 are given in Web Appendix C. We can easily verify that the interval censored mixture distribution (4) for Gamma, Weibull (except when shape parameter of Gamma or Weibull is 1, that is, the exponential distribution) or log-normal distribution satisfies Condition 1 and thus the above two theorems hold for our estimates.

In this section, we study the estimation of generation time based on serial interval and incubation time under proper assumptions. The estimation of generation time only subjects to symptomatic population. Suppose an infector got infected at calendar time T 0 and showed symptoms at T 1 . This infector infected an infetee at calendar time T 2 , and the infectee showed symptoms at T 3 . Let G = T 2 − T 0 denote the generation time, S = T 3 − T 1 denote the serial interval, I 1 = T 1 − T 0 and I 2 = T 3 − T 2 be the incubation period of infector and infectee respectively. It is straightforward to see that G = S + I 1 − I 2 .

If a disease is non-infectious during the incubation period (for example, SARS; Lipsitch et al., 2003) , then we can naturally assume I 1 ⊥ ⊥ S and I 2 ⊥ ⊥ G. Then it follows that

where f G and f S are the pdfs of G and S respectively, and the generation time can be estimated by serial interval without inducing bias. However, such case does not apply for COVID-19 as there were reported asymptomatic infections (Rothe et al., 2020) . Instead, we assume I 1 ⊥ ⊥ G, I 2 ⊥ ⊥ G, I 1 ⊥ ⊥ I 2 . The first part of states that the incubation period of the primary case is independent of its generation time. This is true if the disease is infectious during incubation period, and in addition, the ability to pass the pathogens to susceptible host is independent of whether the symptoms are being developed. The rest is straightforward due to the standard assumption of independence between individuals. In addition, we assume the distributions of incubation period, generation time and serial interval are homogeneous among all individuals. Furthermore, to ensure the observed serial intervals could reflect the serial interval of general population, we assume that the missingness (failure of establishing contact tracing) was independent of the length of serial interval. Hence, we obtain that

where the symbol * represents convolution, f G . f S , f I and f −I are the pdfs of G, S, I and −I respectively. Thus, f G is identifiable through characteristic function (or Fourier 

where i = √ −1, φ I (t) can be approximated through the estimated distribution of I introduced in previous section, and φ S (t) can be estimated by the observed serial intervals, {s 1 , . . . , s n }, along with a proper kernel K(·), that is,

where h n is the bandwidth. Note that G must be positive, so to account for the boundary bias, Karunamuni (2009) proposed to use boundary kernel K c (t; y) = a 0 (y)

at the point y > 0. Denote the Fourier transformation φ Kc (t) = ∞ −∞ e itu K c (u)du. Hence, a

consistent estimator for f G is defined as

where M n → ∞, h n → 0 as n → ∞, and Re is the operator taking the real part of a complex value. This estimator is consistent at any interior point in the support of G, provided that the model for incubation period I is correctly specified (Liu and Taylor, 1989) . It is equivalent to specifying a kernel density or a kernel chf, and possible choices are the Vallée Poussin (Fejér) kernels or Cesàro kernels (Devroye, 1989; Anastassiou, 2000) . Note that the generation time must be positive. To correct the bias for devonvolution at the boundary G = 0, a second order correction to remove the boundary effect was proposed by Karunamuni (2000) and

Karunamuni (2009). The density function f G can also be obtained by imposing a parametric model on generation time and fit the density for serial interval, which relies heavily on model specification. More details about the conditions and properties of deconvolution is shown in Web Appendix D.

In this numerical study, we assess the performances of our proposed method and the following methods in estimation of incubation period:

1. The renewal process based mixture model in Qin et al. (2020) which is denoted as Qin's Method hereafter, note that the original method in Qin et al. (2020) is not suitable to be applied in our simulation as the mixture proportion π was prefixed, hence we alter their method by estimating π simultaneously, as a result the Qin's method here is actually an improved version of the method in Qin et al. (2020) and π over 0 and 0.2. Each setting is repeated for 1,000 times. Table 1 summarizes the estimates of parameters in incubation distribution using the Qin's method, Interval Censoring method and our proposed method. We can see that when π = 0, our proposed method and Qin's method provide similar results. For π = 0.2, our approach has smaller bias in Weibull setting. Due to the fact that the log-likelihood is too flat near the maximum, the estimates may be a little biased in finite sample. With larger sample size, the bias is getting smaller. The IC method does not perform well in our simulation as it does not take the length-biased sampling issue and the cross infection probability π into consideration.

[ For generation time estimation, we assume generation time and incubation period both follow Gamma distributions. The mean and variance of these two periods are listed in Figure   2 . We generate 200 serial intervals. Note that it is possible that some serial intervals are negative. We choose the kernel chf φ K (t) = (1 − t 2 ) 3 + , and according to Karunamuni (2009),

The results are displayed in Figure 2 , note that the figure appears in color in the electronic version of this article, and any mention of color refers to that version. The cyan line is the fitted Gamma density using observed positive serial interval data. The red line is the estimated generation time density by deconvolution. We can see that the estimated density of generation time by deconvolution is more close to the true density than fitting the serial intervals, although the deconvolution estimate may be negative in some area.

[ Figure 2 about here.]

In this section we analyze the real data of COVID-19 outbreak, originated from Wuhan, China. As described in Section 2, the times between departure from Wuhan and symptoms onset were collected for the 1,211 cases who got infected in Wuhan and developed symptoms outside Hubei Province, see Figure 3 for the histogram of the collected observations. Table 2 summarizes the estimates of model parameters as defined in Section 3 and quantiles in the incubation distribution with their 95% confidence intervals (CI) by nonparameteric bootstrap. The last two columns list the log-likelihood (loglik) and goodness-of-fit χ 2 statistic (GoF) of each parametric distribution of the incubation period, with higher loglik and lower GoF means a better fit of the model. The number in the bracket of GoF is the p-value of goodness-of-fit test, and all these three models have a good fit. More details about the goodness of fit test is in Web Appendix E. Likelihood ratio test about π can be conducted based on the mixture distribution of half 0 and half chi-squared distribution with 1 degree of freedom to infer the magnitude of π (Self and Liang, 1987; Susko, 2013) . At significant level 0.05, the critical value is 2.71. Although the point estimate of π is zero, the log-likelihood is flat in the region π ∈ [0, 0.2] which results in a situation where a null hypothesis such as H 0 : π > 0.1 or H 0 : π < 0.1 cannot be reject at significant level 0.05, since 2[max θ (θ, 0) − max θ (θ, 0.1)] < 2.71 (illustrated in Figure 3 ). Our model estimated that about 1% of patients have incubation periods longer than 21 days. This might influence the length of quarantine period in regions with a severe epidemic.

[ [ Figure 3 about here.] Figure 3 plots the twice of log likelihood ratio, 2[max θ,π (θ, π) − max θ (θ, π)], versus π.

The dashed line is at 2.71, the 90% quantile of chi-squared distribution with 1 degree of freedom. In fact, the horizontal ordinate of the crossover point is the 95% upper bound of π by likelihood ratio, since 0.5 + 0.5χ 2 (2.71, 1) = 0.95 (mixed chi-squared distribution), where χ 2 (·, 1) is the cdf of chi-squared distribution with 1 degree of freedom.

From the last two columns in Table 2 we can see that Gamma distribution slightly outperforms among three distributions, having the smallest goodness-of-fit statistic. The corresponding incubation period has an estimated mean of 9.10 days and median of 8.50

days, and possess a heavy tail. About 10% infected individuals would develop symptoms after 14.57 days and 1% after 21.17 days. Although the confidence interval of π is relatively wide, variation of the results on the quantiles of incubation period is not significant as shown in Table 2 . Figure 3 visualizes the estimate on the histogram of the time between leaving Wuhan and symptoms onset.

For the estimation of the distribution of generation time, we choose the kernel chf φ K (t) =

(1 − t 2 ) 3 + in (9) with bandwidth h = 2. The estimated probability density of generation time based on the estimated Gamma incubation period is displayed in Figure 4 . We can see that the distributions of generation time has much smaller variance than the serial interval.

[ 

The point estimate of the basic reproduction number is 2.96 with 95% confidence interval [2.15; 3.86] . Note that the estimate of R 0 is 2.18 by using serial interval data instead of generation time, which severely underestimates the infectiousness ability of the disease.

In this paper, we proposed an estimation for incubation distribution which only requires information on travel histories and dates of symptoms onset. Unlike the approach in Kuk and Ma (2005), our estimation of incubation period is feasible regardless that the disease is infectious or not during the incubation period. It enhances the estimation by increasing available sample size and utilizing censored information. We also took mixture distribution of forward time and complete incubation period and the interval censoring caused by daily reports into consideration, hence the result should be more robust than that in Qin et al. (2020) .

According to the theory of renewal process, the density of forward time should be a decreasing function as it is proportional to the survival function of incubation period. If the denisty of the observed time between departure from Wuhan and symptoms onset is unimodal, it might because of (1) the fact that the observations come from a mixture of forward time and full incubation period; (2) the discretized time. Hence, an estimation using mixture distribution together with the censored intervals is recommended if the observed density is not monotonically decreasing. Mixture distribution is robust in incubation analysis in that the potential problem due to the existence of short-term tourists can be addressed by introducing π into the model. In addition, fewer observations of zeros than ones is still reasonable even if there is no full incubation period mixed in the cohort (when π = 0), as the probability to be captured in our cohort is reduced by half if the 'scheduled' departure from Wuhan and symptoms onset occur on the same day, which can be well reflected in the interval censoring situation since F I (0 + ; θ) − F I (0 − ; θ) is just equal to F I (0 + ; θ).

Compared with the estimated incubation period in Li et al. (2020), Backer et al. (2020) and Linton et al. (2020) , our estimation yields a longer estimate of incubation period. This is possibly because we avoided the selection bias by considering a longer follow-up period after departure from Wuhan and successfully recruiting the cases with long incubation periods. However, a limitation here is raised by the possible violation of assumption that the individuals included in the study were either infected in Wuhan or on the way to their

This article is protected by copyright. All rights reserved. In the previous studies of the basic reproduction number of COVID-19, and Figure 1 . Illustration of complete incubation period and forward time. Red circle: getting infected; Blue column: departure from Wuhan; Red cross: symptoms onset. The shaded area is the period during which our cohort sample departed from Wuhan. This figure shows 5 kinds of individuals. Only those who departed from Wuhan in the shaded area were collected in our cohort. A: symptoms onset in Wuhan, not in our cohort; B and C: captured in our cohort with infection before departure; D: captured in our cohort with infection at departure; E: infection outside Wuhan, not in our cohort. This figure appears in color in the electronic version of this article, and any mention of color refers to that version. Biometrics, 000 0000 . COVID-19 data analysis result. Upper: Twice of log likelihood ratio, 2[max θ,π (θ, π) − max θ (θ, π)], versus π. The dashed line is at 2.71, the 90% quantile of chisquared distribution with 1 degree of freedom. In fact, the horizontal ordinate of the crossover point is the 95% upper bound of π by likelihood ratio, since 0.5+0.5χ 2 (2.71, 1) = 0.95 (mixed chi-squared distribution), where χ 2 (·, 1) is the cdf of chi-squared distribution with 1 degree of freedom. Lower: Incubation estimation. Red line: forward time fit; Blue line: Incubation period fit; Black line: mixed observed time fit (covered by the red line). This figure appears in color in the electronic version of this article, and any mention of color refers to that version. Biometrics, 000 0000 Table 1 Estimation of incubation distribution in simulation. Estimates and standard error. The first panel is our proposed method: mixture distribution with censoring. The second panel is the Qin's method. The third panel is the IC method.

(a) Gamma incubation f I (t; θ) = β α t α−1 e −βt /Γ(α); α = 5, β = 0.8.

Qin 

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CI 95% CI 95% CI 95% CI 95% CI 95% CI 95% CI 95% CI 95% CI

We thank Dr. Dean Follmann from National Institute of Allergy and Infectious Diseases for comments that greatly improved the manuscript. We thank Qiushi Lin at Peking University for collecting the data. We also thank Yuan Zhang at Peking University for finding a condition for continuous inversion formula. 

The data and R codes that support the findings in this paper are openly available on github https://github.com/naiiife/wuhan (Deng et al., 2020) .

This article is protected by copyright. All rights reserved.Web Appendix A, B, C, D and E referenced in Section 3-6, is available with this paper at the Biometrics website on Wiley Online Library.