Modeling incubation period and time of exposure using the four-parameter generalized gamma distribution

The incubation period of infectious diseases, the time from infection with a microorganism to onset of disease, is directly relevant to prevention and control. Since explicit models of the incubation period enhance our understanding of the spread of disease, previous classic studies were revisited, focusing on the modeling methods employed and paying particular attention to relatively unknown historical efforts. The earliest study on the incubation period of pandemic influenza was published in 1919, providing estimates of the incubation period of Spanish flu using the daily incidence on ships departing from several ports in Australia. Although the study explicitly dealt with an unknown time of exposure, the assumed periods of exposure, which had an equal probability of infection, were too long, and thus, likely resulted in slight underestimates of the incubation period.After the suggestion that the incubation period follows lognormal distribution, Japanese epidemiologists extended this assumption to estimates of the time of exposure during a point source outbreak. Although the reason why the incubation period of acute infectious diseases tends to reveal a right-skewed distribution has been explored several times, the validity of the lognormal assumption is yet to be fully clarified. At present, various different distributions are assumed, and the lack of validity in assuming lognormal distribution is particularly apparent in the case of slowly progressing diseases. The present paper indicates that (1) analysis using well-defined short periods of exposure with appropriate statistical methods is critical when the exact time of exposure is unknown, and (2) when assuming a specific distribution for the incubation period, comparisons using different distributions are needed in addition to estimations using different datasets, analyses of the determinants of incubation period, and an understanding of the underlying disease mechanisms.

Accurately determining the exposure time to an infectious pathogen, together with the corresponding incubation period, is vital for identifying infection sources and implementing targeted public health interventions. However, real-world outbreak data often include outliers-namely, tertiary or subsequent infection cases not directly linked to the initial source-that complicate the estimation of exposure time. To address this challenge, we introduce a robust estimation framework based on a three-parameter Weibull distribution in which the location parameter naturally corresponds to the unknown exposure time. Our method employs a γ-divergence criterion-a robust generalization of the standard cross-entropy criterion-optimized via a tailored majorization-minimization (MM) algorithm designed to guarantee a monotonic decrease in the objective function despite the non-convexity typically present in robust formulations. Extensive Monte Carlo simulations demonstrate that our approach outperforms conventional estimation methods in terms of bias and mean squared error as well as in estimating the incubation period. Moreover, applications to real-world surveillance data on COVID-19 illustrate the practical advantages of the proposed method. These findings highlight the method's robustness and efficiency in scenarios where data contamination from secondary or tertiary infections is common, showing its potential value for early outbreak detection and rapid epidemiological response.

Background:Understanding changes in infectiousness during SARS-COV-2 infections is critical to assess the effectiveness of public health measures such as contact tracing.Methods:Here, we develop a novel mechanistic approach to infer the infectiousness profile of SARS-COV-2-infected individuals using data from known infector–infectee pairs. We compare estimates of key epidemiological quantities generated using our mechanistic method with analogous estimates generated using previous approaches.Results:The mechanistic method provides an improved fit to data from SARS-CoV-2 infector–infectee pairs compared to commonly used approaches. Our best-fitting model indicates a high proportion of presymptomatic transmissions, with many transmissions occurring shortly before the infector develops symptoms.Conclusions:High infectiousness immediately prior to symptom onset highlights the importance of continued contact tracing until effective vaccines have been distributed widely, even if contacts from a short time window before symptom onset alone are traced.Funding:Engineering and Physical Sciences Research Council (EPSRC).

Article Figures and data Abstract eLife digest Introduction Results Discussion Materials and methods Appendix 1 Data availability References Decision letter Author response Article and author information Metrics Abstract We collated contact tracing data from COVID-19 clusters in Singapore and Tianjin, China and estimated the extent of pre-symptomatic transmission by estimating incubation periods and serial intervals. The mean incubation periods accounting for intermediate cases were 4.91 days (95%CI 4.35, 5.69) and 7.54 (95%CI 6.76, 8.56) days for Singapore and Tianjin, respectively. The mean serial interval was 4.17 (95%CI 2.44, 5.89) and 4.31 (95%CI 2.91, 5.72) days (Singapore, Tianjin). The serial intervals are shorter than incubation periods, suggesting that pre-symptomatic transmission may occur in a large proportion of transmission events (0.4–0.5 in Singapore and 0.6–0.8 in Tianjin, in our analysis with intermediate cases, and more without intermediates). Given the evidence for pre-symptomatic transmission, it is vital that even individuals who appear healthy abide by public health measures to control COVID-19. eLife digest The first cases of COVID-19 were identified in Wuhan, a city in Central China, in December 2019. The virus quickly spread within the country and then across the globe. By the third week in January, the first cases were confirmed in Tianjin, a city in Northern China, and in Singapore, a city country in Southeast Asia. By late February, Tianjin had 135 cases and Singapore had 93 cases. In both cities, public health officials immediately began identifying and quarantining the contacts of infected people. The information collected in Tianjin and Singapore about COVID-19 is very useful for scientists. It makes it possible to determine the disease's incubation period, which is how long it takes to develop symptoms after virus exposure. It can also show how many days pass between an infected person developing symptoms and a person they infect developing symptoms. This period is called the serial interval. Scientists use this information to determine whether individuals infect others before showing symptoms themselves and how often this occurs. Using data from Tianjin and Singapore, Tindale, Stockdale et al. now estimate the incubation period for COVID-19 is between five and eight days and the serial interval is about four days. About 40% to 80% of the novel coronavirus transmission occurs two to four days before an infected person has symptoms. This transmission from apparently healthy individuals means that staying home when symptomatic is not enough to control the spread of COVID-19. Instead, broad-scale social distancing measures are necessary. Understanding how COVID-19 spreads can help public health officials determine how to best contain the virus and stop the outbreak. The new data suggest that public health measures aimed at preventing asymptomatic transmission are essential. This means that even people who appear healthy need to comply with preventive measures like mask use and social distancing. Introduction The novel coronavirus disease, COVID-19, was first identified in Wuhan, Hubei Province, China in December 2019 (Li et al., 2020b; Huang et al., 2020). The virus causing the disease was soon named severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) (Hui et al., 2020) and quickly spread to other regions of China and then across the globe, causing a pandemic with over 5 million cases and 300,000 deaths at the time of writing (Johns Hopkins University, 2020). In Tianjin, a metropolis located at the north of China, the first case was confirmed on January 21, 2020 (Tianjin Health Commission, 2020). Two days later, the first case was confirmed in Singapore (Ministry of Health Singapore, 2020), a city country in Southeast Asia. As of February 28, 2020, 93 and 135 cases had been confirmed in Singapore and Tianjin (Ministry of Health Singapore, 2020; Tianjin Health Commission, 2020). The first Singapore COVID-19 case was confirmed as an individual who had travelled to Singapore from Wuhan. Many of the initial cases were imported from Wuhan, with later cases being caused by local transmission. Singaporean officials worked to identify potential contacts of confirmed cases; close contacts were monitored and quarantined for 14 days from their last exposure to the patient, and other low-risk contacts were put under active surveillance and contacted daily to monitor their health status. These early outbreaks continue to provide the opportunity to estimate key parameters to understand COVID-19 transmission dynamics. We screened publicly available data to identify datasets for two COVID-19 clusters that could be used to estimate transmission dynamics. In both Singapore and Tianjin, the COVID-19 outbreak occurred within a relatively closed system where immediate public health responses were implemented, contacts were identified and quarantined, and key infection dates were tracked and updated daily. With its experiences in control of the SARS outbreak, the Singaporean government had been adopting a case-by-case control policy from January 2, 2020. Only close contacts of a confirmed case were monitored and quarantined for 14 days. In Tianjin, a number of COVID-19 cases were traced to a department store, where numerous customers and sales associates were likely infected. Additional customers who had potential contact were asked to come forward through state news and social media, as well as asked if they had visited the department store at various checkpoints in the city. All individuals identified as having visited the store in late January were quarantined and sections of the Baodi District where the store is located were sealed and put under security patrol. We estimate the serial interval and incubation period of COVID-19 from clusters of cases in Singapore and Tianjin. The serial interval is defined as the length of time between symptom onset in a primary case (infector) and symptom onset in a secondary case (infectee), whereas the incubation period is defined as the length of time between an infectee's exposure to a virus and their symptom onset. Both are important parameters that are widely used in modeling in infectious disease, as they impact model dynamics and hence fits of models to data. While the pandemic has progressed far beyond these early outbreaks, it remains the case that mathematical modelling, using parameters derived from estimates like these, is widely used in forecasting and policy. The serial interval and incubation period distributions, in particular, can be used to identify the extent of pre-symptomatic transmission (i.e. viral transmission from an individual that occurs prior to symptom onset). There is evidence that pre-symptomatic transmission accounts for a considerable portion of COVID-19 spread (Arons et al., 2020; Baggett et al., 2020; Li et al., 2020a) and it is important to determine the degree to which this is occurring (Peak et al., 2020). Early COVID-19 estimates borrowed parameters from SARS (Wu et al., 2020; Jiang et al., 2020; Abbott et al., 2020), but more recent estimates have been made using information from early clusters of COVID-19 cases, primarily in Wuhan. Depending on the population used, estimates for incubation periods have ranged from 3.6 to 6.4 days and serial intervals have ranged from 4.0 to 7.5 days (Li et al., 2020b; Ki and Task Force for 2019-nCoV, 2020; Backer et al., 2019; Linton et al., 2020; Nishiura et al., 2020); however, it is crucial that the estimates of incubation period and serial interval are based on the same outbreak, and are compared to those obtained from outbreaks in other populations. Distinct outbreak clusters are ideal for understanding how COVID-19 can spread through a population with no prior exposure to the virus. Here, we estimate the portion of transmission that is pre-symptomatic based on estimates of the incubation period and serial interval. We estimate both quantities under two frameworks: first, we use samples as directly as is feasible from the data, for example assuming that the health authorities' epidemiological inferences regarding who exposed whom and who was exposed at which times are correct. Second, we use estimation methods that allow for unknown intermediate cases, such that the presumed exposure and infection events may not be complete. We also separate the analysis of incubation period according to earlier and later phases of the outbreaks, since measures were introduced during the time frame of the data. Results Descriptive analyses Figures 1 and 2 show the daily counts, putative origin of the exposure and individual time courses for the Singapore and Tianjin data. In the Singapore dataset, new hospitalization and discharge cases were documented daily from January 23 to February 26, 2020. 66.7% (62/93) of the confirmed cases recovered and were discharged from the hospital by the end of the study period (Figure 1(a)). The disease progression timeline of the 93 documented cases in Figure 1(c) indicates that symptom onset occurred 1.71 ± 3.01 (mean ± SD) days after the end of possible viral exposure window and cases were confirmed 7.43 ± 5.28 days after symptom onset. The mean length of hospital stay was 13.3 ± 6.01 days before individuals recovered and were discharged. Figure 1 Download asset Open asset Singapore COVID-19 cases. (a) Daily hospitalized cases and cumulative hospitalized and discharged cases. (b) Daily incidence with probable source of infection. (C) Disease timeline, including dates at which each case is unexposed, exposed, symptomatic, hospitalized, and discharged. Not all cases go through each status as a result of missing dates for some cases. Figure 2 Download asset Open asset Tianjin COVID-19 cases. (a) Daily and cumulative confirmed cases, cumulative discharges and daily death cases. (b) Daily incidence with probable source of infection. (c) Disease progression timeline; not all cases go through each status as a result of missing dates for some cases. In the Tianjin dataset, new confirmed cases were documented daily from January 21 to February 22, 2020. 48.1% (65/135) recovered and 2.2% (3/135) had died by the end of the study period (Figure 2(a)). The timeline of the 135 cases is shown in Figure 2(c). Symptom onset occurred 4.98 ± 4.83 (mean ± SD) days after the end of the possible viral exposure window. Cases were confirmed 5.23 ± 4.15 days after symptom onset. The duration of hospital stay of the Tianjin cases is unknown as the discharge date of each case was not available. In both datasets, daily counts decline over time, which is likely a combination of delays to symptom onset and between symptom onset and reporting, combined with the effects of strong social distancing and contact tracing. Incubation period In the Singapore dataset, we find that the median incubation period in our direct analysis (without accounting for intermediate cases) is 5.32 days with the gamma distribution; shape 3.05 (95%CI 2.0, 3.84); and scale 1.95 (1.23, 2.34). The mean incubation period is 5.99 (95%CI 4.97, 7.14) days. In Tianjin, we find a median 8.06 days; shape 4.74 (3.35, 5.72); scale 1.83 (1.29, 2.04). The mean is 8.68 (7.72, 9.7) days. These results are summarised in Table 1, and we also fitted Weibull and log normal distributions; see Appendix 1—table 1. These are consistent with, or slightly longer than, previous estimates, see Appendix 1—table 5 for comparison. Table 1 Mean incubation period, serial interval and pre-symptomatic transmission. Incubation periods are based on the gamma estimates because these are the most convenient for taking the covariation of serial intervals and incubation periods into account (done throughout the table). 95% CIs are provided in brackets. Without intermediatesIncubation (days)Serial interval (days)Mean difference (days)Portion pre-symptomatic(-)Singapore (all)5.99 (4.97, 7.14)4.0 (2.73, 5.57)1.990.74Singapore (early)5.91 (4.50,7.64)1.910.742Singapore (late)6.06 (4.70, 7.67 )2.060.744Tianjin (all)8.68 (7.72, 9.7)5.0 (3.82, 6.12)3.680.81Tianjin (early)6.88 (5.97,7.87)1.880.72Tianjin (late)12.4 (11.1,13.7)7.40.96Account for intermediatesSingapore r=0.054.914.17 (2.44, 5.89)0.770.53Singapore r=0.14.430.260.46Singapore r=0.154.12−0.050.41Singapore r=0.23.89−0.280.38Tianjin r=0.057.544.31 (2.91, 5.72)3.230.79Tianjin r=0.16.892.580.74Tianjin r=0.156.301.990.67Tianjin r=0.25.911.60.64 In Singapore, these estimates are based on a combination of cases for whom last possible exposure is given by travel, and later cases (for whom the presumed infector was used). In Tianjin, social distancing measures were implemented during the outbreak. We find that the estimated incubation period is different, particularly in Tianjin, for cases with symptom onset on or prior to January 31st: see Figure 3 and Figure 4. The estimated median incubation period for pre-Feb one cases in Tianjin is 6.48 days; the q=(0.025,0.975) quantiles are (2.5, 13.3) days. In contrast, post-Jan 31 the median is 12.13 days with q=(0.025,0.975) quantiles (7.3, 18.7) days. The means are 6.88 (5.97, 7.87) days for early cases and 12.4 (11.1, 13.7) days for later cases. Social distancing seems unlikely to change the natural course of infection, but these results might be explained if exposure occurred during group quarantine or otherwise later than the last time individuals thought they could have been exposed. Pre-symptomatic transmission would enable this, if an individual was thought to have been exposed before group quarantine, but in actuality was exposed during quarantine by a pre-symptomatic individual. The time interval in the data would then not be a sample of the incubation period, instead it would be a sample of one or more generation times plus an incubation period. Figure 3 Download asset Open asset Fitted gamma COVID-19 incubation period distributions (without intermediates). Cases are defined as 'early' if they have symptom onset on or prior to January 31, and are classified 'late' otherwise. Figure 4 Download asset Open asset COVID-19 incubation period Kaplan-Meier curves for (a) Singapore and (b) Tianjin. Top panels show unstratified data (all cases with symptom onset given). Bottom panels show 'early' and 'late' cases, where early cases are defined as those with symptom onset on or prior to January 31, and late otherwise. In Singapore, we find the same effect, although much less pronounced. The estimated median incubation time is 5.26, with (0.025, 0.975) quantiles of (1.30, 13.8) days for early cases (also defined as cases with symptom onset on or prior to January 31st) and 5.35 (quantiles (1.22, 14.6)) days for late-arising cases. The means are 5.91 (4.50, 7.64) days for early cases and 6.06 (4.70, 7.67) days for later cases. Fits of gamma and log-normal distributions are similar; see Appendix 1—table 2. Changes in perception of exposure times after control measures were introduced (i.e. people may assume that they must have been exposed prior to control measures), together with pre-symptomatic transmission, could result in missing intermediate transmission events and hence lengthened incubation period estimates. This in part motivates our analysis with intermediate cases. Our estimates of the incubation period with intermediates are similar, under the assumption that intermediates are relatively rare. Results are shown in Figure 5 and Table 1. We find that the median of the bootstrapped mean incubation periods for Singapore with a low (0.05 per day) rate of unknown intermediates is 4.91 days (4.35, 5.69 95% bootstrap CI), compared to a generation time of 3.71 (2.36, 4.91) days. The Tianjin bootstrapped mean incubation period is 7.54 (6.76, 8.56 95% CI) days and the generation time is only 2.82 (1.83, 3.52) days. The estimates are lower when the assumed probability of unknown intermediates is higher. Indeed, if intermediates were present between assumed exposure and onset, naturally the generation time would be shorter than if they were not. The mean generation times are consistently shorter than the mean incubation periods, indicating that infection can occur prior to symptom onset. The difference is particularly pronounced in Tianjin, where long intervals were observed. Figure 5 Download asset Open asset Mean incubation period and generation time estimates from the incubation period intermediates analysis, under the assumption that the scale parameter for both distributions is equal, shown with dependence on the mean number of unknown intermediate cases per day of the empirical time elapsed between exposure and symptom onset. The incubation period is longer than the generation time, so this analysis suggests that symptom onset occurs after infectiousness begins. Top: Singapore. Bottom: Tianjin. The means are the scale times the shape, which is fixed at 2.1 in Singapore and 2.2 in Tianjin. Varying this fixed value for the shape parameter was not found to significantly impact the results. However, this approach makes a number of assumptions and is limited by the fact that if we do not know the true infectors then we are also unlikely to know the true exposure. The data we have is well suited to this method in the sense that there were particular events where exposure is thought to have occurred, and so we can account for intermediates in the manner we have done, but we do not have information for the alternative scenario in which the true exposures were prior to those given in the data. This could happen if, for example, individuals were exposed before attending an event or before known contact, and developed symptoms well after it. Exposure would thus be wrongly attributed to the event or contact. We have accommodated this with uncertainty in the exposure intervals, in particular not insisting that individuals who are likely to be the index case for a cluster (e.g. who developed symptoms on the same day as an event) must have been exposed then, but instead allowing the possibility that they were exposed earlier. Serial intervals Figure 6 represents the empirical serial intervals between all potential transmission case-pairs as noted in the data and represented in Figure 7, split into groups based on date of first symptom onset for each case-pair. The empirical mean serial intervals shorten in the 'late' group in both Singapore and Tianjin; however, the empirically derived 95% confidence intervals overlap (Singapore early 4.44 (-2.81, 11.7) vs. late 3.18 (-1.52, 7.88); Tianjin early 5.48 (-0.968, 11.9) vs. late 4.18 (-2.33, 10.7)). Negative lower bounds are due to the high standard deviation. Figure 6 Download asset Open asset Serial intervals of possible case pairs in (a) Singapore and (b) Tianjin. Pairs represent a presumed infector and their presumed infectee plotted by date of symptom onset. Cases are defined as 'early' if they have symptom onset on or prior to January 31st. Figure 7 Download asset Open asset Network diagram for (a) Singapore (b) Tianjin. Shortening serial intervals are expected as increased quarantine measures are enacted during the course of an outbreak and can be an indication of improved control through successful contact tracing, as seen in SARS (Lipsitch et al., 2003). Our results suggest that serial intervals shortened as the outbreak progressed in both clusters, but they could also be due to right truncation. Accounting for this, we found that the mean serial intervals were 4 and 5 days (Singapore, Tianjin); a Cox regression found no significant difference between the early and late groups' serial intervals. This estimate is made directly from case pairs in the data without accounting for intermediate infectors and co-primary infection, as in the ICC analysis. Table 1 shows our ICC estimates of the mean and standard deviation for the serial intervals, with comparison to other analyses and assumptions in Appendix 1—table 5. The ICC method finds the mean serial interval to be 4.17 (2.44, 5.89 95% bootstrap CI) days (0.882 bootstrap standard deviation) for Singapore and 4.31 (2.91, 5.72) days (0.716 bootstrap sd) for Tianjin, using the first four cases in each cluster. This is consistent with the results with right truncation. Pre-symptomatic transmission We estimated incubation periods and serial intervals with and without accounting for intermediate unknown cases. To estimate the portion of transmission that occurs before symptom onset, we compare the 'direct' (no intermediate) estimates of each, and the 'indirect' (accounting for intermediates) estimates of each. We estimate consistently shorter serial intervals than incubation period, suggesting that there is pre-symptomatic transmission. We took the covariation of incubation periods and serial intervals (and of generation times and incubation periods) into account by sampling the intervals jointly before estimating the fraction of the relevant differences that are negative. Even accounting for correlation, the estimated fraction of pre-symptomatic transmission for Singapore is 0.74 (regardless of early/late split) and for Tianjin is 0.72, 0.96, 0.81 (early, late, based on the direct estimates of the incubation periods and serial intervals also Figure we use the incubation period estimates that account for the pre-symptomatic transmission are in Singapore and in Tianjin, when the assumed of of intermediates is (i.e. when we assume a relatively low rate of unknown intermediates). this rate is the portion of pre-symptomatic transmission but even for we estimate the pre-symptomatic portion to be in Singapore and in Tianjin. Figure Download asset Open asset Pre-symptomatic infection as estimated by samples of interval incubation accounting for Top: Singapore. Bottom: Tianjin. without accounting for pre-symptomatic transmission. In all cases there is pre-symptomatic transmission. These results were obtained under an estimated between the incubation period and serial interval of in Tianjin. instead the were the portion of pre-symptomatic transmission in Tianjin under and is estimated as and With the are and We find that the degree of not impact our estimates of pre-symptomatic transmission. We high estimates of the fraction pre-symptomatic in Tianjin, due to the long incubation It seems likely that these are an of pre-symptomatic transmission during or of other assumptions made about exposures in the of the We that for this data and under we see evidence of at of transmission occurring before symptom onset. In our direct analysis, we estimate that infection occurred on and days before symptom onset of the infector (Singapore, Tianjin). the incubation period is for and cases in our data, on transmission for cases is and days before symptom onset (Singapore, and days before (Singapore, for cases. a low rate of potential unknown intermediate cases into the mean difference to and days (Singapore, we estimate a significant portion of pre-symptomatic transmission as serial intervals are shorter than incubation periods in our analyses These estimates are by the fact that we have estimated both incubation period and serial interval in the same and by the fact that we the same result in two In both of estimates, samples of the incubation period serial interval are with probability or and or and these lower bounds a high rate of unknown intermediates early in the outbreak. This indicates that a portion of transmission may occur before symptom onset Appendix 1 and Figure consistent with the by et al., 2020 and et al., 2020). serial intervals lower number estimates. example, if the at a rate of time of days et al., scenario an estimated number using the mean of the bootstrapped estimates is with a serial interval of 4.17 days and with a serial interval of days In contrast, if a longer serial interval days et al., 2020; Li et al., is used, the estimate is This is based on the between serial and and is a estimate that not into account a and natural of infection and It primarily to how our estimated serial intervals impact in models for COVID-19 dynamics. Discussion Here, we use transmission clusters in two where cases have exposure and symptom onset times to estimate both the incubation period and serial interval of COVID-19. We these datasets available in a convenient they were available publicly but the Singapore was in and the Tianjin cluster was on and in in We that the datasets themselves useful for understanding early spread in these The incubation period and serial interval are key parameters for transmission modeling and for public health modeling remains one of the primary policy in use in local and COVID-19 Serial intervals, together with control the shape and of the et al., the disease's incidence and how quickly an and how quickly methods need to be implemented by public health officials to control the disease et al., et al., In particular, the portion of transmission events that occur before symptom onset is a for infection control et al., and impact the of contact tracing and case (Peak et al., 2020). Singapore and Tianjin officials both quickly when COVID-19 cases and contact tracing and however, there was a difference in the of the measures The first case was identified in Singapore on 2020 and in Tianjin on By Singapore had identified close contacts and implemented a to all to Hubei and all to China, asked to monitor their health for 2 to Singapore, and asked the public to including close contact with people who are and and a mask if they had respiratory symptoms (Ministry of Health Singapore, 2020). by February in Tianjin, contacts were under and the Baodi of 1 million people was under with one person per could 2 days to public were no one could their between and without an to Tianjin were put under and all the and were While Singapore the virus spread relatively well they confirmed cases on cases on 1, cases on 22, and cases on (Ministry of Health Singapore, 2020); Tianjin began to their by and had at confirmed cases as of In Singapore and Tianjin we estimated relatively serial intervals. particular early estimates of for COVID-19 used the SARS serial interval of days et al., 2020; and 2020; et al., 2020). Our serial interval from two those of et al., 2020 and et al., 2020), who estimated a serial interval of and 4.0 days. et al., 2020 a estimate for the serial interval days with 95% but with standard deviation based on cases in we estimate the serial interval to be shorter than the incubation period in both clusters, which suggests pre-symptomatic transmission. This indicates that spread of is likely to be to stop by of cases However, shorter serial intervals also to lower estimates of and our serial intervals if this means that of the need to be to contain We the incubation per

In this study, using the Simple Random Sampling without Replacement (SRSWOR) method, we propose a generalized estimator of population variance of the primary variable. Up to the first order of approximation, the bias and Mean Squared Error (MSE) expressions for the suggested estimator are produced. The suggested estimator's characterizing scalar is optimized, and for this optimal value of the characterizing constant, the suggested estimator's least MSE is also determined. The efficiency criteria of the suggested estimator over the other estimators are determined after a theoretical comparison of the proposed estimator with the other population variance estimators that already exist. Several actual natural populations are used to validate these efficiency parameters. For practical use in various application domains, the estimator with the lowest MSE and the best Percentage Relative Efficiency (PRE) is advised.In this study, using the Simple Random Sampling without Replacement (SRSWOR) method, we propose a generalized estimator of population variance of the primary variable. Up to the first order of approximation, the bias and Mean Squared Error (MSE) expressions for the suggested estimator are produced. The suggested estimator's characterizing scalar is optimized, and for this optimal value of the characterizing constant, the suggested estimator's least MSE is also determined. The efficiency criteria of the suggested estimator over the other estimators are determined after a theoretical comparison of the proposed estimator with the other population variance estimators that already exist. Several actual natural populations are used to validate these efficiency parameters. For practical use in various application domains, the estimator with the lowest MSE and the best Percentage Relative Efficiency (PRE) is advised.In this study, using the Simple Random Sampling without Replacement (SRSWOR) method, we propose a generalized estimator of population variance of the primary variable. Up to the first order of approximation, the bias and Mean Squared Error (MSE) expressions for the suggested estimator are produced. The suggested estimator's characterizing scalar is optimized, and for this optimal value of the characterizing constant, the suggested estimator's least MSE is also determined. The efficiency criteria of the suggested estimator over the other estimators are determined after a theoretical comparison of the proposed estimator with the other population variance estimators that already exist. Several actual natural populations are used to validate these efficiency parameters. For practical use in various application domains, the estimator with the lowest MSE and the best Percentage Relative Efficiency (PRE) is advised.In this study, using the Simple Random Sampling without Replacement (SRSWOR) method, we propose a generalized estimator of population variance of the primary variable. Up to the first order of approximation, the bias and Mean Squared Error (MSE) expressions for the suggested estimator are produced. The suggested estimator's characterizing scalar is optimized, and for this optimal value of the characterizing constant, the suggested estimator's least MSE is also determined. The efficiency criteria of the suggested estimator over the other estimators are determined after a theoretical comparison of the proposed estimator with the other population variance estimators that already exist. Several actual natural populations are used to validate these efficiency parameters. For practical use in various application domains, the estimator with the lowest MSE and the best Percentage Relative Efficiency (PRE) is advised.In this study, using the Simple Random Sampling without Replacement (SRSWOR) method, we propose a generalized estimator of population variance of the primary variable. Up to the first order of approximation, the bias and Mean Squared Error (MSE) expressions for the suggested estimator are produced. The suggested estimator's characterizing scalar is optimized, and for this optimal value of the characterizing constant, the suggested estimator's least MSE is also determined. The efficiency criteria of the suggested estimator over the other estimators are determined after a theoretical comparison of the proposed estimator with the other population variance estimators that already exist. Several actual natural populations are used to validate these efficiency parameters. For practical use in various application domains, the estimator with the lowest MSE and the best Percentage Relative Efficiency (PRE) is advised.In this study, using the Simple Random Sampling without Replacement (SRSWOR) method, we propose a generalized estimator of population variance of the primary variable. Up to the first order of approximation, the bias and Mean Squared Error (MSE) expressions for the suggested estimator are produced. The suggested estimator's characterizing scalar is optimized, and for this optimal value of the characterizing constant, the suggested estimator's least MSE is also determined. The efficiency criteria of the suggested estimator over the other estimators are determined after a theoretical comparison of the proposed estimator with the other population variance estimators that already exist. Several actual natural populations are used to validate these efficiency parameters. For practical use in various application domains, the estimator with the lowest MSE and the best Percentage Relative Efficiency (PRE) is advised.

Estimation of the incubation period of influenza A (H1N1-2009) among imported cases: Addressing censoring using outbreak data at the origin of importation

Primary objective of this work was to determine whether or not a threshold stress and incubation period exists preceeding stress-corrosion cracking of Types 302 and 316 austenitic stainless steel wires. An investigation of the threshold stress and incubation periods was conducted on as-received (bright-annealed wire) and laboratory-annealed wire. The corrosion environment used was boiling 42 weight percent MgCl2 solution. Stress levels varied from 700 to 35,900 psi; however, most testing was conducted at 5000, 10,000 and 20,000 psi. Testing to determine threshold stresses was conducted by exposing stressed wires to the corrosive environment and recording the times-to-failure. Incubation times were determined by exposing stressed specimens to the environment for preselected periods of time and subsequently examining the specimens metallographically to determine average crack depths. A correlation between average crack depth and exposure time was obtained. Results of threshold stress determination testing revealed that no threshold stress was evident for as-received wire, but laboratory annealed wire stressed at 5000 psi did not crack until cross-sectional area had been reduced considerably due to general corrosion. A study of the incubation times revealed that, if there is an incubation period with as-received wire, it is less than ten minutes. For laboratory annealed wires, incubation periods of 35 minutes at 20,000 psi, 100 minutes at 10,000 psi and 500 minutes at 5000 psi were found.

Wistar rats were exposed for 24 mo to diesel emissions containing a low (0.2 ppm, 0.21 mg/m3), medium (1.04 ppm, 1.18 mg/m3), or high (2.96 ppm, 3.05 mg/m3) concentration of NO2 and particles, or diesel emissions containing a medium (1.12 ppm, 0.01 mg/m3) concentration of NO2 without particles. At 6-mo intervals during the exposure period, rats were autopsied, and their lungs were prepared for light- and electron-microscopic examination. Morphological evaluations included examination for hyperplasia of airway goblet cells, shift in the types of glycoprotein of intracellular mucus granules in goblet cells, infiltration of inflammatory cells in the airways, enlargement of the cross-sectional area of an alveolus as a parameter of air space enlargement, and development of alveolar holes, which is considered to be an early hallmark of alveolar destruction. The number of goblet cells with acid-form mucus granules increased with the exposure concentration and time; however, goblet cells did not show any hyperplastic changes. Furthermore, inflammatory cells such as alveolar macrophages, mast cells, plasma cells, neutrophils, and lymphocytes infiltrated the airways and the alveoli, and showed some cell-to-cell contact. Although no significant enlargement of the air space of the lungs was seen in any exposure group, the number of alveolar holes was significantly higher in the high-concentration group in comparison with the control group at each exposure time, and also increased in other exposure groups, even in the low-concentration group at certain exposure times. Morphological changes in the lungs were mild even in the animals exposed to the highest levels of diesel emissions for 24 mo. Elimination of particles from diesel emissions led to reduced morphological changes such as a decreased shift in the types of glycoprotein of mucus granules in goblet cells, decreased infiltration of inflammatory cells in the lungs, and reduced anthracosis.

Exposure to a high-fat diet during intrauterine life and post-birth causes cardiac histomorphometric changes in rats: A systematic review

Impact of UV-B radiation on some biochemical changes and growth parameters in Echinacea purpurea callus and suspension culture

Large-scale radiation emergency scenarios involving protracted low dose rate radiation exposure (e.g. a hidden radioactive source in a train) necessitate the development of high throughput methods for providing rapid individual dose estimates. During the RENEB (Running the European Network of Biodosimetry) 2019 exercise, four EDTA-blood samples were exposed to an Iridium-192 source (1.36 TBq, Tech-Ops 880 Sentinal) at varying distances and geometries. This resulted in protracted doses ranging between 0.2 and 2.4 Gy using dose rates of 1.5–40 mGy/min and exposure times of 1 or 2.5 h. Blood samples were exposed in thermo bottles that maintained temperatures between 39 and 27.7 °C. After exposure, EDTA-blood samples were transferred into PAXGene tubes to preserve RNA. RNA was isolated in one laboratory and aliquots of four blinded RNA were sent to another five teams for dose estimation based on gene expression changes. Using an X-ray machine, samples for two calibration curves (first: constant dose rate of 8.3 mGy/min and 0.5–8 h varying exposure times; second: varying dose rates of 0.5–8.3 mGy/min and 4 h exposure time) were generated for distribution. Assays were run in each laboratory according to locally established protocols using either a microarray platform (one team) or quantitative real-time PCR (qRT-PCR, five teams). The qRT-PCR measurements were highly reproducible with coefficient of variation below 15% in ≥ 75% of measurements resulting in reported dose estimates ranging between 0 and 0.5 Gy in all samples and in all laboratories. Up to twofold reductions in RNA copy numbers per degree Celsius relative to 37 °C were observed. However, when irradiating independent samples equivalent to the blinded samples but increasing the combined exposure and incubation time to 4 h at 37 °C, expected gene expression changes corresponding to the absorbed doses were observed. Clearly, time and an optimal temperature of 37 °C must be allowed for the biological response to manifest as gene expression changes prior to running the gene expression assay. In conclusion, dose reconstructions based on gene expression measurements are highly reproducible across different techniques, protocols and laboratories. Even a radiation dose of 0.25 Gy protracted over 4 h (1 mGy/min) can be identified. These results demonstrate the importance of the incubation conditions and time span between radiation exposure and measurements of gene expression changes when using this method in a field exercise or real emergency situation.

Bovine spongiform encephalopathy

The study of intracellular bacterial pathogens in cell culture hinges on inhibiting extracellular growth of the bacteria in cell culture media. Aminoglycosides, like gentamicin, were originally thought to poorly penetrate eukaryotic cells, and thus, while inhibiting extracellular bacteria, these antibiotics had limited effect on inhibiting the growth of intracellular bacteria. This property led to the development of the antibiotic protection assay to study intracellular pathogens in vitro. More recent studies have demonstrated that aminoglycosides slowly penetrate eukaryotic cells and can even reach intracellular concentrations that inhibit intracellular bacteria. Therefore, important considerations, such as antibiotic concentration, incubation time, and cell type need to be made when designing the antibiotic protection assay to avoid potential false positive/negative observations. Yersinia pestis, which causes the human disease known as the plague, is a facultative intracellular pathogen that can infect and replicate in macrophages. Y. pestis is sensitive to gentamicin and this antibiotic is often employed in the antibiotic protection assay to study the Y. pestis intracellular life cycle. However, a large variety of gentamicin concentrations and incubation periods have been reported in the Y. pestis literature without a clear characterization of the potential influences that variations in the gentamicin protection assay could have on intracellular growth of this pathogen. This raised concerns that variations in the gentamicin protection assay could influence phenotypes and reproducibility of data. To provide a better understanding of the potential consequences that variations in the gentamicin protection assay could have on Y. pestis, we systematically examined the impact of multiple variables of the gentamicin protection assay on Y. pestis intracellular survival in macrophages. We found that prolonged incubation periods with low concentrations of gentamicin, or short incubation periods with higher concentrations of the antibiotic, have a dramatic impact on intracellular growth. Furthermore, the degree of sensitivity of intracellular Y. pestis to gentamicin was also cell type dependent. These data highlight the importance to empirically establish cell type specific gentamicin protection assays to avoid potential artificial data in Y. pestis intracellular studies.

The purpose of this study was to develop and validate a novel method for sleep quality quantification using personal handheld devices. The proposed method used 3- or 6-axes signals, including acceleration and angular velocity, obtained from built-in sensors in a smartphone and applied a real-time wavelet denoising technique to minimize the nonstationary noise. Sleep or wake status was decided on each axis, and the totals were finally summed to calculate sleep efficiency (SE), regarded as sleep quality in general. The sleep experiment was carried out for performance evaluation of the proposed method, and 14 subjects participated. An experimental protocol was designed for comparative analysis. The activity during sleep was recorded not only by the proposed method but also by well-known commercial applications simultaneously; moreover, activity was recorded on different mattresses and locations to verify the reliability in practical use. Every calculated SE was compared with the SE of a clinically certified medical device, the Philips (Amsterdam, The Netherlands) Actiwatch. In these experiments, the proposed method proved its reliability in quantifying sleep quality. Compared with the Actiwatch, accuracy and average bias error of SE calculated by the proposed method were 96.50% and -1.91%, respectively. The proposed method was vastly superior to other comparative applications with at least 11.41% in average accuracy and at least 6.10% in average bias; average accuracy and average absolute bias error of comparative applications were 76.33% and 17.52%, respectively.

INTRODUCTION: The production of an increasing amount of biomass/biowaste is an environmental challenge that our society urgently needs to address. Many residues from industrial processes contain valuable biocompounds, either as such or that can be transformed into such by simple and sustainable transformation processes. Biomasses can be converted into graphite-based carbon (carbon dots, CD) by a hydrothermal process, using only water as a green solvent, i.e. without toxic solvents and with minimal impact on the environment. OBJECTIVE: The purpose of this study is to assess the safety of two green CDs compounds made from waste biomaterials, i.e. Cork powder and Fish scales, on biotechnologists who produce disposable biosensing devices. Cork powder is a residue of the cork stopper production and Fish scales are a residue of the fish processing industry. We performed both a cytotoxicity and a genotoxicity test (the Trypan Blue Staining Assay and the Alkaline Comet Assay, respectively) to investigate the effects of these bio-nanomaterials in Jurkat cells, a human tumor- derived lymphoblastoid cell line defective for P53 activity, exposed to the compounds. The two assays complement each other since the former informs about induction of cell death, while the latter provides a quantitative estimate of DNA damage induction in (surviving) cells. Scaling concentrations (up to 100 ug/ml) were selected for both compounds and prolonged exposure times, up to 72h, were studied. MATERIALS AND METHODS: Green carbon dots from Fish Scale and Gill (FSG-CDs) and Cork (C-CDs) were prepared from the Nile tilapia (Oreochromis niloticus) and cork powder, respectively, by the hydrothermal treatment (described in poster by Dawood et al., ‘Green carbon dots developed from biomass for sensing Marine Biotoxins’). Jurkat cells were cultured in RPMI supplemented with RPMI 1640 supplemented with 10% FBS, 100 U/ml penicillin, and 100 μg/ml and maintained in a humidified incubator at 37 °C and 5% CO2/air. FSG-CDs and C-CDs water solutions (2mg/ml) were sterilized (0,22um filters) and diluted in complete culture medium (1mg/ml) before cell administration. Appropriate volumes were then added to the cell suspensions to obtain the final concentrations of 0, 5, 25, 50, 100 ug/ml, also analysing three times of prolonged exposure (24h, 48h, 72h). In addition to negative control (no treatment), hydrogen dioxide (H2O2) was used as positive control (150uM, 30 mins at 37°C). Trypan Blue Staining Assay was carried out before and immediately after the end of treatments by counting viable and nonviable cells in a hemocytometer. Slide preparation for alkaline Comet Assay was performed at the end of exposure, at +4°C, in the dark. One single electrophoretic run was carried out per each test compound, where all concentrations and incubation times were inserted in the basin and run together. The software “Comet IV” (Instem) was used for comet image analysis, and a minimum of 50 comets / experimental point were measured. RESULTS: No significant induction of cell death was induced by FSG-CDs and C-CDs at the different concentrations and times of exposure. No genotoxic effect was detected for FSG-CDs at all concentrations and exposure times, while a slight but significant increase of DNA damage was induced by C-CDs. An extremely significant increase of DNA damage was recorded in the positive control (H2O2), as expected.