key: cord-289692-fraczoxu
authors: He, Wenqing; Yi, Grace Y.; Zhu, Yayuan
title: Estimation of the basic reproduction number, average incubation time, asymptomatic infection rate, and case fatality rate for COVID‐19: Meta‐analysis and sensitivity analysis
date: 2020-06-09
journal: J Med Virol
DOI: 10.1002/jmv.26041
sha: 
doc_id: 289692
cord_uid: fraczoxu

The coronavirus disease‐2019 (COVID‐19) has been found to be caused by the severe acute respiratory syndrome coronavirus 2 (SARS‐CoV‐2). However, comprehensive knowledge of COVID‐19 remains incomplete and many important features are still unknown. This manuscript conducts a meta‐analysis and a sensitivity study to answer the questions: What is the basic reproduction number? How long is the incubation time of the disease on average? What portion of infections are asymptomatic? And ultimately, what is the case fatality rate? Our studies estimate the basic reproduction number to be 3.15 with the 95% CI (2.41‐3.90), the average incubation time to be 5.08 days with the 95% CI (4.77‐5.39) (in day), the asymptomatic infection rate to be 46% with the 95% CI (18.48%‐73.60%), and the case fatality rate to be 2.72% with 95% CI (1.29%‐4.16%) where asymptomatic infections are accounted for.

Since the first case of the coronavirus disease-2019 (COVID-19) was found in Wuhan, China in December 2019, the disease has rapidly spread in the city of Wuhan, then to Hubei Province, China, and subsequently, across the world. 1 On 11 March 2020, the World Health Organization (WHO) declared COVID-19 to be a pandemic. patients is important for disease surveillance. To determine how deadly the COVID-19 is, it is fundamental to evaluate the case fatality rate which is calculated as the ratio of the number of deaths from to the number of infected cases.

Since the outbreak of the disease, a large body of research on COVID-19 has been done and many articles have been published in scientific journals or shared on platforms such as bioRxir and medRxir. Moreover, COVID-19 data contain substantial errors in that the number of confirmed cases is considerably under-reported, which is attributed to two primary reasons. Insufficient test kits do not allow every potential patient with COVID-19-like symptoms to be tested, and there has been a good portion of asymptomatic COVID-19 carriers who would never be tested and counted as confirmed cases. It is useful to understand the asymptomatic infection rate, defined as the ratio of the number of asymptomatic infections to the number of all infected cases.

To address these issues, we carry out a meta-analysis to synthesize the reported estimates of the basic reproduction number, the average incubation time, and the case fatality rate as well as the asymptomatic rate in a rigorous way by factoring out the variabilities associated with the relevant studies. To accommodate the effects of missing asymptomatic infections on calculating the case fatality rate, we further perform a sensitivity analysis for the estimation of the case fatality rate. Our Table 1 , were identified by the first author (WH) to be included in the analysis, together with Serra 3 and Day, 4 which were found on April 2.

The inclusion criteria are the availability of both point estimates and 95% confidence intervals (95% CIs) (or equivalently, standard deviations) for the basic transmission number, the average incubation time, the asymptomatic rate, or the case fatality rate. Table 1 presents the summary information of the selected articles together with the descriptions of the data used in those articles. We extract the results for the basic reproduction number from 2,5-8,10,11 and the results for the average incubation time from. 5, 6, 9, 13, 16 The results from 3, 4, 20, [21] [22] [23] are extracted for estimation of the asymptomatic infection rate. The estimates for the case fatality rate together with their 95% CIs are taken from. 11, 12, 14, 15, [17] [18] [19] In the articles, 6,7,9 the reported 95% CIs were asymmetric which we suspect were caused by employing a transformation (such as the exponential transformation) to the initial CIs for the reparameterized effective size; for example, some authors may apply the logarithm to reparameterize the basic reproduction number or the average incubation time before performing the analysis.

Using the inverse transformation, we convert the reported asymmetric CIs and work out the associated standard deviations which are used in determining the weights for the meta-analysis.

As shown in the top panel of Figures 1-4 , estimates of the basic reproduction number, the mean incubation time, the asymptomatic infection rate, and the case fatality rate are quite different from study to study. To obtain synthetic results, we perform a meta-analysis to aggregate the information from multiple studies with the same estimand (or effect size of interest) yet different features including the differences in the data collection, the sample size, and the conditions. Suppose K studies report an estimate and the associate standard deviation for an effect size of interest. For the ith study with i = 1, …, K, let Y i denote the effect size of interest and let σ i 2 represent its associated variance estimate. In our analysis here, Y i is taken as the basic reproduction number, the average incubation time, the asymptomatic infection rate, and the case fatality rate, respectively. We calculate a weighted average of the results from those K studies under either the fixed effect model or the random effects model. 24 Under the fixed effect model, the meta mean effect size is given by

and the associated standard deviation is

where w i = 1/σ i 2 is the weight for the ith study.

With the random effects model, the meta mean effect size, denoted Y meta,R , and its standard deviation, denoted sd(Y meta,R ), are determined by the same expression as Equations (1) and (2) except 

The top panel of Figure 1 shows the results for the basic production number reported in the seven studies. The I 2 index for those studies is 97.8%, suggesting that the random effect model should be considered in conducting the meta-analysis. This result agrees with the perception that the basic reproduction number is time-dependent and varies from place to place.

The bottom panel of Figure 1 includes the meta-analysis results. days. This estimated average incubation time is about 2 days shorter than the mean incubation time of 7 days announced by ECDC. 27 We point out that our estimate is obtained by combining the information from those studies before 24 February 2020 with study subjects in Wuhan city or other places in China.

In the top panel of Figure 3 , we display the estimates of the asymptomatic infection rate reported by. 3 

Finally, we are interested in estimating the case fatality rate which measures how deadly COVID-19 is for the infected people. The meta-analysis results derived from seven studies available in the literature, shown in the top panel of Figure 4 , are reported at the bottom panel of Figure 4 , where we assume the random effect models because the I 2 index is 99.5%. The estimated case fatality rate is 3.34%, slightly smaller than 3.4%, the estimate reported on 3 March 2020 by the WHO. 28 The 95% CI suggests that the average case fatality rate can be as small as 2.18% and as large as 4.49%.

These results are obtained from the data up to 21 March 2020 which contain five studies with subjects in China and two studies with worldwide subjects.

We comment that the true average case fatality rate is ex- 

To better understand what the true case fatality rate may be, we further conduct two sensitivity studies. In the first study, we repeat the meta-analysis of the case fatality rate in Section 3.5 by further including the results calculated from the data of the Princess Diamond cruise. 29 This analysis is driven by the consideration that the case fatality rate derived from the cohort of the cruise passengers is highly likely to be accurate, because the number of confirmed cases from the cruise is very likely to be close to the true number of infections. The bottom of Figure 5 reports the meta-analysis results In our second sensitivity analysis, we revise the results in Let p R = D/C R be the reported case fatality rate and let p T = D/C be the true case fatality rate. If we assume that C = C R + C A , then the reported case fatality rate and the true case fatality differ by the factor 1 − r A :

Estimates of the case fatality rate that have been reported in the current literature are merely directed to p R rather than p T .

To sensibly estimate the true case fatality, we use Equation (3) to adjust the reported results of the seven studies listed at the top panel of Figure 4 . Specifically, we may multiply the factor 1 − r A with an estimate for the reported rate p R as well as its standard deviation for each study and then run a meta-analysis. However, the exact value of the asymptomatic infection rate is unavailable, and we only have its estimates from various studies displayed at the top panel of Figure 3 .

To assess how the uncertainty of not knowing the true value of r A , we use two ways to set a value for r A to modify the reported fatality rates for the studies listed at the top panel of Figure 4 for running a new meta-analysis.

First, taking r A as one of seven reported estimates listed at the top panel of Figure 3 , we modify the reported results provided by each study listed at the top panel of Figure 4 using Equation (3), and report the meta-analysis results at the top panels of Figure 6 . In the second analysis, we take r A as the synthesized estimate reported in Figure 3 , that is, r A is set as 46%, and then run the meta-analysis for these adjusted case fatality rates under the random effects model.

We report the results at the bottom panel of Figure 6 , which shows that the estimate of the case fatality rate is 1.8% with the 95% CI ranging from 1.18% to 2.43%.

We carry out a meta-analysis and sensitivity study for estimating the basic reproduction number, the average incubation time, the asymptomatic infection rate, and the case fatality rate for COVID-19.

Examining the published results between 24 January 2020 and 

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Estimation of the basic reproduction number, average incubation time, asymptomatic infection rate, and case fatality rate for COVID-19: Meta-analysis and sensitivity analysis

The authors declare that there are no conflict of interests.

WH identified the articles that were screened by the third author, extracted the results from individual studies, conducted all the data analyses, and prepared the initial draft. GY discussed the analysis methods with WH and wrote the manuscript. YZ searched the literature, provided the candidate articles to WH for further examination.

http://orcid.org/0000-0002-8913-9273