Meta-analysis of several epidemic characteristics of COVID-19.

As the COVID-19 pandemic has strongly disrupted people's daily work and life, a great amount of scientific research has been conducted to understand the key characteristics of this new epidemic. In this manuscript, we focus on four crucial epidemic metrics with regard to the COVID-19, namely the basic reproduction number, the incubation period, the serial interval and the epidemic doubling time. We collect relevant studies based on the COVID-19 data in China and conduct a meta-analysis to obtain pooled estimates on the four metrics. From the summary results, we conclude that the COVID-19 has stronger transmissibility than SARS, implying that stringent public health strategies are necessary.

Introduction

Coronavirus disease (COVID-19) is an infectious disease caused by severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2), a newly discovered coronavirus, which leads to respiratory illness and can be transmitted from person to person. Ever since December 2019, when the first case of COVID-19 in Wuhan, P. R. China (or mainland China) was reported, the novel coronavirus has hit most of the countries in the world, with the United States (U.S.) being the one having the largest number of confirmed cases (Worldometer, 2020). On March 11, 2020, the World Health Organization (WHO) declared COVID-19 a worldwide pandemic. As of May 9, 2020, the WHO has reported a total of 3,855,788 confirmed cases all over the world and the total number of deaths has reached 265,862.

The COVID-19 pandemic has significant negative impacts on both the global health and the economy. In the U.S., for example, the unemployment rate has jumped up to 14.7% in April, 2020, reaching the highest rate and the largest monthly increase since January 1948 (U.S. Bureau of Labor Statistics, https://www.bls.gov/bls/newsrels.htm). Continuous efforts have been made by every country in the world to slow the spread of the disease and mitigate the associated negative impacts on various aspects of the society.

So far, great amount of scientific research has been conducted on COVID-19, ranging from ongoing clinical trials that evaluate potential treatments to statistical analyses on the characteristics of this infectious disease. With more and more COVID-19 data and studies available, it is vitally important to aggregate the information and pool the statistical findings. Hence, here we conduct a meta-analysis based on published studies of the COVID-19 outbreak in mainland China.

Our meta-analysis that accounts for between-study heterogeneity concentrates on four common epidemic metrics that characterize the transmission of the COVID-19:

Basic reproduction number: Often referred to as R0, the basic reproduction number measures the contagiousness or transmissibility of infectious agents and is interpreted as the expected number of infections caused directly by one case in a completely susceptible population.

Incubation period: This metric is defined as the number of days between when an individual was actually infected and when this person starts to show symptoms. Understanding the incubation period of the COVID-19 is crucial as it provides guidelines on deciding a reasonable length of the quarantine period.

Serial interval: Defined as the time between the start of symptoms in the primary patient (infector) and onset of symptoms in the patient being infected by the infector (the infectee), the serial interval is critical in the calculation of R0.

Epidemic doubling time: This metric measures the period of time needed for the total number of cases in the epidemic to double, and is an important factor that reflects the speed at which the COVID-19 is spreading.

The rest of this manuscript is organized as follows. We first provide details on the data collection process in Section 2. Then in Section 3, we introduce the key ingredients for a meta-analysis, where the modeling and estimation methods are outlined. Section 4 lists a detailed summary for the four epidemic metrics as given above. Statistical results from the meta-analysis are reported in Section 5, and sensitivity analysis is given in Section 6. Concluding remarks are then disclosed in Section 7.

Study selection

We conduct a comprehensive literature screening for the articles published on scientific journals (including early versions) or preprint platforms like medRxiv and bioRxiv. The key words of our searching are COVID-19 (SARS-CoV-2 or 2019-nCOV), the epidemic characteristics of interest (basic reproduction number, incubation period, serial interval and epidemic doubling time), and China, where the selection criteria is relaxed for regional labels. Specifically, we include the studies on the whole country of China, China except for Hubei province (when the city of Wuhan is located), a list of provinces and municipalities, a collection of cities or even a specific region, in order to increase the sample size and the potential statistical power. The variability and heterogeneity across the selected studies are accounted when adopting appropriate methodologies in the meta-analysis.

In addition, we only include the studies reporting unambiguous estimates and the associated 95% confidence intervals or standard deviations. If the standard deviation of an estimator is given, we calculate the 95% confidence interval under the assumption of normal approximation provided that the fitted model is valid. This selection criterion is needed to ensure the consistency of analysis results.

Some studies included here provide more than one estimates that are obtained from various methods or models. For those studies, we select one estimate and its associated confidence interval in our analysis. We illustrate our choices and the corresponding reasons in details in Section 5.

Meta-analysis

Meta-analysis is a statistical procedure aiming to combine scientific results from multiple comparable studies or trials. It is one of the most popular analytical tools in the statistical analysis, which derives a pooled estimate by aggregating relevant information, thus increasing the statistical power. In particular, when comparing different studies addressing the same question, the crux is to measure the standardized difference across various results, i.e. the effect size. In this sense, we rely on meta-analysis to improve the estimates of the effect size of an intervention or an association and examine variability to detect inconsistent results across studies.

In general, there are two methods to pool effect sizes from multiple studies: the fixed-effects model (FEM) and the random-effects model (REM). A FEM assumes all included studies come from the same population, whereas a REM is constructed based on the assumption that data collected comes from different population. Although we restrict our searching of COVID-19 research with data only from mainland China, difference in the population may still exist, due to possible variations in the data samples as well as sampling errors. Hence, we choose to adopt the REM in our analysis.

Let k index the selected studies and Y be the estimator of the k-th study. A REM assumes a normal-normal hierarchical model: Here θ is the parameter of interest, and is the variance of Y. The hyper-parameters μ and τ2 correspond to the true effect and the across-study variance that reflects heterogeneity of the population, respectively.

The inference for REM is done sequentially: we first estimate the heterogeneity variance (denoted as ), then with given, we obtain estimates of the effect μ. There are a variety of estimation methods for τ2; see Veroniki et al. (2016) for a concise survey. We adopt the Hartung-Knapp-Sidik-Jonkman (HKSJ) method (Hartung and Knapp, 2001; Sidik and Jonkman, 2002), which is shown to be more robust when the number of studies is small and there is a substantial heterogeneity in the population.

Given , the conditional maximum-likelihood of effect estimate becomes where the weights are specified as Then we use the normal approximation to construct confidence intervals, where the standard deviation of is given by Later in Section 5, we also provide prediction intervals, which are equally significant components in a meta-analysis. In the presence of a substantial across-study heterogeneity, prediction intervals are preferred as they not only quantitatively provide a range for the effect size of a new study, but also measure the uncertainty of the estimate in a way that acknowledges the heterogeneity.

Epidemic characteristics

In this section, we briefly introduce the epidemic characteristics that are investigated in this manuscript.

Basic reproduction number

In epidemiology, the basic reproduction number, denoted R0, is the expected number of cases caused by one case in a completely susceptible population. It is a critical metric to describe the contagiousness or transmissibility of infectious diseases (Delamater et al., 2019). The estimation of R0 is primarily based on compartmental models, where the susceptible-infectious-recovered (SIR) model and its extensions are the most commonly used. A variety of methods have been developed to estimate R0, such as maximum likelihood methods and Bayesian approaches; we refer the interested readers to Dietz (1993) for a concise survey and to Nikbakht et al. (2019) for a practical comparison of the methods. For the users of statistical software R, the package R0 includes most standard methods for the estimation of R0.

Since the outbreak of COVID-19, R0 has been one of the most critical metrics receiving substantial interest in the community, as it provides a basic benchmark (with threshold 1) to define a pandemic. Besides, R0 helps indicating the potential severity of an epidemic outbreak. It is evident that R0 is closely related to the fraction of the number of infected individuals out of a population once the outbreak ends (Holme and Masuda, 2015). Furthermore, R0 is an indispensable component when estimating the effective reproduction numbers (denoted as R), which are often used to assess the effectiveness of the intervention procedure to mitigate the spread of an epidemic.

Incubation period

The second characteristic that we investigate here is the incubation period. The incubation period of an epidemic is the period from the time of the contact of a transmission source (susceptible or confirmed infector) and the time of symptom onset. The incubation period provides important information during an outbreak, as it helps to determine when the infected individuals who are symptomatic are most likely to spread the disease, and signal necessary public health activities such as monitoring, surveillance and intervention.

From a statistical perspective, the incubation period is a crucial factor to model the current and future trends of an epidemic, as well as evaluate intervention strategies. One of the biggest challenges of estimating the incubation period is that data are often coarsely observed, in addition to several other issues such as censoring and selection bias. The estimation of incubation period is generally based on the methods developed in Reich et al. (2009) and the extensions.

Serial interval

The serial interval is defined as the time duration between the onset of symptoms in the primary patient and the onset of symptoms in the secondary patient who receives the disease from that primary one (Lipsitch et al., 2003). It is the sum of the latent period and the duration of infectiousness. Being another crucial factor for constructing epidemiological models, the serial interval is one of the fundamentals for computing and estimating R0; see Fine (2003) for a summary of the importance of serial interval in epidemiological studies.

The estimation of serial interval is based on the generation time distribution, which is closely related to the infection rate of a epidemic. Standard estimation procedures are given in Diekmann et al. (2013, Chapter 13). In practice, the R package R0 collects functions that estimate serial intervals via maximum likelihood methods (White et al., 2009) and the estimation of serial interval in the package EpiEstim is based on the method developed in Cori et al. (2013).

Epidemic doubling time

The epidemic doubling time (or simply doubling time) is another important index in epidemic studies, as it measures the length of the period during which the number of confirmed cases is doubled. It is evident that the doubling time is reversely related to another epidemic parameters of interest: case-fatality rate. Hence, learning doubling time helps epidemiologists not only understand the transmissiblity of an infection disease but also evaluate its severity. In the course of pathology, the doubling time is useful for analyzing the growth rate of viruses and antigens. Doubling time also can be used to assess the effectiveness of public health interventions and protocols, as an increase in the doubling time usually indicates a slowdown in epidemic transmission.

The estimation of the epidemic doubling time is generally model based, where exponential growth models are the most frequently adopted (Galvani et al., 2003; Du et al., 2020b). In this manuscript, we conduct a meta-analysis on the doubling time to gauge the growth rate of COVID-19.

Results

In this section, we apply the methods introduced in Section 3 to estimate the epidemic characteristics of interest listed in Section 4. The analysis is primarily done in R, where several standard packages for meta-analysis are utilized; for instance, meta, metafor, dmetar among others.

Relevant studies used for our meta-analysis are summarized in Table 2, where a total of 12 research articles are included. Note that the sample size is not large, so we depict a funnel plot in Figure 1 to see whether there is any bias owing to the small sample size. In a funnel plot, a significantly asymmetry pattern indicates publication bias. A more rigorous method is to exploit the Egger’s test (Egger et al., 1997), for which a small p-value suggests rejecting the null hypothesis (i.e., some bias caused by the small sample size does exist). The p-value here is 0.47, showing that no further procedure is needed for correcting the effect size for the present meta-analysis.

Table 2:

Estimates and confidence intervals of R0 for COVID-19 in China in the literature

	R0
Source	estimate	lower bound	upper bound
Cao et al. (2020)	4.08	3.37	4.77
Kucharski et al. (2020)	2.35	1.15	4.77
Li et al. (2020a)	2.20	1.40	3.90
Liu et al. (2020a)	2.90	2.32	3.63
Imai et al. (2020)	2.60	1.50	3.50
Read et al. (2020)	3.11	2.39	4.13
Sanche et al. (2020)	5.70	3.80	8.90
Shen et al. (2020)	4.71	4.50	4.92
Sun et al. (2020)	2.94	2.38	3.50
Tian et al. (2020)	3.15	3.04	3.26
Wu et al. (2020a)	2.68	2.47	2.86
Zhu and Chen (2020)	2.69	2.61	2.77

Figure 1:

Funnel plot of meta-analysis for R0

Using the estimation procedure demonstrated in Section 3, we see that the estimate of R0 is 3.16 with a 95% confidence interval [2.60, 3.72] and a 95% prediction interval [1.25, 5.08]. The associated forest plot is presented in Figure 2. The index of heterogeneity, I2, which ranges from 0 to 1, is used to quantify the dispersion of effect sizes. Here we have I2 = 97%, indicating a substantial heterogeneity in the population. The conclusion from I2 is also consistent with a small p-value from the Cochran’s Q test (less than 10−4). Hence, the REM is indeed more appropriate than the FEM for our meta-analysis.

Figure 2:

Forest plot of meta-analysis for R0

With high heterogeneity in the underlying population, prediction intervals that incorporates heterogeneity is more informative than confidence intervals which focuses only on summary estimates. We hence recommend using the prediction interval as an interval estimator when inferring R0.

Based on the results from our meta-analysis, we conclude that the basic reproduction number (R0) of COVID-19 appears to be greater than that of SARS (point estimate around 3), as reported by WHO in “Consensus document on the epidemiology of severe acute respiratory syndrome (SARS)”. However, it is not as large as an average-based estimate (3.28) for COVID-19 reported in Liu et al. (2020b). Compared with the results from other two meta-analyses on COVID-19, our estimate is slightly greater than 3.15 reported in He et al. (2020a), and moderately greater than 3.05 reported in Dong et al. (2020). Nonetheless, we do not observe statistically significant difference in either case.

Analogous to the previous part, we give the funnel plot in Figure 3, from which a (roughly) symmetric pattern is observed. This is consistent with the p-value (0.915) from the Egger’s test. Hence, again, publication bias is not present here.

Figure 3:

Funnel plot of meta-analysis for incubation period

The estimate of the incubation period of COVID-19 based on our meta analysis is 5.35 (days), with a 95% confidence interval [4.29, 6.42] and a 95% prediction interval [1.97, 8.73]; see Figure 4. Our result is greater than the median (4 with interquartile range (IQR) from 2 to 7) of the incubation period estimated by Guan et al. (2020), which is not included in our meta-analysis as no 95% confidence interval is provided therein. One possible reason is that the study period of Guan et al. (2020) is between December 11, 2019 and January 29, 2020, which is considered as a relatively early stage of the COVID-19 outbreak in mainland China. Besides, our estimate is larger than that (5.08) from another meta-analysis in He et al. (2020a).

Figure 4:

Forest plot of meta-analysis for incubation period

Our meta-analysis also suggests that the incubation period of COVID-19 is a bit longer than that of SARS (commonly ranging from 3 to 5 according to WHO), and our finding is close to the estimate of the incubation period of COVID-19 by the U.S. Centers for Disease Control and Prevention (CDC). Our result provides evidence in support of the 14-day monitoring and quarantine periods in implementation.

For this part, we collect 12 studies to proceed along our meta-analysis. Albeit the funnel plot in Figure 5 displaying an asymmetric pattern, the p-value from the Egger’s test is 0.29, suggesting that it is not required to implement a correction procedure. The estimate of the serial interval is 5.35 with a 95% confidence interval [4.63, 6.07] and a 95% prediction interval [3.13, 7.57].

Figure 5:

Funnel plot of meta-analysis for serial interval

The estimate of the serial interval of COVID-19 based on our meta-analysis is close to that of SARS (5–6 days according to WHO’s report in “Consensus document on the epidemiology of severe acute respiratory syndrome (SARS)”). A shorter serial interval of COVID-19, together with a shorter mean incubation period, suggests higher possibility that a transmission is completed before the onset of symptoms. Therefore, reducing the source of transmission (by hospitalizing infected individuals or implementing “stay-at-home” protocols to susceptible individuals) and reasonably extending the quarantine period are extensively helpful to slow the progression of COVID-19.

The number of the collected studies for epidemic doubling time is 8 (less than 10). As a small sample size is likely to cause bias in meta-analysis (Lin, 2018), the visualization of the funnel plot in Figure 7 exhibits asymmetry.

Figure 7:

Funnel plot of meta-analysis for epidemic doubling time

We consider the trim-and-fill procedure (Duval and Richard, 2000) based on the imputations to reduce the bias owing to a small sample size, and the estimation results before and after implementing the trim-and-fill procedure are given in Figure 8. The estimate without correction is 4.86 (with a 95% confidence interval [3.26, 6.45]), which is larger than the trimmed estimate of value 3.48 (with a 95% confidence interval [1.60, 5.35]). Hence, even though the p-value of the Egger’s test is not significant, a remedy is still necessary, as an extensively small sample size usually jeopardizes the statistical power of the Egger’s test. In contrast, when we apply the trim-and-fill procedure to the previous three epidemic metrics, no significant difference has been detected.

Figure 8:

Forest plots of meta-analysis for epidemic doubling time before (top) and after (bottom) implementing the trim-and-fill procedure

Although there seems to be no official report on the epidemic doubling time of SARS from WHO or CDC, we find that an estimate of the doubling time of SARS from a published article (Galvani et al., 2003) is 16.3 through literature search. The estimate of the doubling time of SARS is three times more than the counterpart of COVID-19 based on our analysis, suggesting that COVID-19 is a more contagious disease.

Sensitivity analysis

Meta-analysis and sensitivity analysis usually go hand in hand. Meta analysis places focus on the summary of a systematic review of relevant studies, whereas the sensitivity analysis is used to assess the robustness of the results from the meta-analysis. In practice, the sensitivity analysis is a repeat of the meta-analysis, substituting alternative studies or the results from unclear studies. In other words, the goal of sensitivity analysis is to explore the impact of the meta-analysis by including or excluding studies in the meta-analysis based on some criteria as in Higgins et al. (2019, Section 9.7). In this section, we present a couple of sensitivity analyses for the basic reproduction number R0. Analogous studies can be carried out for other epidemic characteristics, done mutatis mutandis. For the sake of brevity, we present the details about the sensitivity analysis of R0 only in this section.

In the first sensitivity analysis, we only include the published articles (a total number of 7 left) in the new analysis, by leaving out preprints that have not yet been through the peer review process. The new estimate of R0 is 2.80 with a 95% confidence interval [2.05, 3.54]. It is unnecessary to use the prediction interval for inference in the new analysis since we have I2 = 19%, which is interpreted as unimportant to estimation consistency (Higgins et al., 2019, Section 9.5).

Having observed a small I2, we conduct a follow-up sensitivity analysis which includes only the 7 published articles and fit a FEM. The estimate of R0 decreases slightly to 2.70, and the 95% confidence interval gets narrower: [2.62, 2.77].

Now instead of leaving out ambiguous results, we expand the scope of our study to the East Asia. We add several studies from Japan, Korea and the Diamond Princess Cruise, which are listed in Table 6. The estimate of R0 for this analysis is 2.80 with a 95% confidence interval [2.05, 3.54] and a prediction interval [0.92, 4.69]. With 1 = 98%, we recommend using the prediction interval for interval-based inference.

Table 6:

Additional estimates and confidence intervals of basic reproduction number of COVID-19 for sensitivity analysis

		R0
Source	region	estimate	lower bound	upper bound
Kuniya (2020)	Japan	2.60	2.40	2.80
Sugishita et al. (2020)	Japan	1.99	1.89	2.09
Shim et al. (2020)	South Korea	1.50	1.40	1.60
Zhang et al. (2020b)	Diamond Princess Cruise	2.28	2.06	2.52
Zhao et al. (2020)	Diamond Princess Cruise	2.20	2.10	2.40

Concluding remarks

In this meta-analysis, we study a collection of recent studies on the COVID-19 pandemic, focusing mainly on four epidemic characteristics: or the basic reproduction number, the incubation period, the serial interval, and the doubling time of the epidemic. We summarize our numerical findings in Table 1 and include corresponding comparisons with SARS, which is also a viral respiratory illness caused by a coronavirus in 2003. From Table 1, we see that compared to SARS, the COVID-19 has a larger R0, a longer incubation period and much shorter doubling time, thus suggesting this novel coronavirus be more contagious and stringent public health strategies be necessary.

Table 1:

Summary of our numerical findings with respect to R0 or the basic reproduction number, the incubation period, the serial interval and the doubling time of the epidemic.

Metric	Est.	95% CI	95% PI	≥ or ≤ SARS
Basic Rep. Num.	3.16	[2.60, 3.72]	[1.25,5.08]	≥
Incub. Period	5.35	[4.29, 6.42]	[1.97,8.73]	≥
Serial Int.	5.35	[4.63, 6.07]	[3.13, 7.57]	≈
Epi. Doub. Time	3.48	[1.60, 5.35]	[−2.78, 9.74]	≤

The numerical results also provide insights on further studies. For example, with pooled estimates of R0 and the doubling time available, we can then compare them with the effective reproduction number (or R) or the doubling time after the nationwide lockdown protocol has been implemented in China to see the effectiveness of the public health strategies. The estimates of the incubation period and the serial interval reassure the necessity of reinforcing a 14-day quarantine period to prevent the spread of the disease.

It is also worthwhile noting the potential limitations of the current meta-analysis. To ensure the accuracy of the estimates collected, we do not include publications that report estimates with a 90% confidence interval or estimates with quartiles. This may lead to some loss of precision or publication bias for our estimates in meta-analysis. Meanwhile, the study on the COVID-19 may not be only restricted to the four characteristics listed in the manuscript. Other metrics, e.g., the case-to fatality rate and the testing capacity, and factors, e.g., significant clusters and patients’ underlying chronic medical conditions, are also of great importance in the future endeavors to mitigate the negative impacts of the COVID-19 pandemic.

Table 3:

Estimates and confidence intervals of incubation period of COVID-19 in China in the literature

	incubation period
Source	estimate	lower bound	upper bound
Backer et al. (2020)	6.40	5.70	7.70
He et al. (2020b)	2.30	0.80	3.00
Lauer et al. (2020)	5.10	4.50	5.80
Leung (2020)	6.90	5.50	8.30
Li et al. (2020a)	5.20	4.10	7.00
Linton et al. (2020)	5.00	4.40	5.60
Liu et al. (2020a)	4.80	2.20	9.40
Men et al. (2020)	5.84	2.91	8.75
Qin et al. (2020)	8.13	7.37	8.91
Sanche et al. (2020)	4.20	3.50	5.10
Xia et al. (2020)	4.90	4.40	5.40

Table 4:

Estimates and confidence intervals of serial interval of COVID-19 in China in the literature

	serial interval
Source	estimate	95% lower bound	upper bound
Du et al. (2020a)	6.30	5.20	7.60
Du et al. (2020b)	3.96	3.53	4.39
Du et al. (2020c)	5.29	4.72	5.86
He et al. (2020b)	5.20	4.10	6.40
Li et al. (2020a)	7.50	5.30	19.00
Li et al. (2020b)	6.27	5.62	6.98
Shao et al. (2020)	4.40	2.90	6.70
Tindale et al. (2020)	4.22	3.43	5.01
Wu et al. (2020b)	7.50	5.80	8.10
Xia et al. (2020)	4.10	0.80	7.40
You et al. (2020)	5.40	4.50	6.20
Zhang et al. (2020a)	5.10	1.30	11.60

Table 5:

Estimates and confidence intervals of epidemic doubling time of COVID-19 in China in the literature

	epidemic doubling time
Source	estimate	lower bound	upper bound
Du et al. (2020b)	7.31	6.26	9.66
Kraemer et al. (2020)	4.00	3.60	5.00
Lau et al. (2020)	4.00	3.50	4.30
Li et al. (2020a)	7.10	3.00	20.50
Muniz-Rodriguez et al. (2020)	1.80	1.50	2.30
Volz et al. (2020)	6.60	4.00	12.70
Wu et al. (2020a)	6.40	5.80	7.10
Wu et al. (2020b)	5.20	4.60	6.10