A Meta Analysis for the Basic Reproduction Number of 1 COVID-19 with Application in Evaluating the Eﬀectiveness of 2 Isolation Measures in Diﬀerent Countries

10 COVID-19 is quickly spreading around the world and carries along with it a signiﬁcant threat 11 to public health. This study sought to apply meta-analysis to more accurately estimate the 12 basic reproduction number ( R 0 ) because prior estimates of R 0 have a broad range from 1.95 13 to 6.47 in the existing literature. Utilizing meta-analysis techniques, we can determine a more 14 robust estimation of R 0 , which is substantially larger than that provided by the World Health 15 Organization (WHO). A susceptible-Infectious-removed (SIR) model for the new infection cases 16 based on R 0 from meta analysis is proposed to estimate the eﬀective reproduction number R t . 17 The curves of estimated R t values over time can illustrate that the isolation measures enforced 18 in China and South Korea were substantially more eﬀective in controlling COVID-19 compared 19 to the measures enacted early in both Italy and the United States. Finally, we present the daily 20 standardized infection cases per million population over time across countries, which is a good 21 index to indicate the eﬀectiveness of isolation measures on the prevention of COVID-19. This 22 standardized infection case determines whether the current infection severity status is out of 23 range of the national health capacity to care for patients. 24

by countries and even by states and local governments/institutions in the United States (US). 23 The duration and aggressiveness of isolation necessarily depend on the stage of the outbreak 24 for the affected country. For example, China has reached a buffer period by using early and 25 aggressive quarantine measures. As of this writing, most of Europe is currently at its peak 26 period, and the US remains in a period of exponential case growth which may be due to late, 27 inconsistent, and relatively permissive isolation measures enacted by states in absence of an early 28 and unified federal response. 29 The number of people who are infected during the peak period depends mainly on the 30 efficacy of a quarantine in the absence of a vaccine, and so a quarantine has been carried out to 31 decrease the effective reproduction number of COVID-19. From the term of epidemic principles, 32 the virus usually has an initial basic transmissibility R 0 . The basic reproduction number R 0 is 33 an important index to determine the epidemic intensity, and so many studies have been carried continues with the assumption of no resurge of the epidemic, the reproduction number will 37 drop below one. This means that each individual will, on average, infect less than one other 38 individual. After the effective reproduction number reaches one or less than one, the epidemic 39 will subsequently die off in a gradual manner. Also, the peak of the infection cases can be delayed 40 or reduced after government intervention by reducing the effective reproduction number R t , and 41 accordingly, it reduces the strain on healthcare systems which are set to run at near-capacity in 42 absence of an epidemic. 43 Therefore, the above epidemic scenarios motivated us to investigate the effectiveness of the 44 isolation policy implemented across different countries with real data because it is important for 45 public health to identify effective measures to prevent the spread of COVID-19. This study has 46 three purposes. First, since estimates of R 0 range widely (1.95 to 6.47) in the existing literature, 47 we utilize meta analyses to determine a more robust estimate for R 0 . Second, we apply a 1 susceptible-infected-removed (SIR) model for the new infection cases based on R 0 from our meta 2 analysis to estimate the effective reproduction number R t in order to evaluate the effectiveness 3 of isolation policies. Third, we standardize the infection cases to per million population as a 4 more conducive comparison of the distribution of COVID-19 and more readily show how the 5 infection case is beyond the health system capacity in some countries. We demonstrate that 6 the relative success of the isolation policy to control the effective reproduction number from the 7 statistical model based on real data. To this end, the results can supply some useful guidelines 8 for controlling the rapid spread of COVID-19 in the world. 9 One of the main contributions of this paper is that we give a robust estimator of R 0 from 10 meta analysis. Base on this estimated R 0 , we propose to use a Bayesian approach to estimate the 11 effective reproduction number R t from a SIR model, and then we use the effective reproduction 12 number R t to compare the effectiveness of the isolation measures across the countries. The rest 13 of this article is organized as follows. The proposed models are introduced in Section 2. Section 3 14 demonstrates the results from the proposed models. Conclusions are given in Section 4. 15 2 Method 16 We introduce the related statistical models in this section. Each infectious diseased has a repro-17 duction number. If the reproduction number is higher, the spread of the disease in the absence 18 of quarantine measures (government isolation policy) is greater. The number of infected patients 19 at time t depends on the infected patients at t − 1, the effective reproduction number, and the 20 government isolation policy to stop the virus rapidly spreading from person to person.

21
Let Y t denote the number of the infected cases at time t, and X t is the government isolation policy. We assume that Y t at time t is dependent on Y t−1 at time t − 1, the effective reproduction 23 number R t at the time t, and the government isolation policy X t . The government isolation policy 24 X includes the local government measures such as a balance between freedom and permissiveness. 25 We do not focus on the construction of the function F of Y t based on Y t−1 , R t , and X t in this 26 paper.

27
The basic reproduction number R 0 is an important pandemic index to indicate infection 28 intensity. The higher the basic reproduction number, the more people that will be infected given  Therefore, we propose the meta analysis to estimate it in Section 2.1. To estimate the effective 34 reproduction number, we propose to use an epidemic susceptible-infected model in Section 2.2.

35
This standardized infection case per million population allows us to compare the intervention 36 effects against COVID-19 across countries. If the infection case exceeds the healthcare capacity, 37 the pandemic will cause a higher mortality rate. Therefore, the theory of healthcare capacity is 38 given in Section 2.3, and the relationship between the healthcare capacity and the peak of the 39 infection case in real data is given in Section 3. 6.49. This huge difference of R 0 motivated us to estimate the basic reproduction number by the 8 scientific meta-analysis method, which is a statistical tool that combines the results of multiple 9 scientific studies. 10 It is known that meta-analysis can be used to address the same question in multiple scientific 11 studies, where each individual study reporting measurement was expected to have some degrees 12 of errors. And so one of the advantages of this approach is to allow us to use a meta-analysis 13 approach to derive a pooled estimate closest to the unknown common truth of R 0 . A benefit 14 of this approach was allowing us to aggregate the information leading to a higher statistical 15 power and a more robust point estimate than that is possible from the measure derived from  Based on this, we propose to use the random-effects meta analysis model, which was de-24 veloped by Hedges and Olkin (1985) and DerSimonian and Laird (1986). For K independent 25 studies, the random-effects meta analysis model is specified as

27
where R 0j is the estimate of R 0 from the j th study, µ j ∼ N (0, τ 2 ), j ∼ N (0, σ 2 j ), and j = 28 1, 2, . . . , K. The parameter τ 2 represents the between-study variability and is often referred to 29 the heterogeneity parameter. It represents the variability among the studies, beyond the sampling 30 variability. Our target is to estimate the true basic reproduction number R 0 . We propose to use 31 the following weighted average as the estimator for R 0 : This is a conditional standard error with the known τ 2 and σ 2 j .

37
There are many methods to estimate the between-study variability τ 2 and the within-study 38 variance σ 2 j . Most meteorologists used estimates s 2 1 , · · · , s 2 K of σ 1 2 , · · · , σ K 2 . For example, s 2 j = ((U CI j − LCI j )/2/1.96) 2 , where U CI j is the upper limit of the 95% confidence interval in j 1 study, and LCI j is the lower limit of the 95% confidence interval. Here we compare three main 2 methods to estimate, τ 2 , which causes different results forR 0 and its corresponding standard 3 error s.e.(R 0 ). These methods include the non-iterative methods proposed by Cochran (1954) 4 and DerSimonian and Laird (1986), and an iterative method by Paule and Mandel (1982). The 5 Cochran's ANOVA estimate for τ 2 is The DerSimonian and Laird estimator for τ 2 is where R 0B = K j=1 w j0 R 0j / K j=1 w j0 , and w j0 = 1/s 2 j . The Paule and Mandel estimator for τ 2 10 is the solution to the estimating equation determined through a simple iteration as shown in the paper DerSimonian and Kacker (2007).

28
where ∆Y t+δ = Y t+δ − Y t is the new infected cases over the period δ, β is the contact rate, and 29 R 0 is the basic reproduction number. The period δ is any time period. From this SIR model 4, 30 we obtain the effective reproduction number R t as follows: From Equation (5), the effective reproduction number R t reaches one when the number of new 1 infected cases reaches a peak point. For example, δ = 1 day, then we have a simple equation: For known R 0 and β, we can estimate R t from Equation (6). However, this simple approach is 4 not valid and robust because it relies on the two-day data information at the times of t and t + 1.

5
For example, lags in data reporting and the increase of nucleic acid testing capacity can cause 6 a bias for daily case reports. Therefore, Bettencourt and Ribeiro (2008) proposed a Bayesian 7 approach to estimate R t .

8
Here we give a brief summary of our proposed Bayesian algorithm to estimate the most 9 likely value of R t of COVID-19 based on the basic reproduction number R 0 from our meta 10 analysis in Section 2.1. For simplicity, we let δ = 1 and assuming β = 1. The probability mass 11 function P (∆Y t+1 |R t ) of new cases ∆Y t+1 in terms of R t is assumed to be a discrete probability 12 distribution. For example, a Poisson Distribution with the parameter λ t = exp((R t −1)/R 0 )∆Y t . 13 Using Bayes' rule, we have 15 If the posterior probability of the previous period, P (R t−1 |∆Y t ), is used to substitue the prior 16 probability P (R t ), then Equation (7) can be approximately approached by If iterate Equation (8)

21
In summary, the initial value of R t is estimated from Equation (5)  the hospital's capacity. 13 The different isolation measures have been put in place with the hope of reducing the overall 14 peak of COVID-19 infected cases in different countries. Early and aggressive isolation policies are indicated to be effective if the number of infection cases is represented by the hill curve. 1 Although the areas under both curves are equal, the hill curve never exceeds the capacity of the 2 healthcare system. For example, the areas under both the mountain curve and the hill curve are 3 the same with 10,000 infected cases in Figure 1, but the number of infected cases (1,000) at the 4 peak point in the hill curve was much lower than that (4,000) in the mountain curve. There are 5 not adequate resources to care for those with serious infections as well as those already requiring 6 hospital care for other health conditions if the peak number of patients is in the mountain curve. 7 The Chinese government has taken aggressive isolation measures. Contrary to China, Europe, countries will be given in result Section 3.3.

13
In this section, we apply the proposed methods in Section 2 to the real COVID-19 data in order 14 to answer the research questions addressed in Section 1.

The Effective Reproduction Number
We apply the SIR model by Bettencourt and Ribeiro (2008) to National COVID-19 data to 2 estimate the effective reproduction number R t in this section. The recent article by Vaidyanathan 3 (2020) gave an updated algorithm to estimate R t of COVID-19 based on the SIR model by 4 Bettencourt and Ribeiro (2008). Therefore, we use the updated algorithm of Vaidyanathan 5 (2020) to fit the curve of the effective reproduction number R t across countries. We let R 0 = 3.17 6 from our meta analysis, β = 1, and δ = 1 day in Equation.
We use a Gaussian smoother with a 7 14-day rolling window for the daily new cases by following the smooth approach by Vaidyanathan 8 (2020). 9 We choose the starting date to impose the social distancing isolation measures in each   Italy had the cases with 3,500 per million population as shown in Figure 4, and it was becoming 4 flat. The USA was increasing very fast, and it has reached over 4,500 per million population. 5 The curves of China and South Korea were very flat, indicating that these two countries have 6 controlled the epidemic in an effective way. Compared with South Korea, China has relatively 7 fewer infection cases per million population.

8
As mentioned in Section 2.3, the peak number of infected cases can be delayed or reduced 9 by implementing social distancing measures. However, if the peak curve exceeds the capacity of 10 the healthcare system without a successful intervention, especially with a substantial influx of 11 patients with a serious infection in the intensive care unit, this will result in a higher mortality 12 rate. 13 The daily infected new cases per million population over time across four countries are  This may partially explain why the mortality rate of 11.9% (13,155/110,574) in Italy was much 3 higher than the mortality rate of 3.7% (3,312/88,554) in China as of April 1st, 2020. In other 4 words, government measures, such as city lockdown, have not been put in place to effectively 5 reduce the overall peak of the infection case curve, which has been mentioned in Figure 1