key: cord-0589541-86vrgmvz
authors: Lee, Kwangmin; Kim, Seongmin; Jo, Seongil; Lee, Jaeyong
title: Estimation of World Seroprevalence of SARS-CoV-2 antibodies
date: 2022-01-31
journal: nan
DOI: nan
sha: 80e3b987da656dbd065c081da61c94f2b54eb619
doc_id: 589541
cord_uid: 86vrgmvz

In this paper, we estimate the seroprevalence against COVID-19 by country and derive the seroprevalence over the world. To estimate seroprevalence, we use serological surveys (also called the serosurveys) conducted within each country. When the serosurveys are incorporated to estimate world seroprevalence, there are two issues. First, there are countries in which a serological survey has not been conducted. Second, the sample collection dates differ from country to country. We attempt to tackle these problems using the vaccination data, confirmed cases data, and national statistics. We construct Bayesian models to estimate the numbers of people who have antibodies produced by infection or vaccination separately. For the number of people with antibodies due to infection, we develop a hierarchical model for combining the information included in both confirmed cases data and national statistics. At the same time, we propose regression models to estimate missing values in the vaccination data. As of 31st of July 2021, using the proposed methods, we obtain the 95% credible interval of the world seroprevalence as [38.6%, 59.2%].

At the beginning of December 2019, the first coronavirus disease 2019 (abbreviated COVID-19) patient, due to severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2), was identified in Wuhan, China (Lu et al.; . In the following weeks, the disease rapidly spread all over China and other countries, which caused worldwide damage and is still widespread. According to the official statement, COVID-19 has so far caused more than 317 million infections and 5.5 million deaths globally.

Vaccines are a critical tool for protecting people because of producing antibodies against infectious diseases. Every country in the world is struggling to block the spread of the virus and treat patients. As part of that, countries are administering COVID-19 vaccines, and the majority of people in many countries have been given the vaccines. There are a variety of available COVID-19 vaccines, e.g., AstraZeneca, Johnson & Johnson, Moderna, Novavax, and Pfizer-BioNTech, and candidates currently in Phase III clinical trials (Forni et al.; 2021) .

Seroprevalence is the ratio of people with antibodies, which is produced by previous infection or vaccines, to a particular virus in a population. In this paper, we study the seroprevalence for SARS-CoV-2 infections in people all over the world using information officially reported by countries. The available information includes confirmed cases, the number of people vaccinated, types of vaccines, and serosurvey data.

Recently, there have been various approaches for estimating the seroprevalence of antibodies to SARS-CoV-2. For example, Dong and Gao (2020) proposed a Bayesian method that uses a user-specific likelihood function being able to incorporate the variabilities of specificity and sensitivity of the antibody tests, Stringhini and et al. (2020) utilized a Bayesian logistic regression model with a random effect for the age and sex, and Kline et al. (2020) developed a Bayesian multilevel poststratification approach with multiple diagnostic tests. Lee et al. (2021) presented a Bayesian binomial model with an informative prior distribution based on clinical trial data of the plaque reduction neutralization test (PRNT), a kind of serology test.

Although these approaches are useful to estimate the SARS-CoV-2 seroprevalence, there is a limitation. The approaches have been developed for the populations in certain regions, not global. We here propose a new Bayesian method for estimating the seroprevalence of SARS-CoV-2 antibodies in the worldwide population. The method estimates the percentage of people with antibodies produced from viral infection and vaccines by country and takes a Bayesian hierarchical model to combine the estimated those. Additionally, the method utilizes informative priors constructed from external information. By doing so, we can provide the global seroprevalence estimates that reflect available information and uncertainty.

To the best of our knowledge, this is the first study on statistical modeling for estimation of the unknown seroprevalence of SARS-CoV-2 antibodies in the world population.

The rest of the paper is organized as follows. In the next section, we introduce the serology testing and vaccination datasets for the SARS-CoV-2 and briefly review the model proposed in Lee et al. (2021) for constructing an informative prior. In Section 3, we propose a new Bayesian approach to estimate the world seroprevalence of SARS-CoV-2. Section 4 presents the results of empirical analysis using real data. Finally, the conclusion is given in Section 5.

In this subsection, we introduce the notation used in the rest of the paper and describe datasets for estimating the number of effectively vaccinated people by country.

The datasets include the vaccinations, delivery amount of vaccines, and clinical trial data of vaccines.

We utilize the vaccination data given in Mathieu et al. (2021) , which is collected from official public reports on vaccinations against COVID-19 by country. The dataset contains the cumulative vaccine doses administrated, the cumulative number of fully vaccinated people, the report dates, and the information for vaccine manufacturers. As of 31 July 2021, the number of countries on reports is 182.

We denote the jth report date of the ith country using d i,j where i = 1, 2, . . . , 182 and j = 1, 2, . . . , J i , and the cumulative doses administrated and the cumulative number of fully vaccinated people until the date d ij are denoted by X i,j and Y i,j , respectively, for the jth report of the ith country. Note that X i,j is observed for all i and j, while Y i,j is not observed in some reports. Specifically, Y i,j is not observed at all in two countries, Cote d'Ivoire and Ethiopia, and is partially observed in 113 countries. We denote the set of vaccine manufacturers used at the corresponding date by V i,j . For example, if the vaccines produced by AstraZeneca and Pfizer-BioNTech are only used at the jth report date of the ith country, then V i,j = {AstraZeneca, Pfizer-BioNTech}.

We define X i,j,k as the cumulative doses by vaccines from the kth manufacturer for k = 1, . . . , K, where K is the number of vaccine manufacturers in the whole vaccination data. With this definition, we have X i,j = K k=1 X i,j,k . In the vaccination data we consider, X i,j,k are observed in 32 countries.

As of the 31st July, UNICEF COVID-19 vaccine market dashboard (2021) presents the delivery data, which refer to the amounts of doses that a country has received. The delivery data consist of publicly reported delivered vaccine amounts, including bilateral agreement, COVAX shipment, and donations. Among 182 countries providing vaccination reports (Section 2.1.1), the delivery data are available for 140 countries. We use these data for the estimation of missing values of X i,j,k .

Let D be the set of country indexes having the delivery amount data, and lets i,k be the delivery amount of the kth vaccine in the ith country, i ∈ D. We define s i,k as

which denotes the proportion of the kth vaccine delivered in the ith country. Note that for the case i / ∈ D, this definition is based on the assumption that the delivery amount of the kth vaccine in a country is affected by the total supply of this vaccine.

In the vaccination data introduced in Section 2.1.1, twelve kinds of vaccines are used. The name of the manufacturer identifies these vaccines, and the list is represented in Table 1 .

We divide these vaccines into three groups based on the required doses for one person, and we call these groups type 1, 2, and 3 vaccines. The numbering of the type represents the required doses for the full vaccination of one person.

There are research results on clinical trials for Pfizer, Moderna, AstraZeneca (AZ), Sputnik V, and Janssen. Each clinical trial is a randomized study with placebo and vaccinated groups. Let N (C) and N (V ) be the number of people in the placebo and vaccinated groups, respectively. The numbers of the COVID-19 confirmed cases among N (C) and N (V ) are observed, and denoted by n (C) and n (V ) . We present summarised results of clinical trial data in Table 2 . For the type 2 vaccines, two sets of clinical trials are conducted: one set is for those vaccinated with one dose, and the other set is for those fully vaccinated.

We introduce the serological survey data from SeroTracker, a knowledge hub of COVID-19

serosurveillance (Arora et al.; 2021) . In the serological survey data, we use only nationwide survey data, i.e., we exclude the survey data for sub-population such as a group of health care workers. As of the 31st of July 2021, there are 126 nationwide serological surveys from 48 countries. Each serological survey has its sampling period. The histogram of the last dates in the sampling periods is shown in Figure 1 .

We present a Bayesian method to estimate the seroprevalence. Specifically, we propose the method for estimation of the seroprevalence based on the two parts: the proportions of the effectively vaccinated and of the infected, which are denoted by θ (V ) and θ (I) , respectively.

Recall that the effectively vaccinated are people with antibodies produced from vaccines and that the infected are those who have gotten the antibodies by infection. To propose the method, we define the seroprevalence θ i (t) at t date of the ith country as

where the product terms θ i (t) in next two subsections 3.1 and 3.2, respectively, for each country and report date.

For the estimation of θ at the jth report date in the ith country. Note that the index j in M i,j indicate the report index of the vaccination data (Section 2.1.1), and vaccination reports are not given for

and P i is the population of the ith country. When there is no report in date t, we use the most recent report from date t. Thus, we focus on the estimation of M i,j for the estimation of θ k ∈ [0, 1] be the efficacies of the kth vaccines for the fully vaccinated people and thoses who have at least one dose but have not finished the required doses, respectively. We assume that the distribution of M i,j is

where d(k) denotes the required doses of the kth vaccine. Under the assumption that the numbers of the people vaccinated once and twice are the same, 2(X i,j,k − 3Y i,j,k )/3 is equal to the number of people who have at least one dose of vaccination, but have not finished the required number of vaccination. We are aware that this assumption is not warranted, but since the vaccine requiring 3 doses is used only in one country, Uzbekistan, we believe that the effect of the assumption is not critical.

k are not observed, we need statistical models for these variables. In the following subsections, we describe these models.

We consider a multinomial regression model for X i,j,k given X i,j and s i,k , which are defined in Sections 2.1.1 and 2.1.2, respectively. Let X i,j = (X i,j,1 , X i,j,2 , . . . , X i,j,K ) ∈ R K be the response vector and w i,j = (w i,j,1 , . . . , w i,j,K ) ∈ R K be a covariate vector, which is to be defined with s i,k and X i,j for j ∈ [j], where [n] := {1, 2, . . . , n} for a positive integer n. We assume

for all x, y ∈ [K]. The equation (3) means that the ratio of usage probability of the xth vaccine to that of the yth vaccine, p i,j,x /p i,j,y , is proportional to the ratio of w i,j,x to w i,j,y after logarithm transformation. This assumption is examined via visualization after the definition of w i,j .

We now define w i,j using the variables for delivery amount s i,k and the numbers of

. In the definition of w i,j , we reflect the idea that w i,j,k is positively dependent both on the delivery amount of the kth vaccine in the ith country and the period during which the kth vaccine is used. First, let

where

defined by multiplying the number of doses administrated at the date of the j th report, dX i,j , to the delivery amount of the kth vaccine in the ith country if the kth vaccine is used at this date. Otherwise, we set dw i,j ,k as zero. Then, we define w i,j := j ≤j dw i,j . Figure 2 is the scatter plot for the points in the set {(log(X i,j,x /X i,j,y ), log(w i,j,x /w i,j,y )) :

both of X i,j,x and X i,j,y are observed}, and shows that the linearlity assumption in (3) is reasonable.

We assign a non-informative prior distribution for β (V 1 ) :

Theorem 3.1 shows that the posterior distribution under the flat prior is proper. The proof is given in the supplementary material. 

both of X i,j,x and X i,j,y are observed.}.

There are missing values in Y i,j (the cumulative number of fully vaccinated people at the jth report date of the ith country), and we propose a distribution for the missing values.

To do this, we first present methods for three simple cases in which only one type of vaccines are used in the country i up to the report date d i,j , and then expand those to the method for the general case in which mixed types of vaccines are used in the country i up to the report date d i,j .

In Case 1 in which only type 1 vaccines are used, Y i,j is easily derived from X i,j since the vaccination is completed with only one dose. Thus, we have

In Case 2, in which only type 2 vaccines are used, we employ the Poisson distribution to the random variable X i,j − 2Y i,j . Note that X i,j − 2Y i,j is the number of the doses administrated to people who have gotten one dose but not finished vaccination as of the jth report date of the ith country. We assume that the longer the interval between the first and the last doses is, the larger X i,j − 2Y i,j is. We also assume that the larger the doses recently administrated is, the larger X i,j − 2Y i,j is.

To specify the doses recently administrated, we address the relation between the report index j and the corresponding report date. For each report index j, d i,j is defined as the report date, and d i,j satisfies d i,1 < d i,2 < . . . < d i,J i . In the vaccination data, there exists an index j such that d i,j − d i,j−1 > 1, i.e. the reports are not given for everyday. When we need vaccination data for date d with {d : d i,j = d, j = 1, . . . , J i } = ∅, we use the data from the closest report. Specifically, we define j * (j, δ; i), to indicate the closest report index from date d j − δ, as

for country index i, report index j and positive integer δ. According to the definition of j * (j, δ; i), when there are more than one minimizer in argmin j ≤j−1 |d i,j − d i,j − δ|, we use the smallest index. In this paper, we set δ = 21, and if there is no confusion, we let j * denote j * (j, δ; i). Using the definition of j * , we define

representing the average of daily doses recently administrated, and we define W i,j = Z i,j T approximating the doses administrated for recent T days, where T is the required interval between the first and last doses.

Supposing only one kind of type 2 vaccine is used, we propose the regression model

This model reflects the assumptions that (X i,j − 2Y i,j ) is positively related to the doses administrated for recent T days. Recall that X i,j − 2Y i,j is the number of the doses administrated to people who have gotten one dose but not finished vaccination as of the jth report date of the ith country. We suppose that people who have gotten only one dose had the first dose in recent T days based on the required interval.

The model (6) can be used only when one kind of type 2 vaccine is used. We expand (6) to consider the case when K kinds of type 2 vaccines are possibly used, where K is a positive integer larger than 1. We substitute T in W i,j to the weighted mean of the intervals as K k=1 w * i,j,k T k . Here T k is the required interval between the first and last doses of the kth vaccine. We define w * i,j,k as

Recall the definition of dw i,j ,k in (4). The variable dw i,j ,k is zero when the kth vaccine is not used at the j th report date of the ith country; otherwise, this variable represents the delivery amount of the kth vaccine in the ith country multiplied by the doses administrated at the corresponding date. Thus, w * i,j,k is constructed from the three factors: the delivery amount, the doses administrated during recent d i,j −d i,j * days, and whether the kth vaccine is used. Using the weighted mean of the intervals, we define W (6). We suggest the distribution for Case 2 as

where V (2) is the index set for type 2 vaccines.

Next, we propose a model for Case 3, in which only type 3 vaccines are used, using the similar idea as in Case 2. To do this, we use the random variable X i,j − 3Y i,j instead of X i,j − 2Y i,j . Here the variable X i,j − 3Y i,j represents the doses administrated to people who have not finished vaccination. Then we consider the Poisson model as

w * i,j,k T k , and V (3) is the index set of type 3 vaccines. We can re-express this distribution as

i,j ))).

Finally, we combine the models (5), (8) and (9) to construct the model for general case.

Let q l be the weight of type l vaccines for l = 1, 2, 3 with q 1 + q 2 + q 3 = 1, which are defined as q l = k∈V (l) w * i,j,k for l = 1, 2, 3. By combining (5), (8) and (9), we propose the generalized model as

i,j )))).

We choose the flat prior distribution on β 1 and β 0 ,

The following theorem shows that the prior induces the proper posterior distribution. The proof for this theorem is given in the supplementary material.

Theorem 3.2. Let n be a positive integer with n ≥ 2, and let x 1 , x 2 , . . . , x n ∈ R and y 1 , y 2 , . . . , y n ∈ N. If there exists a pair of indexes i and j such that

In this subsection, we provide a distribution for Y i,j,k given Y i,j and X i,j for j ∈ [j]. This distribution is based on the following three premises:

The first premise is obvious from the definitions of Y ijk and Y ij , and the second premise is based on the definitions of X ijk and Y ijk . When a type 1 vaccine is considered, the number of fully vaccinated people Y i,j,k coincides with the number of doses X j,j,k since only one dose is required for this type of vaccine. Next, we address the third premise. Recall that T k is the interval between first and last doses of the kth manufacturer's vaccine, and

Those who have gotten the first dose of the kth vaccine until the j * (j, T k ; i)th report date are expected to be fully vaccinated until jth report date. Thus, we assume that Y i,j,k is positively dependent on X i,j * (j,T k ;i),k .

Using the premises, we suggest a distribution for Y i,j,k for k / ∈ V (1) . We letỸ ij = (Y i,j,k(1) , Y i,j,k(2) , . . . , Y i,j,k(K) ), which is the vector comprised of Y i,j,k s excluding the type 1 vaccines. Likewise we letX ij * = (X i,j * (j,T k(1) ;i),k(1) , X i,j * (j,T k(2) ;i),k(2) , . . . , X i,j * (j,T k(K) ;i),k(K) ).

GivenX i,j * , Y i,j and X i,j,k , we suggest the distribution forỸ i,j as

We propose a hierarchical model to estimate E k . The hierarchical model extends the Bayesian method in Graziani (2020) to analyze the clinical trial data of vaccines in Section 2.1.3. Here, we review the Bayesian method by Graziani (2020) analyzing a clinical data set. Let N (V ) and N (P ) be the numbers of vaccinated and placebo groups, and let n (V ) and n (P ) be the number of those who confirmed and N (P ) , respectively. We let µ (V ) and µ (P ) denote the expected values of n (V ) and n (P ) , respectively. The efficacy parameter E ∈ [0, 1] is defined from µ (V ) and µ (P ) as Graziani (2020) assumes that n (V ) and n (P ) follow the Poisson distribution, and we have n (V ) ∼ P ois(µ (V ) ), n (P ) ∼ P ois(µ (P ) ),

By introducing a parameter λ := µ (V ) + µ (P ) , the parameters ( We extend the model (11) to analyze more than one clinical data set for different vaccines. Suppose we haveK data sets as (N

k , λ k and E k for k = 1, . . . ,K in the same way of model (11). We propose the hierarchical model as

The difference of this model from (11) is that E k is assumed to follow Beta(α v , β v ), where α v and β v are hyper-parameters. As in model (11), the prior choice on λ k are not significant for the estimation of E k . We use the empirical Bayesian method for the hyper-parameters α v and β v , i.e., we estimate these values as the maximizer of the marginal likelihood.

We analyze the clinical data of vaccines, data in Table 2 of Section 2.1.3, using the hierarchical model. We divide the data into two groups: partially vaccinated and fully vaccinated groups. The fully vaccinated group includes the clinical trial data of Pfizer, Moderna, AstraZeneca, and Sputnik with two doses and Janssen with dose 1. The other data in Table 2 are included in the partially vaccinated group. We apply the model (12) to each group separately, and we obtain efficacies for partially and fully vaccinated. Note that the hyper-parameters α v and β v are also estimated for each group.

Finally, we show how this method is used for the estimation of effectively vaccinated population (1). Recall that, in (1), distributions of E (12)).

In this section, we propose a method to estimate θ (I) i (t) using a hierarchical model, an extension of the model (13) proposed by Lee et al. (2021) ,

where N is the number of subjects in a serosurvey, X is the number of subjects who is testpositive, p + and p − are sensitivity and specificity of the serology test, respectively, and θ is the seroprevalence. While model (13) is used for the analysis of one set of serosurvey in a country, we suggest the hierarchical model to analyze the serosurvey data over countries given in Section 2.2.

First, we introduce a reparameterized form of model (13) in Section 3.2.1, and we propose the hierachical model in Section 3.2.2 using the reparameterized model. We introduce notations for this section. We use 126 serosurveys introduced in Section 2.2, and let N l and X l denote the numbers of survey samples and test-positive samples in the lth serosurvey, respectively, l = 1, . . . , 126. The index i l represents the country index in which the lth serosurvey is conducted, and the index t l indicates the last date in the sampling period of the lth serosurvey.

We reparametrize model (13) since we are interested in the seroprevalence by infection

i l (t l ), l = 1, . . . , 126, where p + l and p − l are the sensitivity and specificity of the serology test used in the lth survey, respectively. Recall that θ (I) i l (t l ) and θ (V ) i l (t l ) denote the seroprevalence by infection and the proportion of the effectively vaccinated, respectively, in the i l th country at t l date, and the product term θ

i l (t l ) represents the cases in which the infected are vaccinated without the knowledge of infection. If a serosurvey is conducted before vaccination, then θ i l (t l ) = θ (I) i l (t l ). Note that among 126 serosurveys, 105 surveys are conducted before vaccination.

We construct a prior distribution on θ (V ) i l (t l ) from the number of effectively vaccinated in (1), divided by the population. Recall that the distribution for the number of effectively vaccinated is derived only for dates when the vaccination report is provided. If there is no vaccination report of the i l th country in date t l , we use the most recent report from the date. The prior distributions for the other parameters are concerned in the next section.

We propose a hierarchical model to analyze the serosurvey data over countries. Let θ (C) i l (t l ) denote the proportion of the cumulative confirmed cases, which is referred to as confirmed ratio in the i l th country at t l date, respectively. We assume that random variable

i l (t l )) is explained by country-specific random effect and country statistics: population density and GDP per capita of the corresponding country. Note that the ran-

i l (t l ) represents the ratio of the number of infected to that of confirmed. We represent this assumption as

where P D i l and G i l are the standardized log population density and log of GDP per capita, and T N (a,b) (µ, σ 2 ) is the truncated normal distribution with mean µ, covariance σ 2 and the range of (a, b). Combining (14) and (15), we construct the hierachical model as

Next, we describe prior distributions on θ

2 , p + l and p − l . As suggested in Section 3.2.1, we use the distribution (1) for the prior on θ (V ) i l (t l ). Gelman et al. (2006) suggested the flat prior for the standard deviation σ in hierarchical models, and they also showed that this prior gives the proper posterior distribution when flat priors are given for other parameters, µ 0 , τ , β (I) 1 and β (I) 2 for our model. For p + l and p − l , we construct prior distributions based on the method in Section 4 of Lee et al. (2021) .

We give the detail in supplementary material.

In this section, we give the results of the Bayesian inference for the regression models and the hierarchical models in Section 3, and we give the results of world seroprevalence estimation. In Section 3.1, we proposed regression models (2) and (10) to estimate missing variables X i,j,k and Y i,j , and we proposed the hierarchical model (12) to estimate the vaccine efficacies. In Section 3.2, we proposed the hierachical model (16) in models (2) and (10). 

The regression coefficients explain the relation between X i,j −2Y i,j and W i,j via (17). Recall that the random variable X i,j − 2Y i,j represents the number of the doses administrated to people who have gotten one dose but not finished vaccination, and W i,j approximates the doses administrated for recent T days, where T is the required interval of the vaccine.

We give the posterior distributions of β 

The left term represents the log ratio of the seroprevalence by infection to the confirmed ratio, and β Next, we give the posterior distributions of the vaccine efficacy parameter E k in Figure   5 . The efficacies of Pfizer, Sputnik V, and Modena with fully vaccinated attain at least 90%. While the difference of efficacies between the partially and fully vaccinated is slight for AstraZeneca, the difference is big for Pfizer.

We derive the predictive posterior distributions of θ i (t) denote the proportion of the effectively vaccinated population and seroprevalence by infection of the ith country at t date, respectively. We also define seroprevalence of the ith country at t date as

The predictive posterior distribution of θ (1), divided by the population P i . Recall that the index j in M i,j indicate report index, and reports are not given for everyday. When there is no report in date t, we use the most recent report from date t. The predictive posterior distribution 

in (15), given θ (C) i (t), P D i and G i . Next, we define the trend of world seroprevalences using θ We also present a treemap in Figure 7 , which shows the posterior means of seroprevalences by country on 31st July 2021. The seroprevalences of China and India are 51% and 62%, respectively, which are similar to the world seroprevalence on this date. France and UK attain over 80% seroprevalence on this date.

We have proposed a novel Bayesian approach to estimate the seroprevalence of COVID-19 antibodies in the global population. The approach first estimated the seroprevalences by infection and vaccination by country and then took a Bayesian hierarchical model to provide the world seroprevalence by combining the estimated those. We also constructed informative priors by utilizing external information such as clinical trial data.

There are many studies on the estimation of seroprevalence in a population. However, these studies focus on estimating the seroprevalence on the date and country in which the sample is collected, and hence the estimation of the world seroprevalence is not apparent. July 2021. Each tile represents a country, and its area is proportional to the corresponding population. The color and the value in each tile represent the seroprevalence θ i (t) when t is 31st July 2021.

Furthermore, the previous works on the vaccination data were mainly on the cumulative doses administrated and the fully vaccinated population, while the method proposed in the paper predicted the effective vaccinated population using the information on the efficacies of vaccines.

The methods in this paper can be improved. First, in the hierarchical model for the seroprevalence of infection, other covariates can be explored and used for the model.

The covariates we used are national statistics which does not depend on the date factor.

Thus, we expect that explanatory power can be improved by adding the date-dependent covariate, such as the daily number of COVID tests in a country. Second, the model can be improved by considering the sampling period since we just use the last day of the sampling period. Finally, the current study is based on the data up to July 2020 and has the limitation of not considering the decline of neutralizing antibodies in vaccinated people. Therefore, the results can be improved by updating the data and incorporating the decline of neutralizing antibodies in the model.

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The first and second authors equally contributed to this work.