key: cord-0861288-1bfd9qvh
authors: Lawless, Jerald F.; Yan, Ping
title: On testing for infections during epidemics, with application to Covid-19 in Ontario, Canada
date: 2021-07-22
journal: Infect Dis Model
DOI: 10.1016/j.idm.2021.07.003
sha: 88c670072241765150fac6a9d2ad5238177ce0aa
doc_id: 861288
cord_uid: 1bfd9qvh

During an epidemic, accurate estimation of the numbers of viral infections in different regions and groups is important for understanding transmission and guiding public health actions. This depends on effective testing strategies that identify a high proportion of infections (that is, provide high ascertainment rates). For the novel coronavirus SARS-CoV-2, ascertainment rates do not appear to be high in most jurisdictions, but quantitative analysis of testing has been limited. We provide statistical models for studying testing and ascertainment rates, and illustrate them on public data on testing and case counts in Ontario, Canada.

Accurate estimation of the extent and location of viral infections during an epidemic is important for understanding transmission, estimating hospitalization and fatality rates, and informing public health actions. This requires effective testing strategies, especially if many infected persons remain asymptomatic. 5 For the novel coronavirus SARS-CoV-2 and Covid-19 disease, official counts of tests, confirmed cases and test positivity rates give only a partial picture. Confirmed case counts and positivity rates depend on testing rates, testing criteria,

individuals' willingness to be tested, and other factors, and a substantial proportion of cases remains unidentified. Our objective is to discuss testing models 10 for SARS-CoV-2 and consider testing strategies, with reference to experience in the Canadian province of Ontario.

Since the arrival of SARS-CoV-2 in Canada in January 2020 (Marchand-Senécal et al., 2020) , provinces and territories have provided daily and weekly updates on the epidemic. All jurisdictions report newly confirmed cases (infec-15 tions), deaths and hospitalizations along with breakdowns by sex, age, presumed exposure and other factors (e.g. see Government of Alberta, 2021; Government of Ontario, 2021a,b). Other public data include the daily numbers of tests processed and the corresponding positivity rates. Testing levels increased greatly between March and December 2020, and in Ontario, have been fairly stable 20 since January 2021. We will examine test and case counts up to the end of January 2021 later in the paper.

Our specific aims are to consider the connection between testing, case counts and infection rates, and to examine variation in testing rates. We provide simple statistical models motivated by the SARS-COV-2 pandemic, discuss testing 25 strategies, and how more detailed data and random testing would enable estimation of ascertainment rates and thus better assessment of strategies. Section 2 introduces the statistical framework and Section 3 discusses testing strategies. Section 4 considers analysis of testing rates and Section 5 illustrates the models and methods using data from the province of Ontario. Section 6 makes We consider test and case counts in some population for time periods indexed by the variable t. We first set notation for an arbitrary time period, and suppress 35 the value of t for convenience: let N denote the population size, n the number of tests with results in the time period, C test the number of positive tests, and C pop the (unknown) number of active but unidentified cases (infections) in the population just before the current time period. We also define rates related to the data: the test positivity rate, P R = C test /n; the testing rate, 40 T R = n/N ; the confirmed case rate CCR = C test /N ; and the proportion of the population infected but not previously identified, p = C pop /N . We will treat T R as the proportion of the population with test results in a specific time period;

similarly we treat C test as the number of persons testing positive. This is a slight idealization, since some individuals may be tested more than once. We 45 typically consider days and weeks for time periods and assume the numbers of repeat tests in a period is negligible. Over longer periods a difficulty in allowing for repeated testing is that many jurisdictions publish only the numbers of tests and positive tests, and not the numbers of distinct persons tested. We note that C test is distinct from the number of new cases reported during the same 50 time period, which we denote by C. The count C includes some persons whose positive test results were in a previous time period, and depends on the time between a case being confirmed in a laboratory, reported to a local public health unit, and then to a higher authority. Conversely, some of the cases included in C test are not reported in the current time period. 55 To address testing effectiveness we define an additional rate, T RI = C test /C pop , which is the proportion of infected but currently unconfirmed persons in the population who are tested (or "test rate for infecteds"). For now we assume that tests have perfect sensitivity and specificity, so that a tested person is infected if and only if the test is positive. Imperfect tests are discussed later in this section.

With a perfect test the confirmed cases are all true cases, and the defined rates 3 J o u r n a l P r e -p r o o f satisfy the relationship

(1)

The rates CCR, T R and P R are known, but p and T RI are not.

The numbers of tests completed on a given day and the number of positive tests are random outcomes that depend on several factors, including personal behaviour in accessing testing and random variation in collecting and processing specimens. For a random individual in the population, excluding persons previously confirmed to be infected, let

The key parameters for the joint distribution of Y, ∆ are

where π 1 and π 0 are the testing rates among infected and non-infected persons, and θ = π 1 p + π 0 (1 − p) is the overall testing rate. Table 1 shows the joint probabilities for Y and ∆; for a given person, Y is known only if they are tested (∆ = 1). 

Letting r = E(CCR), then analogous to (1), we have the relationship

4 J o u r n a l P r e -p r o o f where γ = π 1 p/θ = P (Y = 1|∆ = 1) = E(P R|n) = E(P R). The probability π 1 applies to tests completed in a specific time period, and is different from the overall probability an infection is (eventually) identified by testing, which is discussed below. Data (n, C test ) for a given time period give estimates θ = 65 T R = n/N, γ = P R = C test /n and r = CCR = C test /N ; the rates p and π 1 are not separately estimable.

When repeat tests on individuals in a time period are negligible, the relationships (1) and (2) indicate that a trend in confirmed case counts or positivity rates across successive time periods does not imply a corresponding trend in ac-70 tual cases or p; trends in θ and π 1 also need to be considered. The value of π 1 is unknown in most settings, but we note two extreme situations: (i) all unidentified infected persons get tested, so π 1 = 1, and (ii) those tested are a random sample of the population (excluding confirmed cases), so that π 1 = θ. Neither situation is realistic across a large heterogeneous population, but sometimes ap-75 ply to specific groups within the population. For example, in groups with severe outbreaks everyone might be tested, so π 1 = 1. Situation (ii) applies in the case of random surveillance testing, used to obtain an estimate of the true infection rate (Office of National Statistics U.K., 2020).

A primary aim of testing is to identify a high proportion of cases. Success 80 depends on how effectively testing targets those who are infected, as measured by the rate π 1 . The ascertainment rate, or overall probability a case is identified through testing, is denoted as π * for infections occurring over some specific time period, whereas π 1 applies to tests reported for a given time period. The value of π * depends on the accuracy of tests and on the values of π 1 for about two 85 weeks following the time of infection; we discuss this in the next section.

To consider multiple time periods we denote infection, testing and expected positivity rates for time period t as p(t), θ(t), γ(t) respectively; similarly we have π 1 (t), π 0 (t), Y (t), ∆(t), n(t), C test (t), P R(t) and so on. Days are the 

where d(t) is the proportion of unidentified cases at the end of day t − 1 that remain detectable on day t. In terms of testing and infection rates this relationship becomes

where i(t) = I(t)/N . Thus p(t) is determined by the rate i(t) of newly detectable 95 infections and the rates at which existing infections are either identified through testing or progress to the point where they are no longer detectable.

If the values of π 1 (t) were close to one, most new infections would be identified soon after they become detectable. This is rarely the case because of limited tracing and testing resources, and in most settings the daily values of π 1 (t) are 100 small. They can be related to the overall probability an infection is detected, or the ascertainment rate, π * ; to consider variation over time we let π * (t) denote the probability an infection that first becomes detectable on day t is eventually confirmed. If the maximum time an infection remains detectable is s max days and if π * (s; t) denotes the probability an infection that becomes detectable on 105 day t is confirmed on day s (s = t, t + 1, . . . , t + s max ), then π * (t) = t+smax s=t π * (s; t). The π * (s; t) may vary with s − t; they also vary with s if testing levels and strategies change over time. In the case where dependence on s − t is negligible, π * (s; t) = π 1 (s) and π * (t) = π 1 (t, t + s max ), where π 1 (a, b) denotes the sum of rates π 1 (s) over days a to b.

Ascertainment rates have been estimated in various countries and regions using a number of methods (e.g. Pullano et al., 2021; Shaman, 2021) . Repeated random testing is the only reliable way to estimate p(t) and thus π 1 (t), and some countries have regular random surveillance surveys; we discuss this in Section 3.

Random testing has not been widespread in Canada, but antibody seropreva- 

Confirmed case and positivity rates can vary widely across parts of a population, according to age, residence, place of work and other factors. To consider variation in testing and case rates rates we suppose the population or some part of it has been partitioned into strata. In a population (of size N ) with strata S 1 , S 2 , . . . , S K let z denote which stratum an individual is in. Let N k be the 7 J o u r n a l P r e -p r o o f size of stratum k, so for a randomly chosen person, λ k = P r(z = k) = N k /N .

Suppressing notation for the time period, we now have θ k = P r(∆ = 1 | z = k), p k = P r(Y = 1 | z = k), and expected confirmed case rates r k satisfy r k = p k π 1k = θ k γ k , for k = 1, . . . , K.. The overall expected positivity rate is

and the infection rate is p = P r(Y = 1) = k p k λ k . If a higher proportion of testing is allocated to strata with higher positivity rates, the effect is to increase γ and r. However, information about the π 1k is needed to determine the effect on

and the ascertainment rate. Section 3 considers how testing might be allocated across strata.

Tests are almost never perfect, and may give false positive and false negative results. In this case we observe Y = I(test is positive) for an individual who is tested, and Y replaces Y in Table 1 ; in addition we denote the number of positive tests in some time period as C test and the positivity rate as P R. If α = P r( Y = 1|Y = 1) is the sensitivity of the test and β = P r( Y = 1|Y = 0) is one minus the specificity, or false positive rate, then Table 1 implies that P R, CCR = C test /N and the estimate p = P R(T R)/π 1 have expectations

where C test (t) is the true number of infected persons among those tested. This

suggests the estimate 

PCR tests (also known as viral RNA or nucleic acid tests) for SARS-CoV-2 have very low false positive rates β (high specificity) but false negative rates α can be substantial (e.g. see Burstyn et al., 2020; Mina et al., 2020) . In this case the infection and positivity rates are underestimated by the factor α. More rapid diagnostic tests such as antigen tests (Mina et al., 2020) also have high 140 specificity but lower sensitivity than a PCR test. The sensitivity of most tests depends on the time since infection (Mina et al., 2020) and to integrate this with variation in infection and testing rates over time, we can utilize the framework of Section 2.2. In the context of (4), the probability π * (s, t) an infection that first becomes detectable on day t is confirmed on day s will depend on the 145 test sensitivity α(s − t) on day s as well as the testing rate π(s) on day s. It might also depend further on s − t because as symptoms become more severe, an individual is more likely to seek a test. Detailed modelling and estimation of π * (s, t) requires more comprehensive data than are usually available. In the absence of such information we suggest an approximation π * (s, t) = απ 1 (s),

where α is the average sensitivity. The probability π * (t) that an infection which becomes detectable on day t is eventually confirmed is then given by (4) as απ 1 (t, t + s max ). We discuss ascertainment rates further in Section 3.3.

Jurisdictions usually target certain groups such as health care workers for regular 155 testing, and provide criteria for who should be tested in the general population.

A report from the European Centre for Disease Prevention and Control (ECDC) (2020) discussed five objectives of testing: A -controlling transmission, B -monitoring incidence and trends and assessing severity (of outcomes) over time, Cmitigating the impact of COVID-19 in healthcare and social care settings, D - is scant evidence concerning ascertainment rates; random surveillance testing has been very limited (Waldner et al., 2021) .

The testing objectives above require that infections be identified rapidly; this 170 requires high values for daily rates π 1 (t), especially in places where transmission risk and potential severity of disease are high. We consider testing strategies by assuming a population, group or region is divided into strata that reflect testing criteria and infection levels. Our discussion will focus on accurate PCR tests; rapid tests are discussed at the end of Section 3.2 and in Section 6.

Suppose first the objective is to identify the largest possible number of cases across a population or equivalently, to maximize the population's ascertainment rate. For a given time period, if K strata can be ordered such that p 1 ≥ p 2 ≥ · · · ≥ p K , and if no further subdivision of strata according to likelihood of infection is possible, the best strategy for a total allotment of n tests would 180 be to allot n k tests to S k (k = 1, . . . , K) as follows: test as many persons as possible from S 1 , then if n 1 < n, test as many as possible from S 2 , and so on. That is, we take n 1 = min(n, N 1 ), n 2 = max(0, min(n − n 1 , N 2 )), . . . , n k = max(0, min(n−n 1 −· · ·−n k−1 , N K )). Since the true rates p k are unknown, we could instead order strata according to their recent confirmed case rates.

More refined objectives would prioritize the identification of infections in critical strata with high transmission risk or potential disease severity. This would call for testing even when recent case rates were low. The ability to target testing so as to maximize π 1k (t) in the face of limited resources would remain crucial. Asymptomatic infections are a particular problem. To identify 190 them effectively we must have other predictors of infection that can be used to define strata; main ones are close contact with a known case or exposure to populations with high case rates. If such predictors are not identified and if a 10 J o u r n a l P r e -p r o o f substantial proportion of infections is asymptomatic, a high ascertainment rate will be impossible to obtain without very high testing rates.

Another potential objective is to estimate the proportion p k of a stratum that is currently infected but not yet confirmed. As discussed earlier, a random sample from stratum k would enable this; in that case P R k is an estimate of p k . An estimate of p k enables estimation of the testing rate π 1k under regular non-random diagnostic testing; this is discussed in section 3.3. 

An alternative approach to estimation of ascertainment rates when the population is stratified is through an assumption of conditionally independent testing.

This requires that those tested within each stratum in a given period are a random sample from the stratum or, in the notation of Section 2, that ∆ is 205 independent of Y , given z. We assumed this in our discussion of testing to maximize the number of infections identified in the preceding section. This assumption is restrictive, but may be a reasonable approximation when the strata partition the population into relatively homogeneous groups according to testing criteria, access to testing and likelihood of infection; persons with Covid-like 210 symptoms would be in different strata than those without symptoms. The positivity rate P R k in stratum k then estimates the infection rate p k , and π 1k = θ k , so from (6) the overall ascertainment rate is estimated by

Exact stratum sizes N k are typically unknown so the λ k and θ k are in that case estimates.

To achieve a high overall ascertainment rate it is necessary to have high testing rates in strata with high positivity or case rates, particularly when the strata are large. This was the basis for the optimal testing allocation discussed in Section 3.1. More generally, the relationship (6) with π 1k = θ k shows the effects of stratum-specific random testing rates on overall ascertainment. For factor α and from (6) the effective ascertainment rate is reduced by the same factor. If testing is random within strata then γ k = p k but in any case, the tradeoff between decreased α and increased testing rates θ k can be determined.

If α = 0.5, for example, testing rates would need to be doubled to achieve the same ascertainment rate as when α = 1. 

To estimate ascertainment rates for regular diagnostic testing we need information on the true numbers of infections in a population, which can then be compared with cases identified by regular testing. One approach has been to use unverifiable modelling assumptions involving transmission or fatality rates 250 (e.g. Shaman, 2021 , Dougherty et al., 2021 , but the only reliable method is 12 J o u r n a l P r e -p r o o f An estimate p(t) of the infection rate p(t) for some population or group in time period t can be combined with data from regular diagnostic testing so as to estimate the positive testing rate π 1 (t) for regular testing in period t. By (2) 13 J o u r n a l P r e -p r o o f we have

where CCR(t) = C test (t)/N (t) is from regular testing and C pop (t) = N (t) p(t). 260 We assume here that tests are perfectly accurate. When this is not the case we can replace C test (t) with the estimate (8). Since testing patterns usually vary by day of the week, it is best to consider weekly or bi-weekly rates.

Uncertainty concerning π 1 (t) arises from random variation in p(t) and in regular testing. This can be addressed through either frequentist or Bayesian procedures. The following frequentist approach which uses normal approximations for the estimators is simple to apply. Bayesian approaches can in principle deal more comprehensively with uncertainty in all its forms, including uncertainty around test sensitivity and specificity. However, it has more difficulty dealing with features such as complex survey designs, whose analysis utilizes estimating functions rather than likelihood. We condition on the number of regular tests n(t), in which case C test (t) depends on γ(t). The estimators p(t) and γ(t) = P R(t) are asymptotically independent and so the asymptotic variance of log π 1 (t) is (Boos & Stefanski, 2013) V ar(log π 1 (t)) = γ(t) −2 V ar( γ(t)) + p(t) −2 V ar( p(t)).

The variance of p(t) will depend on the sample design for random testing, which may be complex (Wu & Thompson, 2020) . If C test (t), given n(t), is a binomial 265 random variable then V ar( γ(t)) = γ(t)(1 − γ(t))/n(t). The binomial assumptions here might not be completely satisfactory due to population heterogeneity and clustering affecting testing in a given period; stratification as described in Section 2.3 can be used to mitigate such effects, provided stratum -specific estimates p k are available.

Regular random testing is rare, and a more feasible objective than estimating time-dependent ascertainment rates is to estimate the average ascertainment illustrated for the province of Ontario in Section 2.2, although the accuracy of such estimates depends on antibody persistence and the sensitivity of antibody tests (Accorsi et al., 2021) . Now that vaccination is widespread, this approach is no longer widely feasible. An alternative approach that uses information about symptomatic infection rates could be applied, if appropriate 280 information is collected. In particular, if the proportion q S of infections that are symptomatic were known, and if the number of confirmed cases C test in a time period were split into symptomatic cases C test−symp and asymptomatic cases, then the overall ascertainment rate for the period could be estimated as π * = C test q S π * S /C test−symp , where π * S is the ascertainment rate for symptomatic 285 infections. The rate π * S is unknown, but might often be close to 1. The proportion q S of infections that are symptomatic varies with age and other factors, and there have been wide ranges in estimates from different studies (Yanes-Lane et al., 2020) . For the 2020 Canadian seroprevalence survey (Ab-C Study Investigators, 2021), the proportion of "definitely" seroprevalent adults 290 who were estimated to be symptomatic was 0.32. A similar alternative approach to estimating ascertainment would be to replace "symptomatic" with "hospitalized" in the preceding discussion. In this case q S represents the proportion of infections that lead to hospitalization, and in this case π * S = 1 is reasonable. However, to estimate the hospitalization rate q S we once again need estimates of 295 the true numbers of infections, and not just confirmed cases. We are thus lead back to the need for some level of random testing, or exhaustive testing. For 

Studies have linked factors such as such as race, population density, population mobility, income, and work environments to confirmed case rates, but usually 15 they do not account for testing and ascertainment rates. Analysis of testing rates θ(t) can provide additional insight. Section 3 demonstrated the importance of high diagnostic testing rates in groups or regions with high infection rates, and we consider two types of analysis: comparisons of testing rates across specified regions or groups, and analysis of variation in testing over time. In each case the number of tests n(t) for a specific group and time period is a response variable, and z(t) denotes a vector of observable covariates associated with n(t). Unobserved factors such as the underlying symptomatic infection rate and individual behaviour also affect testing rates, and to accommodate heterogeneity in counts from such factors, we consider negative binomial mixed Poisson models (Lawless, 1987) , which are widely used for modelling count data (Venables & Ripley, 2002, Section 7.4) . We use the log-linear form, with the mean and variance of n(t) given z(t) and the population size N (t) as

where β is a vector of regression coefficients and τ > 0 is the dispersion parameter; vectors are written in column form.

In Section 5 we use the R function glm.nb in the MASS library to fit models (Venables & Ripley, 2002) for different public health regions in Ontario. Ide-305 ally, it would be best to examine separately testing in targeted groups (e.g. health care workers, long term care residents, industrial locations) and in the general population, but published data do not provide breakdowns that allow this. Testing for different age groups is also of interest; in Ontario, tests by age group are available for the province as a whole but are not provided for separate 310 regions.

We analyze data for Ontario, Canada, focusing on variation in testing rates over time and across regional public health units ( (2) if π 1 (t) and p(t) were fairly stable from day to day.

We focus on weekly counts and rates for analysis. Weeks run from Monday We do not include test counts since then in our analysis, but discuss them briefly 335 at the end of this section.

Weekly testing rates have increased three-to four-fold in most PHUs since May 2020. We compare testing rates across PHUs and association with confirmed case rates by splitting time into four periods during which testing rates were reasonably stable: weeks 19 -23, 24 -36, 37 -46, and week 47 -2021 week rates follow a similar pattern to case rates but with more pronounced high rates in the early weeks, when testing rates were low. Figure 2 shows considerable week to week variation in test counts for PHUs and to examine the effects of time-varying factors, we fitted negative binomial regression models for weekly test counts described in Section 4. The expected value of n(t) given a covariate vector z(t) and population size N (t) takes the form

To consider the effect of recent confirmed case rates we defined covariates 350 z 1 (t) = CCR(t − 1) and z 2 (t) = CCR(t − 1) − CCR(t − 2); the time origin t = 0 corresponds to week 20. We also defined binary covariates to reflect holiday periods and changes in testing criteria, as follows: a covariate Holidays(t) that where testing criteria were temporarily relaxed by the province; the rationale for

x 3 is to reflect the effect of the Christmas -New Year period and the transition 360 into 2021. These covariates are included as covariates z 3 (t) to z 6 (t) in (11).

Other factors such as lockdowns were also considered, but not found significant.

We considered alternative functions of t to represent the temporal increase in testing rates, but found the form in (11) to give the smallest value for AIC among parametric models considered.

365 Table 3 shows estimates of model parameters for the five PHUs. A high case rate in the preceding week (z 1 ) has a significant effect on the current testing rate in all PHUs but aside from Ottawa, the change over the previous two weeks (z 2 ) is not significant. Holidays is not significant but the effects of eased testing criteria (x 1 , x 2 ) and the transition into 2021 (x 3 ) are. We refitted models with The Christmas and New Year period saw a significant decrease in testing for all except the KFLA region. pected values µ(t; z(t)) under the fitted models for the five PHUs. The regression models track the actual test counts well. There is evidence of residual negative autocorrelation in counts for certain time periods; this may be due to backlogs related to laboratory capacity. Some regions show temporary aberrations, which may be due to local initiatives. In weeks 26 and 27, KFLA's testing rate 390 was well above the predicted value; this was associated with an outbreak in a Kingston nail salon in weeks 25 -26. In week 47 Ontario began to allow testing outside of laboratories (Hicks Morley, 2020) ; this may be associated with the slightly more rapid than predicted increase in testing over weeks 47 -50 in some

PHUs.

Instead of higher testing rates in regions with high case rates, we see a negative correlation with confirmed case rates. The highest case rates were mainly in regions with large populations, and further analysis could consider factors such as population density, socioeconomic conditions, and ease of access to testing.

At present, public PHU-level data to support such analysis is unavailable. A 400 comparison of ascertainment rates is crucial to truly understand differences in testing rates across PHUs, but since little random testing has been done, true infection and ascertainment rates in PHUs are very uncertain. As discussed earlier, Ontario-wide estimates based on serological surveys (Ontario Agency for Health Protection and Promotion (Public Health Ontario), 2020; Ab-C Study 405 Investigators, 2021) suggest average ascertainment rates of 0.25 or less up to September 2020.

Since January 2021 the province-wide weekly testing rate has been stable at approximately 0.02, but the negative association between testing and confirmed case rates has eased. Testing rates still vary two-fold across regions, with Peel, 

J o u r n a l P r e -p r o o f

In Ontario, per capita testing rates have consistently varied approximately two-415 fold across regions at any given time, whereas confirmed case rates varied by factors of 20 or more. If testing were targeting higher risk areas, we would expect higher testing rates in regions with high case rates; instead there is a moderate negative association between testing and confirmed case rates across regions. Since January 2021 this association has eased, but regions with the 420 highest confirmed case rates still tend to have lower testing rates. Exceptions are often for PHUs with smaller populations, which are perhaps more able to increase testing proportionally when there is an upsurge in confirmed cases.

Investigation of testing rate variation within PHUs would also be of interest, but requires more detailed data than are available. We note that for the Toronto 425 PHU, dashboard maps show wide variation in case rates across neighbourhoods, and much less variation in testing rates. Neighbourhoods with high case rates tend to be ones with features such as a high proportion of essential workers, lower income levels and more crowded living conditions. A more detailed analysis that relates testing rates to demographic and socioeconomic conditions, ease of access 430 to testing, and behavioural factors would be valuable.

Rapid antigen tests (Mina et al., 2020; Larremore et al., 2021) are increasingly used to monitor high risk settings such as workplaces and schools but to be effective they must be be repeated frequently. Testing strategies for schools have been an area of debate in some countries: some argue for frequent rapid 435 testing but others point out pitfalls when tests have variable and sometimes low sensitivity (Deeks et al., 2021) . These discussions reflect the inevitable tradeoffs between sensitivity and frequency of testing in any setting; once again, however, lack of information on ascertainment rates hampers comparisons. 

☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:

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