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The authors have declared that no competing interests exist.

In the current COVID19 crisis many national healthcare systems are confronted with an acute shortage of tests for confirming SARS-CoV-2 infections. For low overall infection levels in the population the pooling of samples can drastically amplify the testing capacity. Here we present a formula to estimate the optimal group-size for pooling, the efficiency gain (tested persons per test), and the expected upper bound of missed infections in pooled testing, all as a function of the population-wide infection levels and the false negative/positive rates of the currently used PCR tests. Assuming an infection level of 0.1% and a false negative rate of 2%, the optimal pool-size is about 34, and an efficiency gain of about 15 tested persons per test is possible. For an infection level of 1% the optimal pool-size is 11, the efficiency gain is 5.1 tested persons per test. For an infection level of 10% the optimal pool-size reduces to about 4, the efficiency gain is about 1.7 tested persons per test. For infection levels of 30% and higher there is no more benefit from pooling. To see to what extent replicates of the pooled tests improve the estimate of the maximal number of missed infections, we present results for 1 to 5 replicates.

We briefly analyse how

Pooled testing, also called group testing, was first introduced by Dorfman (1943) to screen U.S. soldiers for syphilis [

While the observation [

Here we contribute an estimation of the benefits of a simple, easy to implement, one-stage pooling strategy. Our main goal is to compute the optimal group-size for pooled tests and their dependence on the disease prevalence, i.e. the fraction of infected in the target population. We provide a formula for the optimal group-size, i.e. the optimal number of persons pooled into a single test, and study the dependence of this optimum on false positive and false negative rates of the used PCR test. We demonstrate the optimal group-size dependence on the false negative rate of pooled testing. We briefly remark on a subtlety when using replicate measurements in group testing to control for the false negative rate of tests. We ask how the number of test replicates affects the false negative rate of the pooled test. We conclude that the optimal group size, in the considered single-stage pooling approach, should be smaller than the suggested 32-64, for the currently (April 2020) suspected infection levels in general populations. Finally, we comment that testing a pooled sample more than twice will essentially not further decrease the expected maximal number of possibly missed positive cases.

We assume that

a fraction λ of infected people in a population,

tests have a _{+} and a _{−}. If not stated otherwise, we assume that testing a pooled sample does not change the false positive and false negative rates of the test. We discuss the effect of group-size dependent false negative rates at the end of the paper.

We pool samples into groups of size,

To control false negatives we take

If the pooled sample is declared positive, each individual in the respective group is tested separately.

Under these assumptions we compute

the optimal group size, ^{opt},

the effective number of persons that can be tested with one test,

an estimate for the upper bound for the fraction of infected individuals that are missed by the pooled testing procedure (applied to the population). We call it the

and finally, we discuss and demonstrate how false negative rates increasing with group size affect the optimal group-size, false negative rates (FNR), and the pooled testing risk factor (PTRF).

We call a group

Because of limitations in test sensitivity and specificity, tests will be falsely declared positive in (1 − _{+} cases. False positives do not decrease the chances to capture a true positive but only decrease the efficiency in using the available tests. More importantly, tests will miss positive individuals in _{−} cases, on average. Note that _{+} and _{−} might need to be considered carefully with respect to how tests are performed (essay type) and who gets tested (patients with high or low expected viral load).

To see how test replicates affect FNR and PTRF of the pooled test we test a sample

The expected number of tests per person therefore can be estimated by the upper bound,

For

Similarly, one can compute an upper bound for the expected number of cases that we might miss when testing pooled samples,

Note that the expected _{−}. If there are no biases or correlations within or between groups, we get that the number of missed infections will be λ _{−}λ, which does not depend on group-size. It can be checked that

Results for the optimal pool size, ^{opt}, and the persons per test, _{−} = 0.02 and a false positive rate of _{+} = 0.0012, which are sensible estimates for PCR tests that are currently used in Austria (as of March 20) [^{opt} = 11 where it achieves a ^{−3}. For ^{opt} = 15, ^{−3}, whereas for ^{opt} = 16, the ^{−2}. Note that ^{opt} and

(A) Increase of test efficiency in persons per test, ^{opt} for a given infection level (1%) and given false negative and positive rates of the test. Results are shown for _{+} = 0.0012 and _{−} = 0.02.

The group-size dependent PTRF, on the other hand, is again decreasing for ^{opt} on

^{opt}, as a function of the infection level of the population. The inset shows the case for low infection levels between 0 and 3%. The case for

(A) Optimal pool size, ^{opt}, as a function of the infection level of the population. The inset is a blow-up for low infection levels. The cases for _{+} = 0.0012 and _{−} = 0.02. By taking _{−} = 0.05, ^{opt} and

We computed the same values for a false negative rate of _{−} = 0.05. The results for ^{opt} and

To get a better understanding of the effect that group-size dependence has on of false negative rates, we compare three scenarios. We assume that false positive rates are constant (_{+} = 0.0012) and false negative rates increase linearly with group size. We assume that at the maximally considered group size of 100, scenario (1) has the same value as for group size 1 (_{−} = 0.02); (2) has 5 times that value, and (3) has 10 times that value. While optimal group-size and PPT do essentially not change, the

(A), (B), and (C) show that the overall best choice of replicates with respect to

The optimal pool size and efficiency of pooling strongly depends on the infection level of the population. Let’s assume the simplest case of only one test (1 replicate). From

Replicates help to lower the pooled testing risk factor,

When even numbers of replicates are used, majority rule m2 should be implemented, i.e. a pool should be considered positive if at least half of the replicates are positive. For odd numbers of replicates rules m1 and m2 are identical. Rule m1, that there must be more positive than negative replicates, does not essentially change the optimal group-size for pooling. In relative to m2, m1 has a higher pooled testing risk factor, i.e. the maximal number of positive individuals that can be expected to be missed per tested individual. For two replicates,

Let us emphasize that a pooling strategy is most powerful for population-wide screening and mixed samples, for example at airports. Using them for highly biased samples, e.g. for samples from patients already showing symptoms, will be much less effective. Note, that situations with many asymptomatic individuals, with possibly low viral loads, require test protocols that operate in ranges with values of false negative rates larger than 0.02, which would make

We finish with a practical example. For Austria, a country with slightly less than 10 million inhabitants an actual infection level of 0.1% would indicate an optimal pool size of 34. For a level of 1% it would be 11. Assuming the true number of infected to be somewhere between 10,000 and 100,000 this would mean a reasonable choice of pooling sizes of about 20. This number is definitively lower than the suggested sizes reported in [

PONE-D-20-13830

Boosting test-efficiency by pooled testing strategies for

SARS-CoV-2

PLOS ONE

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Reviewer #1: excellent and important practical manuscript. citation which arrives at similar conclusion and provides direct experimental evidence : Lancet Infect Dis 2020

Published Online

April 28, 2020

S1473-3099(20)30362-5

might bbe cited

A limitation which might be briefly discussed is the differential sensitivity and specificity of specific "PCR" tests which use one or two) target sequences, and there are small differences between viral target genes and the abundance of target RNA species. Efficiency of the specific methods used for RNA extraction and reverse transcription are variables that may effect sensitivity and ultimately limit pooling.

Reviewer #2: 1. Is it reasonable to assume the false positive/negative rate does not change when switched from the regular testing to pooled testing? Does it apply to COVID-19 screening? It is hard to believe using a pool of size 32 or 64 can have the same false rates as testing the individuals one-by-one.

2. What is the choice of r in practice? How does it affect test efficiency? Following the majority rule, suppose r=5, then in practice, once you observed 3 of them were positive, there is no need to test the remaining 2 replicates. How does this affect the calculation of number of tests needed?

3. Line 47 on Page 2, I believe it should be "If the pooled sample is declared positive, we test each individual in the group separately" because of the majority rule you proposed.

4. The derivation of (3-5) in-explicitly assumed that given the true infection statues, the test results are mutually independent. Is this assumption supported by COVID-19 tests? If COVID-19 tests declare a sample as positive if the measured viral loads exceed a predetermined threshold, then this assumption does not hold.

5. There is no COVID-19 data supporting the methodology.

6. Literature review is not sufficient. See Kim et al. (2009, Biometrics).

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Dear Editor and Reviewer.

Thank you for you efforts and the points you raised, which we believe has lead to a significant improvement of our manuscript.

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We hope that the improvements of the manuscript now make it acceptable for publication.

Looking forward to hearing from you,

With kind regards

Rudolf Hanel

Submitted filename:

Boosting test-efficiency by pooled testing for SARS-CoV-2 --

formula for optimal pool size

PONE-D-20-13830R1

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PONE-D-20-13830R1

Boosting test-efficiency by pooled testing for SARS-CoV-2 – formula for optimal pool size

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