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## Summary

In the battle to combat the current COVID-19 pandemic, the standing advice of the World Health Organization to all governments around the world is: “*test, test, test” *[1]*.* The problem for many countries, however, is that there are not sufficient facilities to conduct the required number of tests. The main issues are shortage of testing machines and chemicals. An important restriction is that in the standard approach, one individual requires one test.

With relatively simple mathematics, however, we could increase the number of people tested dramatically, without increasing the number of machines or chemicals required. We believe this possibility deserves further investigation.

It is possible to test more people with fewer tests, if we make use of *group testing* methods. These methods are scientifically well understood, and can yield a large increase in testing capacity, especially when the population being tested has a relatively low infection rate (below 15%): If the infection rate is around 1%, a group of 64 individuals can be accurately diagnosed with on average only eight tests (instead of 64). This would increase testing capacity by 700%. If the infection rate is 15% (which is typical), group testing could still increase testing capacity by 100%. We therefore think that it would be a good idea if governments and testing labs take this possibility into consideration when designing their test protocols.

## Group testing

Group testing is not a new concept [2]: it was developed in Word War II, to test large groups of soldiers for siphylis, with the limited resources that were available [3]. We are currently experiencing a similar problem. Since the 1940s, group testing has been developed into a large array of scientifically proved methods (see e.g. [4]). These methods are especially useful when the expected infection rate within the group is relatively low (say 15%). To explain how group testing works, we work out two example strategies. Example 1 is very efficient when the infection rate is around 1%, Example 2 is efficient when the infection rate is around 15%.

### Example 1: test protocol for groups with low infection rate

Suppose we have a large group, 64 persons, say. Within the group the infection rate is around 1%. Think of a group of healthcare workers, prisoners, or refugees. To prevent new hot-spots occurring, we would like to regularly screen such groups.

Start by taking a few samples of the mucous membrane of all 64 individuals. For the first test, combine all samples into one, and test the combined sample for the virus. There are two possibilities. If the test is negative, then no-one within the group is infected, and no further tests are necessary. When the average infection rate is 1%, the probability of a combined negative test is 53%. If the test is positive, then at least one person is infected. The probability of this is 47%. So, in half of all cases, one test suffices to clear an entire group.

If the combined test is positive, then we need to continue testing. We split the group into two groups of 32, make two combined samples, and test both. At least one of these will contain an infected individual. Groups that test negative are cleared. Positively test groups are split into smaller groups, of 16, 8, 4, 2, and finally, 1. This method is known as the *binary splitting algorithm.* If one out of 64 is infected, we can find that person with 13 tests (see Figure 1). If two persons are infected, we need between 13 and 22 tests. The probability of needing more than 32 tests to find all infected individuals is astronomically small, even at an infection rate of 40%.

On average, we would need to use eight tests, using this method. But this does require a low infection rate, and multiple rounds of testing, which takes time.

### Example 2: test protocol for higher infection rates

When the group has a higher infection rate (say 15%), or when time is a factor, we can still make significant reductions by a different mathematical approach. This works as follows:

The idea is again to combine samples. We make several different combinations. The outcomes of these combined tests will be used as a sort of code, that can be used to determine who are infected (or have a high chance of being infected). This way, we can reduce the number of tests by 50%, if we can accept a 5% decrease in test accuracy.

In this example, we test 8 people with 4 tests. The combined tests are as in Figure 2. The outcomes of the four tests form a code: positive = *O*, negative = *X*. If, for example, the outcomes are test A positive, test B negative, test C positive, test D negative, then the code becomes *OXOX*. Via a lookup table we can then determine for each person what probability they have of being infected. In practice, we will mostly see situations where that probability is 100%. Only in 4% of cases is the certainty less. (For these cases, we could improve the accuracy to 100% by performing four more tests.)

The advantage of this approach is that we know with high accuracy who is infected after only one round of tests. This approach is therefore as fast as individual testing, but testing capacity is increased by 100%.

Code: | XXXX | OXXX | XOXX | XXOX | XXXO | OOXX | OXOX | OXXO | XOOX | XXOO | OOOX | … |

Probability of code | 66% | 8% | 8% | 8% | 8% | 1% | 1% | 1% | 1% | 1% | 0.10% | … |

1 | 0% | 0% | 0% | 0% | 0% | 0% | 0 | 90% | … | |||

2 | 0% | 100% | 0% | 0% | 0% | 10% | 100% | 10% | … | |||

3 | 0% | 0% | 0% | 0% | 0% | 90% | 0 | 10% | … | |||

4 | 0% | 0% | 100% | 0% | 0% | 10% | 0% | 0% | … | |||

5 | 0% | 0% | 0% | 0% | 0% | 10% | 0 | 0% | … | |||

6 | 0% | 0% | 0% | 100% | 0% | 0% | 100% | 0% | … | |||

7 | 0% | 0% | 0% | 0% | 0% | 0% | 0 | 10% | … | |||

8 | 0% | 0% | 0% | 0% | 100% | 0% | 0% | 10% | … | … |

**Important considerations**

Group testing is no ‘free lunch’. Test protocols need to be adapted. Multiple samples need to be taken, and combined in different configurations. Some scenarios require multiple rounds of testing, so would take longer. It depends on the goals and the availability of goods and manpower which strategy is most efficient.

Besides this, it is important to determine the accuracy of testing combined samples. The scientific literature on the accuracy of combined SARS-CoV-2 tests is currently limited [5, 6]. According to an Israeli study, combining 64 samples has no significant effect on the accuracy of the test, but they recommend further research.

On the topic of combining serological tests we could not find any scientific literature.

Another point of attention is the way false positives and negatives propagate in group testing. (In the above examples we assumed a 100% accurate individual test.) For example 1 above, the analysis is relatively easy, and shows that false negatives increase slightly, while false positives decrease dramatically. Clinically speaking, false negatives are more important.

### More information

If you would like more information about group testing, or if you have questions, we are happy to explain things further, or think along with you.

**Bureau WO:** Tim Hulshof, Erik de Ruijter en Janne Brok (jannebrok@bureauWO.com)

*Special thanks to Ton Monasso and Sandra Nolten of PBLQ and Yorit Kluitman of Studio Yorit Kluitman.*

#### References

[1] https://www.bbc.com/news/av/world-51916707/who-head-our-key-message-is-test-test-test [2] Wikipedia page ‘Group Testing’, https://en.wikipedia.org/wiki/Group_testing [3] Dorfman, Robert (December 1943), “The Detection of Defective Members of Large Populations”, The Annals of Mathematical Statistics,**14**(4): 436–440 [4] Ding-Zhu, Du; Hwang, Frank K. (1993). Combinatorial group testing and its applications. Singapore: World Scientific. ISBN 978-9810212933. [5] https://aktuelles.uni-frankfurt.de/englisch/pool-testing-of-sars-cov-02-samples-increases-worldwide-test-capacities-many-times-over/ [6] Idan Yelin et al., Evaluation of COVID-19 RT-qPCR test in multi-sample pools, preprint:

https://kishony.technion.ac.il/wp-content/uploads/2020/03/2020.03.26.20039438v1.full_.pdf