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# The mathematical strategy that could transform coronavirus testing

Scientists say that widespread testing is needed to get outbreaks of the new coronavirus under control. But in many regions, there’s a shortage of the chemicals needed to run diagnostics. In several countries, health officials have started using a strategy that was first proposed in the Second World War: group testing. By testing samples from many people at once, this method can save time, chemical reagents and money, say researchers.

“In the current epidemic, there is a need to test an extremely large number of patients, making pooling an attractive option,” says Roy Kishony, a systems biologist at Technion — Israel Institute of Technology in Haifa.

China, India, Germany and the United States are already using group testing.

There are many ways to conduct group testing, and scientists in several countries are experimenting with the best method for doing this during a pandemic. Their ideas largely come from a field of mathematics that has been applied to a wide range of problems — from detecting faulty Christmas-tree lights to estimating the prevalence of HIV in a population. “There has been a flurry of innovation in this field,” says Dror Baron, an information scientist at North Carolina State University in Raleigh.

Nature highlights four methods currently being trialled.

## Method 1 & 2: From syphilis to coronavirus

The most straightforward strategy for group testing was proposed by economist Robert Dorfman in the 1940s to test soldiers for syphilis.

In this method, an equal number of samples — collected from nasal and throat swabs in the case of the SARS-CoV-2 coronavirus — are mixed together (see Method 1, below) and tested once. Groups of samples that test negative are ruled out. But if a group tests positive, every sample in that group is then retested individually. Researchers estimate the most efficient group size — the one that uses the least number of tests — on the basis of the prevalence of the virus in the community.

In May, officials in Wuhan, China, used Method 1 method as part of their efforts to test the vast majority of the city’s population, reaching roughly ten million people in just over two weeks. Samples from some 2.3 million people were group tested, with up to 5 samples in a group, and 56 infected people were identified.

The method is most efficient when there are low levels of infection, in around 1% of the population, because group tests are more likely to be negative, which saves testing many people individually, say researchers.

“This is probably the easiest method,” says Krishna Narayanan, an information theorist at Texas A&M University in College Station. But there are more efficient ways to construct the second stage than testing everyone individually, he says.

A more sophisticated version involves adding further rounds of group tests, before testing each sample separately (see Method 2, above). Adding rounds reduces the number of people who need to be tested individually.

But this approach is slow, because it takes several hours to get the results for each group test, says Wilfred Ndifon, a theoretical biologist at the African Institute for Mathematical Sciences in Kigali, Rwanda. “This is a fast-growing, fast-spreading disease. We need answers much faster than this approach would allow,” he says.

## Method 3: Multi-dimensions

Ndifon and his colleagues have improved on Dorfman’s strategy, which they plan to trial in Rwanda, and so ultimately reduce the number of tests needed. Their first round of group tests is the same as Dorfman’s, but for groups that test positive, they propose a second round that divides samples between groups that overlap.

Imagine a square matrix with nine units, each representing swabs taken from one person (see Method 3, above). The samples in each row are tested as one group, and the samples in each column are tested as one group, resulting in six tests in total, with each person’s sample being in two groups. If a sample contains SARS-CoV-2 viral RNA, both of the group tests will be positive, making it easy to identify the person. Researchers describe the idea in a preprint posted on the arXiv server on 30 April1.

Increasing the number of dimensions, for example from a square to a cube, allows for larger group sizes and higher gains in efficiency, says Neil Turok, a theoretical physicist at the University of Edinburgh, UK, and a study co-author.

Ndifon, who is part of Rwanda’s COVID-19 task force, says group testing is part of the government’s strategy to quickly identify and isolate infected people. He and his colleagues estimate that their method could cut the cost of testing from US\$9 per person to 75 cents. The researchers are carrying out laboratory experiments to see how many samples can practically be included in a group test and still detect a positive result. Leon Mutesa, a geneticist at the University of Rwanda in Kigali, and another co-author who is part of the government task force, says that he has been able to identify one positive sample in a pool of 100 in the lab.

But Sigrun Smola, a molecular virologist at Saarland University Medical Center in Homburg, Germany, who has been testing samples in groups of up to 20, doesn’t recommend grouping more than 30 samples in one test, to ensure sufficient accuracy. Larger groups will make it harder to detect the virus, and increase the chances of missing positives, she says. Smola is also sceptical of the practical application of the cube-slicing technique in routine testing. “If you told this to a technician, they would say, ‘What a mess. I want a simple scheme,’” she adds.

Ndifon says that his team plans to develop software to automate the placement of samples.

## Method 4: One-step solution

Some researchers say that even two rounds of testing is too many when trying to curb a fast-spreading virus such as SARS-CoV-2. Lab technicians must wait for the results from the first round to come through, which slows the process, says Manoj Gopalkrishnan, a computer scientist at the Indian Institute of Technology Bombay in Mumbai.

Instead, Gopalkrishnan proposes doing all the tests in one round, with many overlapping groups. This would increase the number of tests, but would save time — although the initial set-up is time-consuming, because having extra groups means more samples must be pipetted.

Gopalkrishnan’s approach involves mixing samples in different groups, using a counting technique known as Kirkman triples, which sets rules for how the samples should be distributed. Imagine a flat matrix in which each row represents one test, and each column represents one person (see Method 4, above). Generally, every test should include the same number of samples, and every person’s sample should be tested the same number of times.

But Narayanan says that one-step strategies require more tests to ensure the same level of accuracy as for multi-step group testing. One-step approaches also mean working with a large number of samples at once, which can be tricky, he says. “It is impossible for a technician to do this pooling. You would need a robotic system.”

To simplify the process in the lab, Gopalkrishnan and his colleagues have developed a smartphone app that tells users how to mix the samples. In unpublished results from clinical trials in India in Mumbai, Bengaluru and Thalassery, he says, 5 positive samples were successfully identified out of 320, using only 48 tests.

Researchers in Israel are using an automated system and an app to apply a similar one-step system. Moran Szwarcwort-Cohen, who heads the virology lab at Rambam Health Care Campus in Haifa, says her team is currently assessing the system, with promising results.

Nature 583, 504-505 (2020)

doi: https://doi.org/10.1038/d41586-020-02053-6

## References

1. 1.

Mutesa, L. et al. Preprint at https://arxiv.org/abs/2004.14934 (2020).

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