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# USING MATHS TO SOLVE COVID TESTING

MATHEMATICS | COVID-19 |

As the coronavirus pandemic rolls on into winter, countries around the globe are trying to optimise their testing programmes in order to find people who are infected and isolate them away from others. This is a huge logistical challenge. But what if we could test ten times as many people, without needing anymore actual tests, just using maths?

First we need to discuss how coronavirus tests actually work. When a person gets tested, a swab is inserted into their nasal cavity to obtain a sample. That sample is then PCR’d to give either a positive or negative test result.

Now since PCR will detect even very small quantities of viral RNA, one thing we could do is mix two samples together, and if either of those two patients have coronavirus, the test will come back positive. You could even mix together five or ten samples, and if any one person in that group has coronavirus then the test of the whole pool will come back positive.

You may ask why we would want to do this. For instance if a pool tests positive then we still have to go back and test every individual sample which makes up the pool anyway, to find out which individuals have coronavirus.

However this idea of mixing samples together actually turns out to be much more efficient when most of the members in your pool don’t have coronavirus.

Let’s look at an example containing 100 people, grouped into ten groups of ten as shown in the diagram below. In the first example all the individuals do not have coronavirus so each test we do of ten people will come back negative and no more testing is required.

Thus by just carrying out ten tests of the pools of ten people we know there is no coronavirus in the hundred people and no further testing is required. Compared to testing all members individually, we have just saved 90 tests. Now let’s look at an example where 1% of the population has coronavirus. As before we initially pool the population into groups of ten and then test each pool. In this instance nine of the ten pools will come back negative as in the last example, but one will come back positive.

We now test each individual member of the pool that tested positive. So in this instance, we have used 20 tests, which is five times more efficient than the 100 tests we would have used were we testing each member of the population individually.

How effective this strategy depends on how many people in your pool actually have coronavirus. In general, pooling strategies are very effective when the majority of the population you are testing does not have coronavirus and becomes less successful the greater the proportion of people with the virus. For the simple pooling strategy outlined above, this point where the strategy is no longer efficient is when between 5-10% of the population tested has Coronavirus; any higher and it is not a worthwhile strategy.

This pooling strategy is relatively straightforward and is limited to populations with low infection rates, but using this basic principle, it is possible through 3D and higher dimensional pooling to further increase the efficiency of testing, even in populations with high infection rates. With test kits in short supply and the virus spreading there is an urgent need for optimising the testing countries are carrying out. Pooling strategies, and mathematics, can help.

Written by William Brooke, Director of Witherow Brooke

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