Mathematical logic that can help check more people for coronavirus

Rapid testing of patients during a pandemic is of utmost importance. But when there are not enough COVID-19 tests [in Britain] , or testing is slow, is it possible to come up with a way to improve this process? As a mathematician and engineer, I asked myself whether a theoretician can do anything to help physicians meet WHO requirements for checking the maximum possible number of patients.

Perhaps there is a way to test many patients with a small number of tubes. Instead of using a test tube for each test, we could use several tubes to test more samples — if we draw on logic .

The idea is simple. A sample taken from each of our theoretical patients is distributed in half of all the tubes we have in different combinations. If we have ten tubes, we distribute samples from each patient into five different tubes, combining them differently.

Any test tube with a negative result indicates that all patients whose samples fell into it will give a negative result. Tests with a positive result may contain samples of several positive patients - and each individual patient will be considered positive if all the associated tubes give a positive result.

This approach is especially effective in the early stages of the epidemic, when a small number of people can give a positive result.

We change the approach

The more patients become infected, the more difficult it is to determine which of them has the virus, because test tubes with a positive result will contain material from a larger number of patients. To solve this problem, you need to fix the approach as shown in the following example.

Let's say we have six test tubes and 20 patients. The tubes are arranged in order and numbered: No. 1, No. 2, No. 3, No. 4, No. 5 and No. 6. Each patient is assigned a six-digit number of zeros and ones (in the binary system). Each number corresponds to a specific tube - 0 means that the patient’s sample did not enter this tube, and 1 means that the patient did.



For example, patient No. 1 will have the number [0 0 0 1 1 1], meaning that only tubes No. 4, 5, and 6 will contain his material. Patient No. 2 was given the number [0 0 1 0 1 1], which means that his samples contain tubes No. 3, 5 and 6. And so on, for each of the 20 patients.

Having distributed samples from all 20 patients into test tubes, we send them for verification. If after this we receive positive tests for tubes No. 4, 5 and 6, we can say that only patient No. 1 with the number [0 0 0 1 1 1] will get a positive result, since only this patient’s samples were added to three of these tubes, nos. 4, 5 and 6, and to no other.

Now the hardest part. Suppose we got positive results for tubes No. 3, 4, 5 and 6. We can immediately discard patients whose samples were in tubes No. 1 and No. 2, but we don’t have certainty about the rest, because we have these tubes There are several possibilities. Are patients with numbers [0 0 1 0 1 1] and [0 0 1 1 1 0] sick, or with [0 0 0 1 1 1], [0 0 1 0 1 1] and [0 0 1 1 0 1], or are they all together? All these combinations will give a positive result for tubes No. 3, 4, 5 and 6. Is it possible to get a definite answer about which of the patients is sick?

Yes, but only if the test can give not just a positive or negative result, but a certain degree of positivity - for example, determine the number of antibodies in the sample. Then we can compare the amount of positivity in different tubes, which will give us an idea of ​​the correct combination of positive patients.

Degree of positivity

Returning to our example, if the positivity for tubes No. 4 and No. 5 is the same (for example, they have the same number of antibodies), but differs in tubes No. 3 and No. 6, then we can conclude that patients [0 0 0 1 1 1] and [0 0 1 0 1 1] cannot be positive, since the sample from the first patient [0 0 0 1 1 1] is in tubes No. 4 and No. 5, and the sample from the second [0 0 1 0 1 1] in tubes No. 5 (but not No. 4). This means that the positivity in these tubes cannot be the same if both patients are ill (in tubes No. 5 there will be positivity of both patients, and in No. 4 there will be only one).

The same positivity in both tubes No. 4 and No. 6 will be only in patients [0 0 0 1 1 1] and [0 0 1 1 1 0], since samples of both of them were added to both tubes No. 4 and No. 5, which gave the same positiveness in both test tubes.

In the above example, we were able to test 20 patients with just six tubes. However, this method can be scaled to thousands of patients, and much less tubes are needed to test them. And even when machines capable of conducting tests in five minutes are already under development, this approach can still be faster - and cheaper where resources are scarce.

It turns out that with the help of mathematics we can improve the verification of the collected samples, especially in those places and at a time when the verification encounters difficulties. In such cases, this approach could potentially help mitigate the effects of the new coronavirus and save many lives.