In these times of a pandemic the world is changing. On large scales, but also for people in their every day lives. The same holds for me, so I figured I could post something on the blog again, to break with the habit of not doing so. Disclaimer: besides the exercises below being of almost trivial over-simplicity, I’m a data scientist and not an epidemiologist. Please believe specialists and not me!

Inspired by this blog post (in Dutch) I decided to look at simple versions of testing strategies for infection tests (a popular conversation topic nowadays), in a rather quick-and-dirty way. The idea is that if being infected (i.e. testing positive) is rare, you could start out by testing a large group as a whole. As it were, the cotton swabs of many people are put into one tube with testing fluid. If there’s no infection in that whole group you’re done with one test for all of them! If there, on the other hand, *is* an infection, you can cut the group in two and do the same for both halves. You can continue this process until you have isolated the few individuals that are infected.

It is clear, though, that many people get tested more than once and especially the infected people are getting tested quite a number of times. Therefore, this is only going to help if a relatively low number of people is infected. Here, I look at the numbers with very simple “simulations” (of the Monte Carlo type). Note that these are not very realistic, they are just meant to be an over-simplified example of how group testing strategy can work.

A graphical example of why this can work is given below (image courtesy of Bureau WO):

Above the line you see the current strategy displayed: everyobody gets one test. Below, the group is tested and after we found an infection, the group is cut in halves. Those halves are tested again and those halves with an infection gradually get cut up in pieces again. This leads, in the end, to the identification of infected people. In the mean time, parts of the data without infection are not split up further and everyone in those sections is declared healthy.

Normally, by testing people one by one, you would need as many tests as people to identify all infected people. To investigate the gain by group testing, I divide the number of tests the simulation needs by this total number. The number of people in a very large population that can be tested is a factor *gain* higher, given a maximum number of tests, like we have available in the Netherlands.

In this notebook, that you don’t need to follow the story, but that you can check out to play with code, I create batches of people. Randomly, some fraction gets assigned “infected”, the rest is “healthy”. Then I start the test, which I assume to be perfect (i.e. every infected person gets detected and there are no false positives). For different infection rates (true percentage overall that is infected), and for different original batch sizes (the size of the group that initially gets tested) I study how many tests are needed to isolate every single infected person.

In a simple example, where I use batches of 256 people (note that this conveniently is a power of 2, but that is not necessary for this to work), I assume a overall infected fraction of 1%. This is lower than the current test results in the Netherlands, but that is likely due to only testing very high risk groups. This results in a factor 8 gain, which means that with the number of tests we have available per day, we could test 8 times more people than we do now, if the 1% is a reasonable guess of the overall infection rate.

To get a sense of these numbers for other infection rates and other batch sizes I did many runs, the results of which are summarized below:

As can be seen, going through the hassle of group testing is not worth it if the true infected fraction is well above a percent. If it is below, the gain can be high, and ideal batch sizes are around 50 to 100 people or so. If we are lucky, and significantly less than a percent of people is infected, gains can be more than an order of magnitude, which would be awesome.

Obviously, group testing comes at a price as well. First of all, people need to be tested more than once in many cases (which requires test results to come in relatively quickly). Also, there’s administrative overhead, as we need to keep track of which batch you were in to see if further testing is necessary. Last, but certainly not least, it needs to be possible to test many people at once without them infecting each other. In the current standard setup, this is tricky, but given that testing is basically getting some cotton swab in a fluid, I’m confident that we could make that work if we want!

If we are unlucky, and far more than a percent of people are infected, different strategies are needed to combine several people in a test. As always, wikipedia is a great source of information for these.

And the real caveat… realistic tests aren’t perfect… I’m a data scientist, and not an epidemiologist. Please believe specialists and not me!

Stay safe and stay healthy!