If we are going to go back to our offices at some point before there is a vaccine, it is almost certainly going to have to be in a low prevalence SARS-C0V-2 environment accompanied with measures to keep incidence also low. One of those measures is testing and testing frequently. If we are going to back to the office, our partners back to teaching in person on campus, our children to daycare centers and schools, there will have to be testing and lots of it. A short new paper by Augenblick et al. (2020) discusses the promise of group testing for such workplaces. As this type of decision (how often to test, whether to individually or group test, optimal size and composition of the groups to pool) is one that each of our workplaces (and schools and local governments and so on) may potentially have to make very soon (unless everything is virtual in the Fall), I thought it would be good to summarize this paper for our readers. Many of you may end up helping or even being in charge of designing these schemes...
What is group testing and won’t it produce too many false negatives?
There are many forms of more complicated group testing (such as placing people in multiple groups or doing multiple stages of group testing), but the simplest method, and the one the authors focus on in their paper, is where a group of n people are divided into g groups of size n/g. You take samples from everyone and pool them for tests (for example, you can pool samples from 100 individuals into five groups of 20 and conduct five tests). If a group tests negative, they are cleared. If it tests positive, everyone in the group gets tested individually. [I did not get a sense from this paper whether samples are large enough to test twice or whether individuals have to give samples twice, but obviously the former would be better.] This is apparently the simple two-stage “Dorfman” testing, which has been around since WWII. It is commonly used for screening donated blood for infectious disease.
A worry about group testing is sample dilution, such that the infection cannot be detected via PCR testing. The authors assure us that at least three studies found that existing PCR tests performed fine for groups up to 32 or even larger. If we’re willing to more complex groupings, the group size can increase further. In section 6 on practical considerations, the authors also warn that any sensitivity decline from group testing can be made up from increased frequency in testing, as we will discuss below.
The optimal size of groups is well-known, given disease prevalence rate p. In applications such as blood banks, there is no repeated testing and the correlation between samples can be assumed to be zero or very low. So, that is not where the paper’s contribution is going to be. For instance, in the example I gave above, instead of conducting 100 tests, group testing would need only 20 total tests (with an optimal group size of 23 – yes, please ignore divisibility) when p=0.01 (prevalence of 1%). Figure 1 in the paper shows that the gains are large when prevalence is low and they decline as it increases. So, it’s important that the disease is under control when we’re aiming to conduct group testing (and the aim is to prevent any exponential growth).
Contribution 1: extension to repeated testing and the gains from a (mechanical) reduction in prevalence
Now, imagine that you were going to do frequent repeat testing. Suppose that we are in one of the areas where the pandemic is under control and that a random person’s chance from your workplace to test positive over a one-month period is 1%. You could test everyone once a month (100 tests) or you could have as little as 20 tests with group testing. Say, now, you’re interested in testing more frequently, say every 3 days instead of every 30. You will need at least 200 tests even with optimal group testing, right? No, because, the probability that someone tests positive (conditional on being negative in the previous test) is now much lower than 1%, more around 0.1%. The reduction in prevalence at every third day testing means that you test 10 times as frequently, but conduct only about three times as many tests. Authors report that numerical simulations suggest that if you test x times as frequently, the expected cost of group testing is only about sqrt(x) times the original cost, as long as p is not too high (say, p). This is where I would like to see the math in a little more detail, but the authors can be excused for putting out results they are confident in quickly during a pandemic, where many people are struggling with practical implications of restructuring our daily lives. It is likely, however, that more work is needed before designing policies in real life…
Contribution 2: But, hold on: why test more frequently?
Of course, you can test people every three days instead of every thirty and get a mechanical ‘discount’ from reduced prevalence rates at each testing date, but you’re still spending three times as much money on testing than before. Why do it? The answer is substantially lower infection rates.
The authors give an example in Figure 3. Before, the authors had independent probabilities of infection for people at your workplace. Say, each of the 100 of you have a 1% chance of getting infected by someone from outside your work that is independent of your colleagues. Now, imagine that people at work bump into each other randomly but much more frequently than people from the outside and that they cannot (or do not) always social distance and wear masks, etc., so that their infection probabilities are correlated. What happens then?
The authors add a probability to infect someone else in your group if you’re infected at the start of the testing period and your infection is not caught until the end of the testing period and you’re not removed from the population and isolated. You can see that in this scenario, removing someone from the pool 9-10 times quickly may substantially reduce the spread of the virus at your workplace. Figure 3 shows that infection rates decline very steeply going from monthly to weekly testing, when the onward infection probability within your workplace is 5%, it then tapers off and does not yield more benefits past the bi-weekly testing, when the hypothetical infection rates are already around 1%. This additional benefit of frequent testing is such that increasing the frequency may not add much to the cost of group testing monthly at modest frequencies.
The answer to the question of the optimal frequency depends, of course, on the cost of avoiding an extra infection (and also any costs of tracing and monitoring isolation, etc. need to be added to the testing costs), but this is not part of this paper’s remit. The authors speculate that the optimal frequency is more frequent than the commonly-recommended monthly testing or simply testing symptomatic people. This is also likely to depend on the composition of people at the workplace: while we don’t currently know the long-term health effects of non-fatal infection on individuals, it is possible that schools and universities with young people test weekly, while nursing homes test more frequently, and so on…
One last point before we move on to the final contribution of the paper: the correlation of infections within the workplace both reduces the expected number of tests and leads to larger optimal group sizes. Again, in section 4, the authors rely on our intuition as the readers, but more of the math and statistics in an appendix would likely be useful to many…
Contribution 3: How should we group people for testing?
Above, the probability of one of your colleagues to infect anyone else (once infected from outside) was constant. But, naturally, people who work on a team together, David and I having lunch together, the after-seminar coffee crowd form sub-clusters within our work cluster: they are more likely to infect each other than they are to infect someone they don’t know well. Can we identify such groupings and test them together, so that we gain even more efficiency?
Here, the authors show that the workplace can do much better in containing any outbreaks and save money on testing if it can accurately estimate heterogeneous (outside) infection risks among the staff (say, based on age, zip code, in-person credit card use data!, etc.). Furthermore, if they can then identify the people who interact with each other a lot (the intra-workplace correlations), then they can further optimize testing. For example, simple schemes may involve testing people from the same shift, same floor, etc., while more complex ones might use cell phone tracking data or tracing apps.
In section 5 of the paper, the authors discuss some of the approaches that employers may adopt in more data-rich environments to use machine learning – for example, using a high-dimensional social distance measure to identify people with higher likelihood of infecting one another – or using testing itself to learn about the networks.
The authors end by making a back-of-the envelope calculation that daily group testing could be conducted for about $3-5 per person per day. For a university that has 10,000 students, faculty, and staff on campus, such a scheme would cost about $1 million per month, $3-4 million per semester. That is not cheap, but possibly cost-effective – especially if testing frequency and group sizes are optimized: some of our readers have their work cut out over the next couple of months…