As most countries including Singapore are planning to exit lockdowns, there has been a fair amount of discussion about widespread testing, and how best to use limited testing resources.
Locally, there is a plan to test all foreign workers living in dormitories. Of note is the use of pooling tests as a strategy, of 5 workers to a pool.
From the article, there are about 320,000 foreign workers, including approximately 20,000 who have already tested positive and are receiving treatment. So that leaves us with 300,000 workers to test before they return to work.
In our first stage (pooled), we need to conduct 300,000/5 = 60,000 tests
In our second stage, we test every individual in a pool which has tested positive. This presents a range of possibilities: the best case is full correlation and everyone who is positive are pooled together (minimizing number of re-tests), the worst case is every positive pool has only 1 positive case (most number of re-tests).
This depends on the prevalence rate. Let’s consider a fairly bad scenario of 5% (which is higher than the current prevalence thus far):
Total number of positive cases = 15,000
Best case: only 15,000 positive cases are re-tested, total of 75,000 tests needed. This is 25% of the total number of people and we can clear 285,000 healthy people
Worst case: Need an additional 75,000 tests, for a total of 135,000 tests. This is 45% of the total number of people, and 225,000 people are cleared after the initial stage.
This is pretty good! Of course, in practice the numbers will lie somewhere in between, but this suggests that pooling should not be done at random, i.e. should try to pool roommates together.
But what if things are better or worse? From the above, we can see that the best case is simply the sum of the fraction of the size of each pool and the prevalence rate (because the number of tests in the first stage is determined by the pool size and the best case for the second stage is simply the prevalence). The worst case would be, again, the fraction of the size of each pool plus the product of the prevalence rate and the pool size.
In more mathematical terms:
So, there is a trade-off: larger pool sizes are detrimental if your prevalence is unexpectedly high.
In practice, this helps with the number of reagents, but given that nasal swabs are still needed for each worker needed, this might not help if there is a shortage of swabs.