Motivated by the Covid-19 pandemic, we study the possibility of using group testing to help finding the carrier of SARS-CoV-2. (A small detail: Covid-19 is the name of the disease; SARS-CoV-2 is the name of the virus.) The idea is simple: Suppose there are 5 students that we want to test. We can test if each of their saliva specimens contains the virus, which will costs us 5 testing kits. We can also combine the 5 saliva specimens and test once, which will cost us 1 testing kit. If the test result is negative, then we know none of the 5 students have the virus. However, if the test result is positive, then at least one student has the virus. We then 5 more testing kits to find which of these 5 students have virus. Over all, the average cost is $1/5 + p$ testing kits per student we want to test, where $p$ is the probability that the mixture of 5 saliva specimens is positive.
Now, one interesting aspect about testing SARS-CoV-2 is that, in the beginning of the pandemic, this virus is so new and the whole situation is so emergency that the only reliable way is to test if a specimen contains the DNA fragments that belong to SARS-CoV-2. And the only reliable way to do so to tell the polymerase to amplify (which is a fancy way to say duplication) the targeted fragments until the specimen contains nothing but the targeted fragments. For instance, suppose there are three type of DNA fragments, and denote the numbers of copies by $x$, $y$, and $z$, respectively. Suppose the second type is what we want to test; we then tell the polymerase to amplify that so the numbers of copies becomes $(x, 2y, z)$ after one cycle, $(x, 4y, z)$ after two cycles, and so on, until $2^c y$ is too large compared to $x$ and $z$ and it becomes very easy to detect. This is how the PCR testing works.
A byproduct of the PCR testing is that, if a specimen contains a lot of
virus particles, it would naturally contains a lot of DNA fragments and
so the polymerase will need very little cycles amplify that to a degree
that is detectable. On the other hand, if we begin with very few DNA
fragments, it would take the polymerase a lot of cycles. So by
monitoring the amount of DNA fragments, we get an idea of the number of
virus particles in a specimen. The term Ct value (which stands for
cycle threshold value) is used to denote the number of cycles the
polymerase needs. Let $c$ denote the Ct value and $v$ denote the
number of virus particles, then
$ c \approx 40 - \lfloor\log_2(v)\rfloor $
is a relation between $c$ and $v$ subject to errors.
So we want to combine group testing and Ct value. But Ct value is very nasty to work with. Naturally, if we combine a specimen with $1$ virus particle and another specimen with $1000$ virus particles, we get a mixture with $1001$ virus particles. But that’s not how Ct value works, as $1$, $1000$, and $1001$ virus particles correspond to Ct values $40$, $30$, and $30$. That is to say, a smaller Ct value tend to mask a larger Ct value to the point that it completely erases the information.
Therefore, we proposed Tropical Group Testing [TropicalGT22], which is a framework to study the how group testing and Ct values should interplay.
To be elaborated…