Let’s talk about why it’s critical NOT to assume you are immune to covid-19 when you have a positive antibody test.
I will argue that this problem can perhaps be overcome by requiring three positive tests in order to prove immunity.
Let’s make Levi’s assumption that 1 percent of the population actually has antibodies. Having antibodies is a good thing, because it means you already have had the virus and we can hope that this makes you immune.
For rounder numbers, assume that the test gives a false positive reading in 5 percent of cases where people don’t really have antibodies, and also a false negative reading in 5 percent of the cases when people do have antibodies.
Out of 10,000 people, 1 percent will have antibodies. That is 100 people, with the other 9900 not having antibodies.
Of the 100 people who have antibodies, 5 will falsely be reported as not having them. 95 will correctly be reported as having them.
Of the 9900 who do not have antibodies, 495 will falsely be reported as having them.
Altogether, 495 + 95 = 580 people will be reported as having antibodies, but most of these people will not have them! So you would not want to tell all 580 people that they don’t have to worry about getting infected if they go out and play. Most of them in fact can get infected.
My recommended solution to this problem would be to require a second test for those who test positive the first time. The second test also has to be positive in order to say that the person has antibodies. Assuming that the results are independent, the chances of two tests incorrectly reporting positive result is .0025, or 25 out of 10000.
That still might be too high. But we can take the people who test positive on two tests and make them take a third test. Only if this last test is also positive would you give the person freedom to roam.
Note that if the probability of a false positive depends more on the person being tested than on pure chance, this proposed solution will not work.