I'd be interested in whether false positives are statistically independent when you run the test multiple times on the same patient. If they are, this suggests

I'd be interested in whether false positives are statistically independent when you run the test multiple times on the same patient. If they are, this suggests that at some point you ought to simply re-test everyone who tests positive the first time. "At some point" likely only comes once the disease is well under control, maybe down to 3% of current active cases, just as a wild guess? When that happens, the prior of actually having COVID-19 gets really low. Such that even with a positive COVID-19 test the posterior probability is still somewhat low. This all falls apart if multiple tests on the same patient aren't statistically independent in false positives. For example, I'm no biologist, but I'd guess it's possible there are a few people out there whose human genetics by chance have a short sequence in common with the viral genetics of COVID-19. If that's the case, that part of the patient's own genetic code might invariably get picked up and amplified by the PCR test. So these unlucky folks will always test positive for COVID regardless of whether they have it or not.