There is a simple way to estimate how much SARS-CoV-2 PCR testing we need to start diminishing the COVID-19 pandemic. Suppose we test everyone at a rate , with a PCR test with 100% sensitivity, which means we do not miss anyone who is positive but we could have false positives. The number of positives we will find is , where is the prevalence of infectious individuals in a given population. If positive individuals are isolated from the rest of the population until they are no longer infectious with probability , then the rate of reduction in prevalence is . To reduce the pandemic, this number needs to be higher than the rate of pandemic growth, which is given by , where is the fraction of the population susceptible to SARS-CoV-2 infection and is the rate of transmission from an infected individual to a susceptible upon contact. Thus, to reduce the pandemic, we need to test at a rate higher than .
In the initial stages of the pandemic is one and , where is the mean reproduction number, which is probably around 3.7 and is the mean rate of becoming noninfectious, which is probably around 10 to 20 days. This gives an estimate of to be somewhere around 0.3 per day. Thus, in the early stages of the pandemic, we would need to test everyone at least two or three times per week, provided positives are isolated. However, if people wear masks and avoid crowds then could be reduced. If we can get it smaller then we can test less frequently. Currently, the global average of is around one, so that would mean we need to test every two or three weeks. If positives don't isolate with high probability, we need to test at a higher rate to compensate. This threshold rate will also go down as goes down.
In fact, you can just test randomly at rate and monitor the positive rate. If the positive rate trends downward then you are testing enough. If it is going up then test more. In any case, we may need less testing capability than we originally thought, but we do need to test the entire population and not just suspected cases.