Some Mathematics for the Method of Pooled PCR Test

At the time of the worldwide COVID-19 disaster, the author learned about the pooled (RT-) PCR test from the news. From the common sense of individual tests, the idea of mixing multiple samples seems taboo, however in fact many samples can be tested with a smaller number of tests by the method. As a retired researcher of mathematical engineering, the author was deeply interested in the idea and absorbed in the mathematical formulation and intensive analysis of the method. Later, he found that the original basic equation was already proposed in the old (1943) treatise [1] and so many related research works have been done and available as materials on the web [2], although many of those seem to be based on qualitative or intuitive analysis. In that sense, some of the analysis here seems to be already known in the field, but some results might be novel, such as boundary conditions, derivation of limit values, estimation of infection rate and adaptive optimization scheme of pool test, strict extension to multi-stage pool test, and explicit derivation of asymptotic approximate solutions of optimal pooling number and achieved efficiency measure, etc. In any case, he decided to put it together here as a material rather than a formal treatise, hoping that the results here would be useful for deeper mathematical insights into and better understanding of the pool inspection, and also in its actual practice.