  Polymerase chain reaction (PCR) testing is the gold standard for diagnosing
COVID-19. PCR amplifies the virus DNA 40 times to produce measurements of viral
loads that span seven orders of magnitude. Unfortunately, the outputs of these
tests are imprecise and therefore quantitative group testing methods, which
rely on precise measurements, are not applicable. Motivated by the
ever-increasing demand to identify individuals infected with SARS-CoV-19, we
propose a new model that leverages tropical arithmetic to characterize the PCR
testing process. Our proposed framework, termed tropical group testing,
overcomes existing limitations of quantitative group testing by allowing for
imprecise test measurements. In many cases, some of which are highlighted in
this work, tropical group testing is provably more powerful than traditional
binary group testing in that it require fewer tests than classical approaches,
while additionally providing a mechanism to identify the viral load of each
infected individual. It is also empirically stronger than related works that
have attempted to combine PCR, quantitative group testing, and compressed
sensing.
