  A key requirement in containing contagious diseases, such as the Coronavirus
disease 2019 (COVID-19) pandemic, is the ability to efficiently carry out mass
diagnosis over large populations. Some of the leading testing procedures, such
as those utilizing qualitative polymerase chain reaction, involve using
dedicated machinery which can simultaneously process a limited amount of
samples. A candidate method to increase the test throughput is to examine
pooled samples comprised of a mixture of samples from different patients. In
this work we study pooling based tests which operate in a one-shot fashion,
while providing an indication not solely on the presence of infection, but also
on its level, without additional pool tests, as often required in COVID-19
testing. As these requirements limit the application of traditional
group-testing (GT) methods, we propose a multi-level GT scheme, which builds
upon GT principles to enable accurate recovery using much fewer tests than
patients, while operating in a one-shot manner and providing multi-level
indications. We provide a theoretical analysis of the proposed scheme and
characterize conditions under which the algorithm operates reliably and at
affordable computational complexity. Our numerical results demonstrate that
multi level GT accurately and efficiently detects infection levels, while
achieving improved performance over previously proposed one-shot COVID-19
pooled-testing methods.
