Group Testing With Nested Pools

Inés Armendáriz, Pablo A. Ferrari, Daniel Fraiman, José Mario Martínez, Silvina Ponce Dawson

Abstract: In order to identify the infected individuals of a population, their samples are divided in equally sized groups called pools and a single laboratory test is applied to each pool. Individuals whose samples belong to pools that test negative are declared healthy, while each pool that tests positive is divided into smaller, equally sized pools which are tested in the next stage. In the <inline-formula> <tex-math notation="LaTeX">$(k+1)$ </tex-math></inline-formula>-th stage all remaining samples are tested. If <inline-formula> <tex-math notation="LaTeX">$p&lt; 1-3^{-1/3}$ </tex-math></inline-formula>, we minimize the expected number of tests per individual as a function of the number <inline-formula> <tex-math notation="LaTeX">$k+1$ </tex-math></inline-formula> of stages, and of the pool sizes in the first <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula> stages. We show that for each <inline-formula> <tex-math notation="LaTeX">$p\in (0, 1-3^{-1/3})$ </tex-math></inline-formula> the optimal choice is one of four possible schemes, which are explicitly described. We conjecture that for each <inline-formula> <tex-math notation="LaTeX">$p$ </tex-math></inline-formula>, the optimal choice is one of the two sequences of pool sizes <inline-formula> <tex-math notation="LaTeX">$(3^{k} \text {or }3^{k-1}4,3^{k-1}, {\dots },3^{2},3)$ </tex-math></inline-formula>, with a precise description of the range of <inline-formula> <tex-math notation="LaTeX">$p$ </tex-math></inline-formula>&#x2019;s where each is optimal. The conjecture is supported by overwhelming numerical evidence for <inline-formula> <tex-math notation="LaTeX">$p&gt;2^{-51}$ </tex-math></inline-formula>. We also show that the cost of the best among the schemes <inline-formula> <tex-math notation="LaTeX">$(3^{k}, {\dots },3)$ </tex-math></inline-formula> is of order <inline-formula> <tex-math notation="LaTeX">$O\big (p\log (1/p)\big)$ </tex-math></inline-formula>, comparable to the information theoretical lower bound <inline-formula> <tex-math notation="LaTeX">$p\log _{2}(1/p)+(1-p)\log _{2}(1/(1-p))$ </tex-math></inline-formula>, the entropy of a Bernoulli<inline-formula> <tex-math notation="LaTeX">$(p)$ </tex-math></inline-formula> random variable.