We iterate Dorfman's pool testing algorithm \mbox{%DIFAUXCMD dorfman previous stage. We compute the mean and variance of the number of tests per individual as a function of the pool sizes m=(m_1,\dots,m_k) in the first k stages; in the (k+1)-th stage all remaining individuals are tested. Denote by D_k(m,p) the mean number of tests per individual, which we will call the cost of the strategy m. The goal is to minimize D_k(m,p) and to find the optimizing values k and m. We show that the cost of the strategy (3^k,%DIFDELCMD < \dots%%% ,3) with k\approx \log_3(1/p) is of order p\log(1/p), and differs from the optimal cost by a fraction of this value. To prove this result we bound the difference between the cost of this strategy with the minimal cost when pool sizes take real values. We conjecture that the optimal strategy, depending on the value of p, is indeed of the form (3^k ,%DIFDELCMD < \dots%%% , 3 ) or of the form (3 ^{k-1}4,3^{k-1}  \dots,3  ), with a precise description for k. This conjecture is supported by inspection of a family of values of p. Finally, we observe that for these values of p and the best strategy of the form (3^k,\dots,3), \big(\big) the standard deviation of the number of tests per individual is of the same order as the cost . As an example, when p=0.02 , the optimal strategy is k=3, m=(27,9,3) . The cost of this strategy is 0.20 , that is, the mean number of tests required to screen 100 individuals is 20.
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. This scheme is called adaptive, because the composition of the pools at each stage depends on results from previous stages, and nested because each pool is a subset of a pool of the previous stage. Is the infection probability p is not smaller than 1-3^{-1/3 compute the mean D_k(m,p) and the variance of the number of tests per individual as a function of the pool sizes m=(m_1,\dots,m_k) in the first k stages; in the (k+1)-th stage all remaining samples are tested. The case k=1 was proposed by Dorfman in his seminal paper in 1943. The goal is to minimize D_k(m,p) , which is called the cost associated to~ m. We show that %DIFDELCMD < \dots%%% for p\in (0, 1-3^{-1/3 (3^k %DIFDELCMD < \dots%%% \text{ or 3 ^{k-1}4,3^{k-1} , \dots,3 ^2,3 ), with a precise description of the range of p's where each holds. We then focus on schemes of the type (3^k,\dots,3), and estimate that the cost of the best scheme of this type for p, determined by the choice of k=k_3(p), is of order O\big(p\log(1/p)\big). This is the same order as that of the cost of the optimal scheme, and the difference of these costs is explicitly bounded . As an example, for p=0.02 the optimal choice is k=3, m=(27,9,3) , with cost 0.20 ; that is, the mean number of tests required to screen 100 individuals is 20.
R
We iterate Dorfman's pool testing algorithm \mbox{%DIFAUXCMD dorfman
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. This scheme is called adaptive, because the composition of the pools at each stage depends on results from previous stages, and nested because each pool is a subset of a pool of the
R
We
Is the infection probability p is not smaller than 1-3^{-1/3
R
and
D_k(m,p) and the
R
individuals
samples
R
Denote by D_k(m,p) the mean number of tests per individual, which we will call the cost of the strategy m.
The case k=1 was proposed by Dorfman in his seminal paper in 1943.
R
and to find the optimizing values k and
, which is called the cost associated to~
D
the cost of the strategy (3^k,
R
,3) with k\approx \log_3(1/p) is of order p\log(1/p), and differs from the optimal cost by a fraction of this value. To prove this result we bound the difference between the cost of this strategy with the minimal cost when pool sizes take real values. We conjecture that the optimal strategy, depending on the value of p, is indeed of the form
for p\in (0, 1-3^{-1/3
D
,
R
,
\text{ or
D
) or of the form (3
A
,
A
^2,3
R
for k. This conjecture is supported by inspection of a family of values of p. Finally, we observe that for these values of p and the best strategy of the form
of the range of p's where each holds. We then focus on schemes of the type
A
and estimate that the cost of the best scheme of this type for p, determined by the choice of k=k_3(p), is of order O
A
p\log(1/p)
A
. This is
D
standard deviation of the number of tests per individual is of the
R
the cost
that of the cost of the optimal scheme, and the difference of these costs is explicitly bounded
R
when
for
R
, the optimal strategy
the optimal choice
R
. The cost of this strategy is
, with cost
R
,
;