Citation: Jiang, H.; Ahn, H.; Li, X.
Group Testing with Consideration of
the Dilution Effect. Mathematics 2022,
10, 497. https://doi.org/10.3390/
math10030497
Academic Editors: Sławomir
Nowaczyk, Rita P. Ribeiro and
Grzegorz Nalepa
Received: 29 December 2021
Accepted: 2 February 2022
Published: 3 February 2022
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Article
Group Testing with Consideration of the Dilution Effect
Haoran Jiang, Hongshik Ahn * and Xiaolin Li
Department of Applied Mathematics and Statistics, Stony Brook University, Stony Brook, NY 11794-3600, USA;
haoran.jiang@stonybrook.edu (H.J.); xiaolin.li@stonybrook.edu (X.L.)
* Correspondence: hongshik.ahn@stonybrook.edu
Abstract:
We propose a method of group testing by taking dilution effects into consideration. We
estimate the dilution effect based on massively collected RT-PCR threshold cycle data and incorporate
them into optimizing group tests. The new constraint helps find a robust solution of a nonlinear
equation. The proposed framework has the flexibility to incorporate geographic and demographic
information. We conduct a Monte Carlo simulation to compare different group testing approaches
under the estimated dilution effect. This study suggests that increased group size adversely impacts
the false negative rate significantly when the infection rate is relatively low. Group tests with optimal
pool sizes improve the sensitivity over group tests with a fixed pool size. Based on our simulation
study, we recommend single group testing with optimal group sizes.
Keywords: dilution effect; group testing; optimal group size; sensitivity; sequential test
1. Introduction
Group testing, also known as pooled testing or batch testing, works by amalgamating
specimens from individuals into pools and performing tests on these pools. If the group is
tested negative, all of its members are declared negative. If the group is tested positive, each
member has the remainder of his/her original specimen tested separately to determine the
positive/negative outcome. Its implementation has the potential to greatly accelerate the
rate of testing and increase the test capacity especially when the prevalence rate is relatively
low. The concept of group testing was first introduced for detecting syphilis in US soldiers
during World War II [
1
]. Group testing was studied as an efficient method to detect com-
munity transmission [
2
]. During the COVID-19 outbreak in 2020, Stanford Medical Center,
the University of Nebraska, and the Clinical Reference Laboratory applied group testing as
the screening strategies for the general population [
2
,
3
]. Meanwhile, several universities,
including Duke University, Michigan State University, the State University of New York,
and Syracuse University implemented group testing as their campus screening strategy.
Group testing was discussed with test errors in detail, and it was confirmed that
Dorfman’s method has lower sensitivity than individual testing [
4
]. This drawback was
mitigated [
5
] by a new multi-step group testing followed by possible sequential individ-
ual tests.
There are two important considerations for applying group testing: group size and
dilution effects. Pooling optimal number of specimens together does not adversely affect
the detection of positive specimens and achieved 57% fewer tests on average compared to
individual testing [6].
The optimal group size [
7
] was determined by incorporating the dilution effect and
the expected cost calculated under Dorfman’s procedure. The concentration determines the
group testing sensitivity [
8
]. Ordered pooling is the most efficient way to group patients if
the function of the dilution effect is concave [
9
]. This conclusion generalized the ordered
pooling algorithm [10] from no testing errors to testing errors with dilution effects.
Viral load, also known as viral burden, is a numerical expression of the quantity of a
virus in a given volume of fluid. Viral load (viral RNA concentration) in patient samples
and the rate of successful isolation of virus from clinical specimens in cell culture are the
Mathematics 2022, 10, 497. https://doi.org/10.3390/math10030497 https://www.mdpi.com/journal/mathematics
