In the group testing problem we aim to identify a small number of infected individuals within a large population. We avail ourselves to a procedure that can test a group of multiple individuals, with the test result coming out positive iff at least one individual in the group is infected. With all tests conducted in parallel, what is the least number of tests required to identify the status of all individuals? In a recent test design [ 2016] the individuals are assigned to test groups randomly, with every individual joining an equal number of groups. We pinpoint the sharp threshold for the number of tests required in this randomised design so that it is information-theoretically possible to infer the infection status of every individual. Moreover, we analyse two efficient inference algorithms. These results settle conjectures from [ 2014, 2019].
The most interesting regime for the group testing problem is when the number @math of infected individuals scales as a power @math of the entire population. Mathematically this is because in the linear regime @math the optimal strategy is to perform @math individual tests @cite_6 . Thus, for @math linear in @math there is nothing interesting to do. But the sublinear case is also of practical relevance, as witnessed by Heap's law in epidemiology @cite_42 or biological applications @cite_23 .
@cite_42 @cite_23 @cite_6
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We consider nonadaptive probabilistic group testing in the linear regime, where each of n items is defective independently with probability p in (0,1), where p is a constant independent of n. We show that testing each item individually is optimal, in the sense that with fewer than n tests the error probability is bounded away from zero. Group-testing refers to the problem of identifying (with high probability) a (small) subset of @math defectives from a (large) set of @math items via a “small” number of “pooled” tests (i.e., tests that have a positive outcome if at least one of the items being tested in the pool is defective, else have a negative outcome). For ease of presentation in this paper, we focus on the regime when @math for some @math . The tests may be noiseless or noisy , and the testing procedure may be adaptive (the pool defining a test may depend on the outcome of a previous test), or non-adaptive (each test is performed independent of the outcome of other tests). A rich body of the literature demonstrates that @math tests are information-theoretically necessary and sufficient for the group-testing problem, and provides algorithms that achieve this performance. However, it is only recently that reconstruction algorithms with computational complexities that are sub-linear in @math have started being investigated. In the scenario with adaptive tests with noisy outcomes, we present the first scheme that is simultaneously order-optimal (up to small constant factors) in both the number of tests and the decoding complexity ( @math in both the performance metrics). The total number of stages of our adaptive algorithm is “small” ( @math ). Similarly, in the scenario with non-adaptive tests with noisy outcomes, we present the first scheme that is simultaneously near-optimal in both the number of tests and the decoding complexity (via an algorithm that requires @math tests and has a decoding complexity of @math . Finally, we present an adaptive algorithm that only requires two stages, and for which both the number of tests and the decoding complexity scale as @math . For all three settings, the probability of error of our algorithms scales as @math . For each of the statements mentioned earlier about the order of the number of measurements, decoding complexity, and probability of error, we provide explicitly computed “small” universal factors in our theorem statements. Many models for the spread of infectious diseases in populations have been analyzed mathematically and applied to specific diseases. Threshold theorems involving the basic reproduction number @math , the contact number @math , and the replacement number @math are reviewed for the classic SIR epidemic and endemic models. Similar results with new expressions for @math are obtained for MSEIR and SEIR endemic models with either continuous age or age groups. Values of @math and @math are estimated for various diseases including measles in Niger and pertussis in the United States. Previous models with age structure, heterogeneity, and spatial structure are surveyed. The group testing problem is to identify a population of K defective items from a set of n items by pooling groups of items. The result of a test for a group of items is positive if any of the items in the group is defective and negative otherwise. The goal is to judiciously group subsets of items such that defective items can be reliably recovered using the minimum number of tests, while also having a low-complexity decoder. We describe SAFFRON (Sparse-grAph codes Framework For gROup testiNg), a non-adaptive group testing scheme that recovers at least a (1 − ϵ)-fraction (for any arbitrarily small ϵ > 0) of K defective items with high probability with m = 6C(ϵ)K log 2 n tests, where C(ϵ) is a precisely characterized constant that depends only on o. For instance, it can provably recover at least (1 − 10−6)K defective items with m ≃ 68K log 2 n tests. The computational complexity of the decoding algorithm is O(K log n), which is order-optimal. Further, we describe a systematic methodology to robustify SAFFRON such that it can reliably recover the set of K defective items even in the presence of erroneous or noisy test results. We also propose Singleton-Only-SAFFRON, a variant of SAFFRON, that recovers all the K defective items with m = 2e(1+α)K log K log 2 n tests with probability 1 − O(1 Kα), where α > 0 is a constant. Our key intellectual contribution involves the pioneering use of powerful density-evolution methods of modern coding theory (e.g. sparse-graph codes) for efficient group testing design and performance analysis.