Error Correction Codes for COVID-19 Virus and Antibody Testing: Using Pooled Testing to Increase Test Reliability
11 Error Correction Codes for COVID-19 Virus andAntibody Testing: Using Pooled Testing to IncreaseTest Reliability
Jirong Yi, Myung Cho, Xiaodong Wu, Weiyu Xu, and Raghu Mudumbai
Abstract —We consider a novel method to increase the relia-bility of COVID-19 virus or antibody tests by using speciallydesigned pooled testings. Specifically, to increase test reliability,instead of testing nasal swab or blood samples from individualpersons, we propose to test mixtures of samples from manyindividuals. Group testing has traditionally been used for thepurpose of reducing the number of tests required to diagnose alarge number of individuals, but, in contrast, the pooled sampletesting method proposed in this paper also serves a differentpurpose: for increasing test reliability and providing accuratediagnoses even if the tests themselves are not very accurate.Our method uses ideas from compressed sensing and error-correction coding to correct for a certain number of errorsin the test results. The intuition is that when each individual’ssample is part of many pooled sample mixtures, the test resultsfrom all of the sample mixtures contain redundant informationabout each individual’s diagnosis, which can be exploited toautomatically correct for wrong test results in exactly the sameway that error correction codes correct errors introduced innoisy communication channels. While such redundancy can alsobe achieved by simply testing each individual’s sample multipletimes, we present simulations and theoretical arguments thatshow that our method is significantly more efficient in increasingdiagnostic accuracy. In contrast to group testing and compressedsensing which aim to reduce the number of required tests, thisproposed error correction code idea purposefully uses pooledtesting to increase test accuracy, and works not only in the“undersampling” regime, but also in the “oversampling” regime,where the number of tests is bigger than the number of subjects.The results in this paper run against traditional beliefs that, “eventhough pooled testing increased test capacity, pooled testings wereless reliable than testing individuals separately.”
I. I
NTRODUCTION
In the absence of a vaccine to the SARS-CoV-2 coronavirus,the experience of public health authorities in several countrieshas shown that large-scale shutdowns can (only) be safelyended if a systematic “test and trace” program [1, 2] isput in place to control the spread of the virus. This, inturn, is predicated on the widespread availability of massdiagnostic testing. However, most countries including the USare currently experiencing a scarcity [3] of various medicalresources including tests [4]. Pooled sample testing has beenproposed as a method for increasing the effective capacity ofexisting testing infrastructure using the classical method of
Jirong Yi, Raghu Mudumbai, Xiaodong Wu and Weiyu Xu are fromDepartment of Electrical and Computer Engineering, University of Iowa, IowaCity, IA, 52242.Myung Cho is from Department of Electrical and Computer Engineering,Penn State Behrend, Erie, PA, 16563.Email: [email protected], [email protected] group testing. However, group testing requires highly accuratetest results to be effective; a single false negative test result canpotentially cause many infected individuals to be incorrectlydiagnosed which could lead to further propogation of the virus.Of couirse, test accuracy can be increased by testing eachsample multiple times, but this defeats the main purpose ofgroup testing which is to reduce the number of tests.In this paper, we propose a more sophisticated version ofpooled sample testing that also has the ability to increase thediagnostic accuracy of existing tests even if the individualtests are not highly accurate without requiring an increase inthe number of tests. In other words, our proposed method ofpooled sample testing can deliver highly accurate diagnosticresults for individuals with very low rates of false positives andfalse negatives, even if the tests themselves are highly error-prone. Our method achieves this using mathematical ideasfrom the theories of compressed sensing and error-correctioncoding.
A. Background: COVID-19 virus and antibody tests
The most common tests for the COVID-19 virus currentlyused in the US and recommended by the CDC are swabtests. These tests use the Reverse Transcription PolymeraseChain Reaction (RT-PCR) process to selectively amplify DNAstrands produced by viral RNA specific to the Covid-19 virus.The RT-PCR process which is considered the gold standardfor the detection of mRNA consists of three distinct steps:(1) reverse transcription of RNA into cDNA, (2) selectiveamplification of a target DNA fragment using the PolymeraseChain Reaction (PCR), and (3) detection of the amplificationproduct. While the simple “end-point” version of PCR onlyallows binary detection (presence or absence) of a target RNAsequence, the real-time or quantitative version of the PCRprocess (qPCR) [5] or recent innovation digital PCR (dPCR)also allows the quantification of the RNA i.e. it produces anestimate of the quantity of the RNA material present in thesample [6].Some researchers [7] have proposed the Reverse Transcrip-tion Loop-Mediated Isothermal Amplification (RT-LAMP) asa potentially cheaper and faster alternative to RT-PCR for swabtests. While we focus on tests based on the RT-qPCR process,the methods proposed in this paper are also compatible withRT-LAMP [8] and other DNA amplification methods.The PCR-based virus tests are highly sensitive (i.e. have lowrates of false negatives) as well as specific (i.e. successfully a r X i v : . [ q - b i o . Q M ] J u l differentiates between the Covid-19 virus and other pathogensand therefore shows low false positive rates). However, pooledsampling methods require sample dilution and additionalpreparation that may potentially result in degraded sensitivityas well as specificity.In addition to tests for an active COVID-19 viral infection,there has also been interest in testing for the presence ofantibodies to the COVID-19 virus. The antibody tests mightshow whether a person in the past was infected with theCOVID-19 virus. Virus and antibody tests complement eachother.Antibody tests typically use blood samples (unlike virustests that use nasal swabs), and can use an enzyme immunoas-say process such as ELISA (enzyme-linked immunosorbentassay) [9]. ELISA’s tests typically show high sensitivity; how-ever, some of the early antibody tests that were commerciallyintroduced for COVID-19 may have issues with selectivity [9]. B. Group Testing for Increasing Testing Capacity
One simple method to increase the effective testing capacityis by testing pooled samples of a number of test subjectscollectively instead of testing samples from each person indi-vidually. In a simple version of this “group testing” [10] idea,a single negative test result on a pooled sample immediatelyshows that all individuals in that pool are infection-free. Thus,individual tests only need to be performed when a specificpooled test sample yields a positive test result. When the rateof infection in the population is low, this method allows us toreduce the total number of tests per subject so the throughputof the existing testing infrastructure is increased [11]. Poolingtests have been successfully used for diagnostic testing forinfectious diseases in the past [12, 13].The current testing bottleneck in the COVID-19 crisis hasled to a resurgence of interest in using group testing methodsfor COVID-19 diagnosis. Specifically, there have been recentstudies [14–16] into adapting pooling methods similar to [10]for Covid-19 testing. Preliminary studies on the COVID-19virus also show that pooling samples [17] can be effectivewith existing RT-PCR tests.In a recent work [18], we proposed a different approachbased on the compressed sensing theory [19–21] for detectionof viruses and antibodies using quantitative PCR test results(for example, from qPCR) for pooled sample testing. Acompressed sensing approach for virus detection was also re-ported in [22]. The compressed sensing method can potentiallyachieve higher efficiencies and better performance than grouptesting. In fact, group testing can be seen as a special case ofthe more general compressed sensing method, where the testresult is more than just binary.The basic idea behind the compressed sensing pooled sam-pling method is to prepare a set of mixtures of several individ-uals’ swab specimens, where the mixtures are carefully chosento be different from each other in such a way that, under theassumption that only a small fraction of the individual sampleshave non-zero viral RNA, each individual’s diagnostic statuscan be determined by testing a number of mixtures muchsmaller than the number of individuals.
C. Pooled Sample Testing for Increasing Testing Accuracy
Our simulations in [18] show that the compressed sensingmethod is effective in achieving a significant increase in testingcapacity. In this paper, we take this idea further and showthat the compressed sensing method can also increase theaccuracy of diagnostic tests by taking advantage of redundancyin the pooled sample test results to correct for some numberof incorrect test results.To motivate this idea, consider a population of N individu-als. Let b i ∈ { , } , i = 1 . . . N represent the infection statusof the i ’th individual in the population i.e. b i = 1 indicatesindividual i is infected with the virus. The information vector b . = [ b b . . . b N ] ∈ { , } N represents the infection statusof the population as a whole.Let p denote the infection rate in the population: p . = E (cid:16) N (cid:80) Ni =1 b i (cid:17) . While the information vector b can be rep-resented by the N information bits b i , i = 1 . . . N , anelementary result from information theory shows that theentropy of the information vector is much smaller than N bits, when the infection rate is low: h ( b ) ≡ − N p log ( p ) − N (1 − p ) log (1 − p ) (cid:28) N, if p (cid:28) (1)where we assumed that each individual in the populationindependently has a probability p of being infected. The en-tropy h ( b ) represents the number of bits required to losslesslyrepresent the information in b .Thus, (1) can be interpreted as a theoretical justification forpooled sample testing: in theory, we only need tests that delivera total of N t = h ( b ) bits of information in order to fullyrecover the infection status b i of every individual in the pool. Ifthe tests are binary i.e. only indicate positive/negative infectionstatus and are completely error-free, then in theory we canfully diagnose all N individuals with as few as h ( b ) such tests.If the test provide richer non-binary results (e.g. quantificationof viral RNA concentration from RT-qPCR tests), in theory thenumber of tests needed may be much smaller than h ( b ) .In this sense, pooled sample testing methods such as thecompressed sensing method, can be thought of as data com-pression codes. However, the tools of information theory allowus to design codes that have much more powerful capabilitiesthan just lossless data compression. In particular, we cangeneralize from lossless data compression to codes that canperform error corrections. In the context of virus testing, thismeans a class of pooled sample testing techniques that canachieve accurate diagnostic results even with tests that areindividually highly error-prone.
We show in this paper a class of pooled sample testingmethods that do exactly this: increased diagnostic accuracy(error correction) without requiring an increased number oftests. In other words, we demonstrate a method of pooledsample testing that requires no larger number of tests inaggregate, yet delivers more accurate diagnostic results thanseparately testing each individual. In this paper, our focus isnot on increasing test capacity by reducing the number ofrequired tests, but is instead on purposefully creating pooled samples to increase test reliability or to increase tests’ robust-ness against test errors. There are several recognizable distinc-tions between this work on error correction and recent grouptesting/compressed sensing works for virus/antibody detection:1) The main purpose of error correcting pooled testing is toincrease test reliability, not to reduce required test numbers asin group testing/compressed sensing, even though this paper’sproposed approach provides error correction capability also inthe case of using a reduced number of tests; 2) In our proposederror correction codes using pooled testing, the testings canoperate in an “oversampling” regime where the number ofperformed tests is larger than the number of subjects; 3) Inaddition, in error correction codes using pooled testing, we donot necessarily require the prevalence to be low or require theinvolved signal to be sparse: the signal considered can be afully dense signal.We remark that the results in this paper run against tra-ditional wisdoms that grouping samples together for test-ing would lower test accuracy or reliability compared withindividual separate testings, due to factors such as sampledilutions and pipetting errors, even though pooled testingscould increase test capacity. Our results show this conventionalwisdom can be wrong, and, in some cases, we can purposefully perform pooled testing to significantly increase, rather thandecrease, test accuracy or reliability.
D. Related Works
Our work is most related to [20], one of the seminal papersin compressed sensing, which uses linear programming toperform decoding under channel errors. Compared with [20],in this paper, we purposefully design pooled testing matriceswith ‘0’ or ‘1’ elements to increase test reliability, instead ofbeing given a particular channel (linear transformation) as in[20]. In addition, we are working with non-negative signals,which bring additional structures for sensing and inference[23, 24]. Compared with recent works which aim to boosttest robustness against noisy measurements in group testingand compressed sensing [25, 26], our work considers not only“undersampling” regime, but also “oversampling” regime; notonly sparse signals, but also dense signals. We purposefullydesign/use pooled testing to increase test reliability, instead oftrying to increase the reliability of group testing/compressedsensing used in the “undersampling” regime mainly for in-creasing test capacity.II. P
ROBLEM S TATEMENT
In this section, we will give a mathematical formulationof performing robust virus testing through error correctioncode. We will focus on describing the idea of error correctioncode for virus testing through quantitative pooled testing, eventhough the idea of error correction code can be extendedto traditional qualitative pooled testing. We also focus ourdescription on virus testing, and the mathematical principlesextend to antibody testing.The quantitative modeling of the pooled testing problemrequires the application of real-time quantitative polymerasechain reaction (real-time qPCR) which is built on top of the PCR and conducted in a thermal cycler. The real-time qPCRcan give quantitative measurements of the amplified DNAcopies by using fluorescent reporters in multiple PCR cycles.In each PCR cycle, the DNA template can be doubled, and thestrength of the signal from fluorescent reporter is proportionalto the number of amplified DNA molecules. One can use thethreshold cycle C t , which is defined as the number of cyclesrequired for the fluorescent signal to cross a threshold, todetermine the quantity of DNAs in the qPCR.Assume that we get n samples for n subjects with onesample for each, and we will perform m tests to determine thequantities of COVID-19 viruses in these samples. We denoteby x ∈ [0 , ∞ ) n the quantity of the DNA that can be generatedfrom the subjects’ viral RNAs. In each of the m tests, we willcreate a pooled sample by mixing the samples from multiplesubjects. We use a matrix P ∈ { , } m × n to denote theparticipation of n samples in m tests, i.e. the sample of the j -th ( ≤ j ≤ n ) subject participates in the i -th ( ≤ i ≤ m )test if P ij = 1 , and it will not be used in the j -th test if P ij = 0 . This means that the number of ’s in the i -th columnof P is the number of tests that the sample of i -th subject willparticipate in. We will model the allocation of the subjects’samples by a matrix W ∈ [0 , m × n , and each W ij is thefraction of the j -th sample used in the i -th test, meaning thesum of each column in W is no bigger than 1. With thosesetup above, we get a measurement matrix as A := P (cid:12) W , (2)where (cid:12) represents the Hadamard multiplication.The corresponding m mixed samples will go through m quantitative PCR to quantify the amount of DNAs. Dueto potential background noises and gross errors caused byfactors such as dilutions, sample and reagent contamination,and operational mistakes, the final quantitative measurements y ∈ R m from the real-time PCR can be modeled as y = f ( Ax ) + v + e , (3)where f ( · ) : R n → R m , v ∈ R m , and e ∈ R m represent thetrue signal, the observation noise, and the possible gross error.For example, in the ideal case f ( Ax ) = A x , we have y = Ax + e + v . (4)Our goal is to recover the sample measurements x ∈ [0 , ∞ ) n for n subjects from m tests measurements y ∈ R m underpossible outliers.In practice, according to [18, 28], a measurement matrixfrom the expander bipartite graph can achieve good practicalperformance with theoretical justifications, and we can specifythe matrix P as such matrices, i.e., a sparse binary matrix.The sparsity of matrix P is determined by taking practicalconsiderations such as reducing the operational complexityof pooling, and reducing the risk of dilution resulting frompooling. Due to the above constraints, one can design thematrix P based on bipartite expander graphs [29, 30]. Thoughone has the freedom to design the allocation matrix W , onecan use an even-allocation scheme. Thus, if the j -th subjectis involved in c tests, then the j -th column of P has only c Fig. 1: Quantitative relation between relative fluorescence andcycle number [27].1’s, and the j -th column of A will have nonzero values at thecorresponding locations being c .The low prevalence among population in practice allowsus to assume that the signal x ∈ [0 , ∞ ) n is sparse orapproximately sparse, i.e., most of its entries are zero (orextremely close to zero). We also assume that that the grosserror v ∈ R m is sparse, due to relative rareness of operationalmistakes, chemical reaction failure, and sample dilution.Under all these assumptions, we can formulate the problemof recovering x ∈ R n from y ∈ R m (we remark here that m can be bigger than n ) as minimize (cid:107) z (cid:107) + λ (cid:107) y − Az − u (cid:107) , subject to (cid:107) u (cid:107) ≤ (cid:15), z ≥ , (5)where (cid:107) z (cid:107) is the number of nonzero elements in z , λ ∈ R isa tuning parameter for controlling the tradeoff between (cid:107) z (cid:107) and (cid:107) Az − y − u (cid:107) , the (cid:107) u (cid:107) is the (cid:96) norm of u , (cid:15) ≥ isa parameter controlling the tolerance for noise, and the x ≥ means that every element of x is nonnegative. In (5), weused z as an estimate for x and u as an estimate for v , and y − Az − u is an estimate for e .Due to combinatorial nature of (cid:107) · (cid:107) , solving (5) is some-times computationally challenging, and the norm (cid:107) · (cid:107) can beused as a relaxation technique instead in practice to achievegood performance without much computational complexity[18, 28]. Thus, we can reformulate (5) as minimize (cid:107) z (cid:107) + λ (cid:107) y − Az − u (cid:107) , subject to (cid:107) u (cid:107) ≤ (cid:15), z ≥ , (6)where (cid:107) z (cid:107) is the sum of the absolute value of all the elementsin z , and we will refer (6) as (cid:96) − (cid:96) minimization. Once theestimate for x is obtained, If z j ≥ τ where τ is the thresholdvalue, then we claim the j -th subject is infected and tests“positive”; otherwise, we declare “negative” test result for thesubject.There is a large volume of literature which proposed al-gorithms for solving (6) under certain conditions such asthe restricted isometry property and the null space condition.These ideas range from using off-the-shelf softwares such asCVX [31], to algorithms specifically designed for (cid:96) − (cid:96) minimization such as the homotopy method and iterativelyreweighted least square algorithm [28]. In this paper, we willuse the CVX [31].III. N UMERICAL E XPERIMENTS
In this section, we conduct numerical experiments in orderto evaluate the performance of our proposed method, which isthe error correcting pooled testing introduced in (6). We focuson models for COVID-19 virus testing. In pooled testing, weuse Bernoulli matrices each element of which is either 1 or 0.The numbers of people tested are set to 25 and 40, i.e., n = 25 and . We consider a scenario where k out of n people haveCOVID-19 virus by setting randomly chosen k elements in x ∈ R n to be positive and other n − k elements to zero.The value of each of the non-zero elements in x is chosenuniformly at random from [5 , . We consider the sparsitylevel k from 1 to 6 in the simulations.In pooled testing, the Gaussian noise vector v in (4) is gen-erated i.i.d. according to the Gaussian distribution N (0 , σ ) ,where the noise level σ is varied from 5e-1 to 2e0. Forthe outliers, we generate the sparse outlier vector e as in(4) by having each element of e be a non-trivial (non-zero)outlier with probability P out . If an outlier indeed happens atthe i -th measurement, we generate the outliers for the i -themeasurement in the following way: 1) If the corresponding ( Ax ) i is positive, with 95% probability, we set the outlier e i as − ( Ax ) i and reset v i = 0 such that y i = 0 (this isto simulate a “false negative” measurement); with the other5% probability, we set the outlier e i as × q + 2 , where q follows the standard Gaussian distribution N (0 , , and keepthe originally generated v i ; 2) if the corresponding ( Ax ) i isequal to 0, we will set e i as × | q | + 2 , where q followsdistribution N (0 , , and keep the originally generated v i . Ifthere is no non-trivial outlier for the i -the measurement, and ( Ax ) i = 0 , we set v i = 0 and e i = 0 , to simulate this testas a “negative” test revealing 0 virus. Since the measurementresults y must be a non-negative vector, for the i -th test, welet y i = max { ( Ax ) i + e i + v i , } to make sure the generatedmeasurement y be a non-negative vector (to avoid randomlygenerated e and v being too small dragging ( Ax ) i + e i + v i tonegative region). In the numerical evaluations, we respectivelyconsider three probabilities of the outlier error, denoted by P out , to be , , and . For pooled testing, we use (6)to recover x , and use threshold τ = 1 to decide whether asubject tests positive or negative. We introduce this thresholdto suppress the noise in x may caused by outlier error andGaussian noise.We compare the pooled testing against the individual testingmodel, where the individual samples of subjects are testedseparately (possible multiple times). In the individual testing,the i -th test is modeled as y i = x mod ( i − ,n )+1 + e i + v i , i = 1 , , ..., m (7)where y i is i -th measurement result, mod ( · , · ) means mod-ule operation, e i is the outlier, and v i is Gaussian noisefollowing distribution N (0 , σ ) . We generate the noises andoutliers in the same way as described for the pooled testing, except for in individual testing, we generate outliers and noisesbased on x mod ( i − ,n )+1 instead of ( Ax ) i . For example, for n = 25 and i = 27 , y i is the measurement result for the 2ndsubject (this subject has been tested once already in the 2ndtest), and the outlier and noise simulated for the -th testis randomly generated based on x . In individual testing, if m < n , there must be some subjects who do not get qPCRtested at all; and in those cases, and for our simulations, weconsider these subjects as COVID-19 negative. Additionally, inindividual testing, if a subject is tested multiple times, and aslong as one result is identified as being COVID-19 positive,we consider the subject as COVID-19 positive. This comesfrom the motivation of not missing COVID-19 positive cases.The number of measurements, denoted by m , is varied from10 to 50 in n = 25 and from 10 to 80 in n = 40 . Thus, inour individual testing scenario, the maximum number of testsfor a subject is two.For both the pooled testing and the individual testing, werun 100 random trials for each parameter setup, and recordthe False Negative Rate (FNR) and the False Positive Rate(FPR), which are computed on average out of 100 trials asfollows: FNR = Number of negative cases in people with COVID-19 virusNumber of people having COVID-19 virus , FPR = Number of positive cases in people without COVID-19 virusNumber of people not having COVID-19 Virus . Hence, the FNR represents the percentage of people identifiedas COVID-19 negative among people infected with COVID-19 viruses, which can be a critical error in COVID-19 testing.For the FPR, it is interpreted as the percentage of peoplewho are diagnosed as having COVID-19 virus among peoplenot infected with COVID-19 viruses. The FPR and FNR canhappen because of limited sensitivity, limited specificity, sam-ple contamination, sample dilutions, failed chemical reactions,operational mistakes and other factors. The FPR can be an im-portant concern in COVID-19 antibody testing, while the FNRis an important concern in COVID-19 virus testing. Lowerboth FPR and FNR represent the better testing performancein detecting virus and checking antibody. Additionally, if onemethod (say, Method A) achieves the same FNR and FPRbut with a smaller number of tests than another method (say,Method B), then we deem Method A better than MethodB. This is because the number of tests is related to thethroughput of testing, and a high-throughput testing allows usto increase the capacity of testing in a limited time. Therefore,we will compare the FNR and the FPR of the pooled testingagainst those of the individual testing under various setupsof parameters including the number of tests, noise levels andprobability of outlier occurrences.
A. Different probabilities of outlier errors
In Figures from 2 to 5, (a), (b), and (c) show the FNR ofthe pooled testing and the individual testing in log-scale withdifferent probabilities of outlier error varied from from to , and (d), (e), and (f) describe the corresponding FPR.Here, the number of people tested is set to 25, i.e., n = 25 , and the number of people having COVID-19 virus is variedfrom 1 to 6 out of 25, i.e., k = 1 , ..., . The noise level is fixedto e . From various simulations as shown in Figures from 2and 5, the pooled testing lowers both the FNR and the FPRas the number of measurements increases. This is because asthe number of measurements increases, we can recover moreaccurate results x and e via (cid:96) − (cid:96) minimization introducedin (6). Unlike the pooled testing, the individual testing canreduce the FNR as the number of measurements increases,while there is a slight increase in FPR at the same time. This isbecause the individual testing diagnoses the COVID-19 virustest “positive” once we have one positive test result amongmultiple tests.From these various simulation results with different proba-bilities of outlier errors, in many cases, we demonstrate thatthe pooled testing can simultaneously have (significant) lowerFNR and FPR than those of the individual testings. In a limitednumber of cases under m < n , the individual testing provideslower (although not significantly lower) FPR than that of thepooled testing, because only a few people are tested underindividual testing, and fewer false positive mistakes are made(recall that the tested subjects are assumed to be diagnosedas “negative” ) Additionally, since, under m < n , there aresimply untested subjects in individual testing, and, in general,a false negative outlier can have more impact on an individiual,the individual testing has relatively higher FNR across all theparameter setups of m and n .Furthermore, we demonstrate the outperformance of thepooled testing in the COVID-19 testing against the individualtesting with more test subjects. Figures from 6 to 9 showthe comparison in both FNR and FPR as the number ofmeasurements increases, between the pooling testing and theindividual testing, for n = 40 . In Figures from 6 to 9, (a), (b),and (c) show the FNR of the pooled testing and the individualtesting with different probabilities of outlier error from to and different sparsity level from k = 1 to k = 6 .Correspondingly, in Figures from 6 to 9, (d), (e), and (f)indicate the FPR of the both testing. Through the simulationresults shown in Figures from 6 to 9, with even larger n , itis shown that the pooled testing can identify people havingCOVID-19 virus more accurately than the individual testingwith small number of measurements. Therefore, the pooledtesting can simultaneously have higher throughput and higheraccuracy than the individual testing. B. Different noise levels
In order to check the impact of Gaussian noises on thedetection performance, we further run simulations by varyingnoise levels. We vary the Gaussian noise level from 5e-1 to2e0. We randomly choose 100 trials and record the FNR andthe FPR of the pooled testing and the individual testing. Herein the simulations, we set the sparsity level to , i.e., k = 3 ,and consider the two probability of outlier error and .Figures 10 and 11 illustrate the simulation results in log-scalewith P out = 0 . when n = 25 and n = 40 respectively.In addition, Figures 12 and 13 show the simulation resultsin log-scale with P out = 0 . when n = 25 and n = 40 (a) FNR ( P out = 0 . ) (b) FNR ( P out = 0 . ) (c) FNR ( P out = 0 . )(d) FPR ( P out = 0 . ) (e) FPR ( P out = 0 . ) (f) FPR ( P out = 0 . ) Fig. 2:
Simulations for different probabilities of outlier errors. False Negative Rate (FNR) and the corresponding False Positive Rate (FPR)with n = 25 , k = 1 , and Gaussian noise level 1e0. (a) FNR ( P out = 0 . ) (b) FNR ( P out = 0 . ) (c) FNR ( P out = 0 . )(d) FPR ( P out = 0 . ) (e) FPR ( P out = 0 . ) (f) FPR ( P out = 0 . ) Fig. 3:
Simulations for different probabilities of outlier errors. False Negative Rate (FNR) and the corresponding False Positive Rate (FPR)with n = 25 , k = 2 , and Gaussian noise level 1e0. (a) FNR ( P out = 0 . ) (b) FNR ( P out = 0 . ) (c) FNR ( P out = 0 . )(d) FPR ( P out = 0 . ) (e) FPR ( P out = 0 . ) (f) FPR ( P out = 0 . ) Fig. 4:
Simulations for different probabilities of outlier errors. False Negative Rate (FNR) and False Positive Rate (FPR) with n = 25 , k = 3 , and Gaussian noise level 1e0. (a) FNR ( P out = 0 . ) (b) FNR ( P out = 0 . ) (c) FNR ( P out = 0 . )(d) FPR ( P out = 0 . ) (e) FPR ( P out = 0 . ) (f) FPR ( P out = 0 . ) Fig. 5:
Simulations for different probabilities of outlier errors. False Negative Rate (FNR) and the corresponding False Positive Rate (FPR)with n = 25 , k = 6 , and Gaussian noise level 1e0. (a) FNR ( P out = 0 . ) (b) FNR ( P out = 0 . ) (c) FNR ( P out = 0 . )(d) FPR ( P out = 0 . ) (e) FPR ( P out = 0 . ) (f) FPR ( P out = 0 . ) Fig. 6:
Simulations for different probabilities of outlier errors. False Negative Rate (FNR) and the corresponding False Positive Rate (FPR)with n = 40 , k = 1 , and Gaussian noise level 1e0. (a) FNR ( P out = 0 . ) (b) FNR ( P out = 0 . ) (c) FNR ( P out = 0 . )(d) FPR ( P out = 0 . ) (e) FPR ( P out = 0 . ) (f) FPR ( P out = 0 . ) Fig. 7:
Simulations for different probabilities of outlier errors. False Negative Rate (FNR) and the corresponding False Positive Rate (FPR)with n = 40 , k = 2 , and Gaussian noise level 1e0. (a) FNR ( P out = 0 . ) (b) FNR ( P out = 0 . ) (c) FNR ( P out = 0 . )(d) FPR ( P out = 0 . ) (e) FPR ( P out = 0 . ) (f) FPR ( P out = 0 . ) Fig. 8:
Simulations for different probabilities of outlier errors. False Negative Rate (FNR) and False Positive Rate (FPR) with n = 40 , k = 3 , and Gaussian noise level 1e0. (a) FNR ( P out = 0 . ) (b) FNR ( P out = 0 . ) (c) FNR ( P out = 0 . )(d) FPR ( P out = 0 . ) (e) FPR ( P out = 0 . ) (f) FPR ( P out = 0 . ) Fig. 9:
Simulations for different probabilities of outlier errors. False Negative Rate (FNR) and the corresponding False Positive Rate (FPR)with n = 40 , k = 6 , and Gaussian noise level 1e0. respectively. In those figures, (a) and (d) are for noise level e − , and (b) and (e) are for noise level e , and (c) and(f) are for noise level e . As shown in the figures, as thenoise level increases, both FNR and FPR of the pooled testingand the individual testing become worse. However, the pooledtesting still outperforms the individual testing with variousnoise levels in term of the FNR for every examined valueof m , and in term of the FPR for m ≥ n . C. Different sparsity levels
In this subsection, we further run simulations by varyingthe sparsity level, i.e., the number of people having COVID-19 viruses. For these simulations, we set the noise level to e − and the probability of outlier error P out to . . Wevary the sparsity level k from 1 to 6. Figures 14 and 15 showthe FNR and FPR of both the pooled testing and individualtesting with different sparsity level when n = 25 and n = 40 respectively. D. Discussion
The overall takeaway from the Figures 6 to 9 is that thepooled sampling method achieves significantly higher accu-racy compared to individual testing. Also in absolute terms, thepooled sampling method is able to provide accurate diagnosticresults even when individual test results are highly noisy. Somespecific observations from the simulations are as follows. • In most of the simulations, the pooled sampling methodsimultaneously achieves lower FPR and FNR than in-dividual sampling.
We did not observe even a singleinstance when the opposite was true i.e. where individualtesting outperformed the pooled sampling method in bothFPR and FNR. • The FPR for the individual sampling method actuallygets worse with increased number of measurements. Thisis simply an artifact of the individual testing method inconservative strategy in order to prevent miss in COVID-19 positive case. The overall accuracy of the individualtesting method does always improve with increased num-ber of measurements when FNR is taken into accountalong with FPR. • For the pooled sampling method, both FPR and FNRalways monotonically decrease with increased number ofmeasurements. (The apparent non-monotonicity in e.g.Fig. 14(f) is simply an artifact of the randomness in thesimulations.) (a) FNR (Noise level: 5e-1) (b) FNR (Noise level: 1e0) (c) FNR (Noise level: 2e0)(d) FPR (Noise level: 5e-1) (e) FPR (Noise level: 1e0) (f) FPR (Noise level: 2e0) Fig. 10:
Simulations for different noise levels. False Negative Rate (FNR) and the corresponding False Positive Rate (FPR) with n = 25 , k = 3 , P out = 0 . , and noise level varied from 5e-1 to 2e0. (a) FNR (Noise level: 5e-1) (b) FNR (Noise level: 1e0) (c) FNR (Noise level: 2e0)(d) FPR (Noise level: 5e-1) (e) FPR (Noise level: 1e0) (f) FPR (Noise level: 2e0) Fig. 11:
Simulations for different noise levels. False Negative Rate (FNR) and the corresponding False Positive Rate (FPR) with n = 40 , k = 3 , P out = 0 . , and noise level varied from 5e-1 to 2e0. (a) FNR (Noise level: 5e-1) (b) FNR (Noise level: 1e0) (c) FNR (Noise level: 2e0)(d) FPR (Noise level: 5e-1) (e) FPR (Noise level: 1e0) (f) FPR (Noise level: 2e0) Fig. 12:
Simulations for different noise levels. False Negative Rate (FNR) and the corresponding False Positive Rate (FPR) with n = 25 , k = 3 , P out = 0 . , and noise level varied from 5e-1 to 2e0. (a) FNR (Noise level: 5e-1) (b) FNR (Noise level: 1e0) (c) FNR (Noise level: 2e0)(d) FPR (Noise level: 5e-1) (e) FPR (Noise level: 1e0) (f) FPR (Noise level: 2e0) Fig. 13:
Simulations for different noise levels. False Negative Rate (FNR) and the corresponding False Positive Rate (FPR) with n = 40 , k = 3 , P out = 0 . , and noise level varied from 5e-1 to 2e0. (a) FNR ( k = 1 ) (b) FNR ( k = 3 ) (c) FNR ( k = 6 )(d) FPR ( k = 1 ) (e) FPR ( k = 3 ) (f) FPR ( k = 6 ) Fig. 14:
Simulations for different sparsity levels. False Negative Rate (FNR) and the corresponding False Positive Rate (FPR) with n = 25 , P out = 0 . , and noise level e − , and k varied from to . (a) FNR ( k = 1 ) (b) FNR ( k = 3 ) (c) FNR ( k = 6 )(d) FPR ( k = 1 ) (e) FPR ( k = 3 ) (f) FPR ( k = 6 ) Fig. 15:
Simulations for different sparsity levels. False Negative Rate (FNR) and the corresponding False Positive Rate (FPR) with n = 40 , P out = 0 . , and noise level e − , and k varied from to . R EFERENCES[1] V. Lee, C. Chiew, and W. Khong, “Interrupting transmission of COVID-19: lessons from containment efforts in Singapore,”
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