Simulation of Pool Testing to Identify Patients with Coronavirus Disease 2019 under Conditions of Limited Test Availability

FIGURE 1 | cont.

Optimal pool size and corresponding efficiency, probability of false negative, savings, and expected number of false negatives


This decision analytical model study did not require institutional review board review because it used simulation-based research, per Common Rule 45 CFR §102 (e). This study followed the Strengthening the Reporting of Empirical Simulation Studies (STRESS) reporting guideline. 

We considered a two-stage pool testing2,3 in the presence of imperfect testing, in which is prevalence, Se is test sensitivity, and Sp is specificity. We assumed that the probability of a true-positive result pool test equals Se, the probability of a false-positive result equals 1 − Sp, tests Se and Sp would be unaffected by the number of patients in a pool (k), and all tests are identically distributed. Let be a random variable denoting the number of tests needed to complete the pooling strategy. Let be the total number of patients, then there are N/k subgroups, each with members. Assuming independent Bernoulli trials, let Pk be the probability of having at least 1 positive test in patients. If 1 patient of the subgroup has RT-PCR results positive for SARS-CoV-2, then there will be (+ 1) tests necessary; otherwise, only one test will be needed. The expected number of tests for each subgroup with patients is (+ 1)Pk + (1 − Pk). Then, for N/k subgroups, the expected number of tests needed is E(Z) = (N/k)[(+ 1)Pk + (1 − Pk)] = (N/k)[kPk + 1], in which Pk = (1 − Sp)(1 − p)k Se(1 − (1 − p)k), which incorporates the effects of test sensitivity and specificity. The optimal pool size is achieved for that minimizes E(Z). To characterize the pooled test strategy, we define efficiency of the pool as Ef E(Z)/= [kPk + 1]/and expected number of false negatives (ENFN) as kp(1 − S2e). This method does not include a dilution effect on the pooling strategy sensitivity, PS S2e, as a function of pool size, where S2e provides the upper bound of the pooling sensitivity.


For demonstration purposes, we reported the results for a typical number of RT-PCR tests for 94 patients, a specificity of 100 percent, a prevalence from 0.001 percent to 40 percent, and sensitivities from 60 percent to 100 percent (Figure 1). Mathematical simulations showed that a pool testing strategy was an improvement over individual testing for a prevalence less than 30 percent and that the optimal pool size, k0, was approximately 1 + 1/√(pSe),∈ (0.1 percent, 30 percent). For a realistic scenario, such as a sensitivity of 70 percent and prevalence of 1 percent, the optimal strategy required 13 patients per subgroup. With this optimal pool size, only 16 percent as many tests would be required by subgroup tests than by individual tests. Figure 2 shows test efficiency, cost savings, probability of false negatives, and the expected number of false negatives for and p.

FIGURE 2 | Operating characteristics of pooled testing stratified by prevalence

Operating characteristics of pooled testing stratified by prevalence


This decision analytical model study found that pool testing efficiency varied with prevalence, test sensitivity, and patient pool size. Therefore, pool testing may be considered as an alternative, especially in circumstances of limited SARS-CoV-2 test availability and a COVID-19 prevalence less 30 percent. One potential limitation of pool testing is that the false-negative rate may increase owing to dilution of positive samples. The mean viral load of more than 1.5 × 104 RNA copies per mL in nasal swabs4 spikes within the first week after clinical onset and reaches 1.5×107 RNA copies per mL.5 In our example with an optimal pool size of 13 patients, high-sensitivity RT-PCR assays with a lower detection limit of more than 1,100 RNA copies per mL will detect SARS-CoV-2 in pooled samples. If sensitivity of the test at hand is a concern, test swabs can be collected and eluted into one virus transportation medium container, thereby increasing reliability. However, this requires re-collecting individual samples if a pool tests positive. The mathematical underpinning of the proposed method is generic and can be applied to other infectious diseases.6,7

Meet The Experts


Alhaji Cherif head shot

Senior Mathematician, Renal Research Institute

Nadja Grobe head shot

Supervisor, Laboratory Research, Renal Research Institute

Xiaoling Wang head shot

Senior Research Scientist, Renal Research Institute

Peter Kotanko head shot

Senior Research Scientist, Renal Research Institute


  1. Yelin I, Aharony N, Shaer Tamar E, et al. Evaluation of COVID-19 RT-qPCR test in multi-sample pools. Clin Infect Dis. Published online May 2, 2020;ciaa531. doi:10.1093/cid/ciaa531.
  2. Dorfman R. The detection of defective numbers of large populations. Ann Math Stat. 1943;14:436-440. doi:10.1214/aoms/1177731363.
  3. Johnson NL, Kotz S, Wu X. Inspection errors for attributes in quality control. London, New York: Chapman & Hall, 1991.
  4. Wang W, Xu Y, Gao R, et al. Detection of SARS-CoV-2 in different types of clinical specimens. JAMA. 2020;323(18):1843-1844. doi:10.1001/jama.2020.3786.
  5. Zou L, Ruan F, Huang M, et al. SARS-CoV-2 viral load in upper respiratory specimens of infected patients. N Engl J Med. 2020;382(12):1177-1179. doi:10.1056/NEJMc2001737.
  6. Zenios SA, Wein LM. Pooled testing for HIV prevalence estimation: exploiting the dilution effect. Stat Med 1998;17(13):1447-67. doi:10.1002/(SICI)1097- 0258(19980715)17:13<1447::AID-SIM862>3.0.CO;2-K.
  7. Cherif A, Grobe N, Wang X, Kotanko, P. Simulation of pool testing to identify patients with coronavirus disease 2019 under conditions of limited test availability. JAMA Network Open 2020 May;3(6):e2013075. doi: 10.1001/jamanetworkopen.2020.13075.

Get The Full Report

Get the full 2020 Global Annual Medical Report, along with Fresenius Medical Care North America updates throughout the year. Click the link below to learn more.