Boosting test-efficiency by pooled testing for SARS-CoV-2-Formula for optimal pool size

Abstract

In the current COVID19 crisis many national healthcare systems are confronted with an acute shortage of tests for confirming SARS-CoV-2 infections. For low overall infection levels in the population the pooling of samples can drastically amplify the testing capacity. Here we present a formula to estimate the optimal group-size for pooling, the efficiency gain (tested persons per test), and the expected upper bound of missed infections in pooled testing, all as a function of the population-wide infection levels and the false negative/positive rates of the currently used PCR tests. Assuming an infection level of 0.1% and a false negative rate of 2%, the optimal pool-size is about 34, and an efficiency gain of about 15 tested persons per test is possible. For an infection level of 1% the optimal pool-size is 11, the efficiency gain is 5.1 tested persons per test. For an infection level of 10% the optimal pool-size reduces to about 4, the efficiency gain is about 1.7 tested persons per test. For infection levels of 30% and higher there is no more benefit from pooling. To see to what extent replicates of the pooled tests improve the estimate of the maximal number of missed infections, we present results for 1 to 5 replicates.

Fig 1. Group test efficiency.

(A) Increase of test efficiency in persons per test, PPT. The maximum of this curve indicates the optimal pool size, ω^opt for a given infection level (1%) and given false negative and positive rates of the test. Results are shown for r = 1, 2, 3, 4 and 5 replicates of testing the pooled sample. the maximum efficiency gain is naturally found for r = 1 and is about 5.1 persons per test, followed by r = 2 with a gain of approximately 3.6. (B) The pooled test risk factor for the pooled sample, PTRF. The result shows that taking more replicates decreases the false negatives. However, note that this also decreases the efficiency in terms of PPT. γ₊ = 0.0012 and γ₋ = 0.02.

Fig 2. Infection level dependence.

(A) Optimal pool size, ω^opt, as a function of the infection level of the population. The inset is a blow-up for low infection levels. The cases for r = 1, 3, and 5 replicates is shown in blue, red, and orange, respectively. (B) Efficiency gain of persons per test, PPT; the inset shows low infection levels. (C) The pooled testing risk factor PTRF. It is clear that taking more replicates does practically not lower PTRF, except for r = 2. γ₊ = 0.0012 and γ₋ = 0.02. By taking γ₋ = 0.05, ω^opt and PPT remain practically unchanged, FNPT doubles for all infection levels in this case (not shown).

Fig 3. Group-size dependence of the false negative factor seen in three scenarios of false negative rates that increase linearly with group-size; no group-gize dependence (A,D), a doubling (B,E), and a quadrupling (C,F) of the false negative rate values at group-size 20.

(A), (B), and (C) show that the overall best choice of replicates with respect to PTRF, are r = 1 and r = 2. The insets are blow-ups for low infection levels. Panels (D), (E), and (F) show that this remains true if we consider the optimal PTRF at a given gain in persons per test, PPT, except maybe for very low infection levels, corresponding to PTRF below 0.5%, i.e. optimal group sizes larger than 20.