On the True Number of COVID-19 Infections: Effect of Sensitivity, Specificity and Number of Tests on Prevalence Ratio Estimation

Abstract

In this paper, a formula for estimating the prevalence ratio of a disease in a population that is tested with imperfect tests is given. The formula is in terms of the fraction of positive test results and test parameters, i.e., probability of true positives (sensitivity) and the probability of true negatives (specificity). The motivation of this work arises in the context of the COVID-19 pandemic in which estimating the number of infected individuals depends on the sensitivity and specificity of the tests. In this context, it is shown that approximating the prevalence ratio by the ratio between the number of positive tests and the total number of tested individuals leads to dramatically high estimation errors, and thus, unadapted public health policies. The relevance of estimating the prevalence ratio using the formula presented in this work is that precision increases with the number of tests. Two conclusions are drawn from this work. First, in order to ensure that a reliable estimation is achieved with a finite number of tests, testing campaigns must be implemented with tests for which the sum of the sensitivity and the specificity is sufficiently different than one. Second, the key parameter for reducing the estimation error is the number of tests. For a large number of tests, as long as the sum of the sensitivity and specificity is different than one, the exact values of these parameters have very little impact on the estimation error.

Keywords: Covid-19; SARS-CoV-2; cross-sectional studies; false positive and false negative probabilities; molecular, serological and medical imaging diagnostics; number of infections; policy-making and testing campaigns; prevalence ratio; sensitivity and specificity.

Figure 1

A SARS-CoV-2 test represented by a random transformation from $\{I,S\}$ into $\{I,S\}$ via the conditional probability distribution $P_{Y|X}$.

Figure 2

Relation between the input vector $\mathit{x}=(x_1,x_2,\dots ,x_n)\in {\{I,S\}}^n$ (state of the individuals); the output vector $\mathit{y}=(y_1,y_2,\dots ,y_n)\in {\{I,S\}}^n$ (result of the tests); the calculation of the fraction of positive tests ${\overline{P}}_Y^{\left(n\right)}\left(I\right)$ in (3); and estimation of the prevalence ratio ${\widehat{P}}_X^{\left(n\right)}\left(I\right)$ in (8a).

Figure 3

Population in which the fraction of individuals infected with SARS-CoV-2 is P_X = 0.4 and n = 10, 000 individuals are tested (Example 1).

Figure 4

Population in which the fraction of individuals infected with SARS-CoV-2 is P_X = 0.4 and n = 10, 000 individuals are tested (Example 1).

Figure 5

Population in which the fraction of individuals infected with SARS-CoV-2 is P_X = 0.4 and n = 100, 000 individuals are tested (Example 2).

Figure 6

Population in which the fraction of individuals infected with SARS-CoV-2 is P_X = 0.4 and n = 100, 000 individuals are tested (Example 2).

Figure 7

Population in which the fraction of individuals infected with SARS-CoV-2 is P_X = 0.4 and n = 100, 000, 000 individuals are tested (Example 3).

Figure 8

Population in which the fraction of individuals infected with SARS-CoV-2 is P_X = 0.4 and n = 100, 000, 000 individuals are tested (Example 3).