Table 1.
Overview of methods for the estimation of cumulative incidence.
Fig 1.
Point-estimates of cumulative incidence using the cutoff-based methods and the mixture model.
Each violin represents 50 in silico serosurveys conducted with cohorts of 10, 000 virtual individuals and the points represent the median values. (A) Point-estimates of the cumulative incidence. The dashed line indicates the true cumulative incidence we assumed in the simulations. Please note that the scale of the y-axis differs between the sub-figures. (B) Size of the 95% uncertainty intervals. For the Rogan-Gladen and mixture-model estimates, the uncertainty intervals are the 95% confidence intervals, which we calculated with the bootstrap method (see Methods). For the Bayesian estimators, the uncertainty interval are the 95% credible intervals.
Fig 2.
Estimated fold increases in cumulative incidence for the cutoff-based methods and the mixture model.
In the simulated serosurveys, we assumed the cumulative incidence to increase from 1.5% to 15%, resulting in a true fold increase of 10 (dashed line). The violins show the distribution of 50 in silico serosurveys for both cumulative incidence levels conducted with cohorts of 10, 000 individuals and a test with an AUC-ROC value of 0.975. The dots indicate the median value.
Fig 3.
Statistical power of the mixture model.
In all simulations, the number of control and case data is fixed to 5, 000 each and the true cumulative incidence level is 8%. (A) Statistical power versus the number of individuals in the serosurvey for varying levels of test accuracy (AUC-ROC). The power is calculated as the fraction of simulated serosurveys that resulted in a cumulative incidence estimate that is within 25% of the true cumulative incidence and for which the true cumulative incidence level lies within 2 standard deviations of the estimated value. Each point in the graph represents the result of 3, 000 in silico serosurveys. (B) The minimal number of virtual individuals necessary to obtain a statistical power of 0.9 over a range of AUC-ROC values.
Fig 4.
Effect of varying the number of control and case sera used to calibrate the serological test.
(A) An example of the true distribution (solid lines) of the control (grey) and case (orange) sera, the data simulated from those distributions (histograms) and the inferred densities (dashed lines) used in the inference of the cumulative incidence. Here, 150 control and case sera have been simulated and the AUC-ROC value of the test is equal to 0.975. (B) Point estimates of cumulative incidence for various numbers of control and case sera used to calibrate the serological test and three AUC-ROC values. Each violin shows the distribution of the estimated cumulative incidence of 50 in silico serosurveys conducted with cohorts of 10, 000 virtual individuals. The red line shows the true cumulative incidence we assumed in the simulated serosurveys (8%). (C) Size of the 95% confidence intervals of the estimated cumulative incidences. (D) Statistical power versus the number of control and case sera used in the validation data for varying levels of test accuracy (AUC-ROC). Each point in the graph represents the result of 3, 000 in silico serosurveys. (E) The minimal number of virtual individuals necessary to obtain a statistical power of 0.9 over a range of number of control and case sera in the validation data.
Fig 5.
Conceptual figure on how a discrepancy between the test validation and serosurvey data can be detected.
(A) Histograms of simulated validation data from controls and severe cases. (B) Histograms of simulated validation data from controls and severe and asymptomatic cases. (C) Histogram of simulated serosurvey data when all infections in a population are severe. (D) Histogram of simulated serosurvey data when one third of all cases is asymptomatic and two thirds severe.
Fig 6.
Estimates of the cumulative incidence in a population where individuals have been uninfected, as well as symptomatically and severely infected.
The x-axes represent the AUC-ROC value between the asymptomatic and severe case distribution. The AUC-ROC value between the control and the severe case distributions is 1. Each violin represents the result of 50 simulated serosurveys with 10, 000 individuals per serosurvey. The true total cumulative incidence of severe and asymptomatic infections is 10%, of which 20% are asymptomatic. (A) Cyan violins show estimates of the total cumulative incidence based on an inferred case distribution containing only severe case sera, whereas purple violins show estimates where the case distribution is containing both asymptomatic and severe case sera. (B) The estimated cumulative incidence of the mild (light purple) and the severe (dark purple) cases, where the case sera distribution is only based on severe cases, but the likelihood equation also estimates the shape of the asympotomatic cases and their relative prevalence.
Fig 7.
Conceptual diagram of the distribution of the quantitative test measures for control and case sera.
(A) Hypothetical probability density of quantitative test measures of control sera and three possible case sera distributions. (B) ROC-curves corresponding to the distribution of quantitative test measures of the control sera and each of the possible distributions for the case sera. (C) Visualization of the ‘maximal Youden’ and ‘high specificity’ cutoffs. (D) Visualization of the ‘maximal Youden’ and ‘high specificity’ cutoffs in the ROC curves.