Fig 1.
Overview of different approaches for nowcasting the effective reproduction number Rt.
(A) Case counts by date of report are not biased by right truncation and can be used to estimate Rt via a direct (no truncation adjustment) approach, assuming knowledge of the delay between infection and report. (B) Case counts by date of symptom onset are only delayed by the incubation period but subject to right truncation. The truncation can be adjusted for by using a stepwise (additional adjustment step) or generative (integrated truncation model) approach. (C) When line list data are incomplete, cases with missing onset date must be accounted for. This can be achieved using a stepwise (additional imputation step) or generative (integrated missingness model) approach. (I–V) Models used in the steps of the different approaches. To ensure comparability, the model components for Rt estimation, truncation adjustment, and missing date imputation were designed to be maximally similar across all approaches (see S1 Appendix B for full model definitions).
Fig 2.
Nowcasts of Nt on line list data of a simulated first wave scenario using different approaches of adjusting for right truncation.
Shown are the true number of cases by symptom onset date Nt (black), the number of cases reported until the nowcast date (grey bars), and point nowcasts with 95% credible intervals (CrI) in four different phases of the epidemic wave, obtained through i) a stepwise approach using cases by date of symptom onset with a truncation adjustment step (blue), and ii) a generative approach using cases by date of symptom onset with an integrated truncation and renewal model (red). The direct approach using cases by date of report cannot produce nowcasts of Nt. Shown below each phase is the weighted interval score (WIS, lower is better) for Nt nowcasts of each approach during a selected week (grey shade) over 50 scenario runs (see Table 2 for exact figures). Colored vertical bars show average scores, decomposed into penalties for underprediction (crosshatch), dispersion (circles), and overprediction (stripes). The horizontal bar below shows the percentage of times each approach achieved the lowest WIS out of 50 scenario runs, respectively. Results are shown for nowcasts made at different lags from the selected week (vertical dotted lines), i. e. at the end of the selected week (top row), one week later (middle row), and two weeks later (bottom row).
Fig 3.
Nowcasts of Rt on line list data of a simulated first wave scenario using different approaches of adjusting for right truncation.
Shown are the true Rt (black) and point nowcasts with 95% credible intervals (CrI) in four different phases of the epidemic wave, obtained through i) a direct approach using cases by date of report with no truncation adjustment (yellow), ii) a stepwise approach using cases by date of symptom onset with a truncation adjustment step (blue), and iii) a generative approach using cases by date of symptom onset with an integrated truncation and renewal model (red). Shown below each phase is the weighted interval score (WIS, lower is better) for Rt nowcasts of each approach during a selected week (grey shade) over 50 scenario runs (see Table 3 for exact figures). Colored bars show average scores, decomposed into penalties for underprediction (crosshatch), dispersion (circles), and overprediction (stripes). The horizontal bar below shows the percentage of times each approach achieved the lowest WIS out of 50 scenario runs, respectively. Results are shown for nowcasts made at different lags from the selected week (vertical dotted lines), i. e. at the end of the selected week (top row), one week later (middle row), and two weeks later (bottom row).
Fig 4.
Nowcasts of Nt on incomplete line list data of a simulated first wave scenario using different approaches to account for missing onset dates.
Shown are the true number of cases by symptom onset date Nt (black), the number of cases reported until the nowcast date (dark grey bars for known, light grey bars for missing onset dates), and point nowcasts with 95% credible intervals (CrI) in four different phases of the epidemic wave, obtained through i) a stepwise approach using an independent imputation step (green), ii) a stepwise approach using a backward imputation step (blue), and iii) a generative approach using an integrated missingness model (red). All approaches used a generative model for nowcasting. Shown below each phase is the weighted interval score (WIS, lower is better) for Nt nowcasts of each approach during a selected week (grey shade) over 50 scenario runs (see Table 4 for exact figures). Colored bars show average scores, decomposed into penalties for underprediction (crosshatch), dispersion (circles), and overprediction (stripes). The horizontal bar below shows the percentage of times each approach achieved the lowest WIS out of 50 scenario runs, respectively. Results are shown for nowcasts made at different lags from the selected week (vertical dotted lines), i. e. at the end of the selected week (top row), one week later (middle row), and two weeks later (bottom row).
Fig 5.
Nowcasts of Rt on incomplete line list data of a simulated first wave scenario using different approaches of accounting for missing onset dates.
Shown are the true Rt (black) and point nowcasts with 95% credible intervals (CrI) in four different phases of the epidemic wave, obtained through i) a stepwise approach using an independent imputation step (green), ii) a stepwise approach using a backward imputation step (blue), and iii) a generative approach using an integrated missingness model (red). All approaches used a generative model for nowcasting. Shown below each phase is the weighted interval score (WIS, lower is better) for Rt nowcasts of each approach during a selected week (grey shade) over 50 scenario runs (see Table 5 for exact figures). Colored bars show average scores, decomposed into penalties for underprediction (crosshatch), dispersion (circles), and overprediction (stripes). The horizontal bar below shows the percentage of times each approach achieved the lowest WIS out of 50 scenario runs, respectively. Results are shown for nowcasts made at different lags from the selected week (vertical dotted lines), i. e. at the end of the selected week (top row), one week later (middle row), and two weeks later (bottom row).
Table 1.
Overview of main results for different nowcasting approaches for complete and incomplete line list data.
Shown is a summary, based on an evaluation on synthetic and real-world data, of the qualitative behaviour of i) different approaches for Rt estimation and truncation adjustment, and ii) different approaches for missing data imputation.
Table 2.
Performance of nowcasts of Nt on line list data of a simulated first wave scenario using different approaches of adjusting for right truncation.
Shown is the performance of Nt nowcasts in four different phases of the epidemic wave, at different lags of a selected week (same week [0–6 days], one week after [7–13 days], and two weeks after [14–20 days]). Nowcasts were obtained through i) a stepwise approach using cases by date of symptom onset with a truncation adjustment step, and ii) a generative approach using cases by date of symptom onset with an integrated truncation and renewal model. Performance is measured by the weighted interval score (WIS, lower is better), shown is the average score over 50 scenario runs () and the percentage of runs in which each approach achieved the best score (%best).
Table 3.
Performance of nowcasts of Rt on line list data of a simulated first wave scenario using different approaches of adjusting for right truncation.
Shown is the performance of Rt nowcasts in four different phases of the epidemic wave, at different lags of a selected week (same week [0–6 days], one week after [7–13 days], and two weeks after [14–20 days]). Nowcasts were obtained through i) a direct approach using cases by date of report with no truncation adjustment, ii) a stepwise approach using cases by date of symptom onset with a truncation adjustment step, and iii) a generative approach using cases by date of symptom onset with an integrated truncation and renewal model. Performance is measured by the weighted interval score (WIS, lower is better), shown is the average score over 50 scenario runs () and the percentage of runs in which each approach achieved the best score (%best).
Table 4.
Performance of nowcasts of Nt on incomplete line list data of a simulated first wave scenario using different approaches to account for missing onset dates.
Shown is the performance of Nt nowcasts in four different phases of the epidemic wave, at different lags of a selected week (same week [0–6 days], one week after [7–13 days], and two weeks after [14–20 days]). Nowcasts were obtained through i) a stepwise approach using an independent imputation step, ii) a stepwise approach using a backward imputation step, and iii) a generative approach using an integrated missingness model. Performance is measured by the weighted interval score (WIS, lower is better), shown is the average score over 50 scenario runs () and the percentage of runs in which each approach achieved the best score (%best).
Table 5.
Performance of nowcasts of Rt on incomplete line list data of a simulated first wave scenario using different approaches to account for missing onset dates.
Shown is the performance of Rt nowcasts in four different phases of the epidemic wave, at different lags of a selected week (same week [0–6 days], one week after [7–13 days], and two weeks after [14–20 days]). Nowcasts were obtained through i) a stepwise approach using an independent imputation step, ii) a stepwise approach using a backward imputation step, and iii) a generative approach using an integrated missingness model. Performance is measured by the weighted interval score (WIS, lower is better), shown is the average score over 50 scenario runs () and the percentage of runs in which each approach achieved the best score (%best).
Fig 6.
Nowcasts of Nt on incomplete hospitalization line list data during the COVID-19 pandemic in Switzerland.
Shown are nowcasts with 95% credible interval (CrI) for the number of hospitalizations with COVID-19 by date of symptom onset Nt in different phases of the first and second wave, obtained through i) a fully stepwise approach using cases by date of symptom onset with a backward imputation step and a truncation adjustment step (blue), and ii) a fully generative approach using an integrated missingness, truncation and renewal model (red). The direct approach using cases by date of report cannot produce nowcasts of Nt. Also shown are the number of cases by symptom onset date reported until the respective nowcast date (grey bars), and consolidated point estimates (7–14 days after maximum delay, averaged over all models) of the true Nt (black dashed lines). Shown below each phase is the weighted interval score (WIS, lower is better) for Nt nowcasts of each approach during a selected week (grey shade) evaluated on the consolidated point estimates. The scores (colored bars) are decomposed into penalties for underprediction (crosshatch), dispersion (circles), and overprediction (stripes). Results are shown for nowcasts made at different lags from the selected week (vertical dotted lines), i. e. at the end of the selected week (top row), one week later (middle row), and two weeks later (bottom row).
Fig 7.
Nowcasts of Rt on incomplete hospitalization line list data during the COVID-19 pandemic in Switzerland.
Shown are nowcasts with 95% credible interval (CrI) for the effective reproduction number Rt in different phases of the first and second wave, obtained from hospitalization line list data through i) a direct approach using cases by date of report with no truncation adjustment (yellow), ii) a fully stepwise approach using cases by date of symptom onset with a backward imputation step and a truncation adjustment step (blue), and iii) a fully generative approach using cases by date of symptom onset with an integrated missingness, truncation, and renewal model (red). Results are shown for nowcasts made at different lags from the selected week (vertical dotted lines), i. e. at the end of the selected week (top row), one week later (middle row), and two weeks later (bottom row).