Performance in forecasting weekly dengue and influenza incidence

Figs 1 and 2 show weekly dengue and ILI nowcasts for NobBS and the HH approach over multiple seasons for both diseases. Table 1 summarizes the point and probability-based accuracy metrics for each, where higher accuracy is indicated by higher average scores, lower MAE, RMSE, and rRMSE, and lower distance from 0.95 for the 95% PI coverage. Because the NobBS model accounts for both under-reporting and the autocorrelated progression of transmission across successive weeks, it makes predictions even in weeks when there are no cases reported for the week. Conversely, the HH model does not make nowcasts for weeks in which there are no initial case reports (common in the dengue Puerto Rico data), hence the nowcasts in Figs 1C and 2C appear as discontinuous lines. To compare models despite these differences, we report accuracy metrics between NobBS and the HH approach for both (1) the full time series of the data and (2) weeks when at least one case was reported in the first week, i.e. the subset of weeks for which both models could make predictions (Table 1). We also computed error metrics for the HH model for the full time series by assigning point estimates of 0 cases for nowcasts in weeks without predictions.

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Fig 1. Weekly dengue fever nowcasts for December 23, 1991 through November 29, 2010 using a 2-year moving window.

(A) Weekly NobBS nowcasts along with (B) point estimate and uncertainty accuracy, as measured by the score and the prediction error, are compared to (C) weekly nowcasts by the HH approach with (D) corresponding scores and prediction errors. For nowcasting, the number of newly-reported cases each week (blue line) are the only data available in real-time for that week, and help inform the estimate of the total number of cases that will be eventually reported (red line), shown with 95% prediction intervals (pink bands). The true number of cases eventually reported (black line) is known only in hindsight and is the nowcast target. Historical information on reporting is available within a 104-week moving window (grey shade) and used to make nowcasts. The score (brown line) and the difference between the true and mean estimated number of cases (grey line) are shown as a function of time.

https://doi.org/10.1371/journal.pcbi.1007735.g001

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Fig 2. Weekly ILI nowcasts for June 30, 2014 through September 25, 2017 using a 6-month moving window.

(A) Weekly NobBS nowcasts along with (B) point estimate and uncertainty accuracy, as measured by the score and the prediction error, are compared to (C) weekly nowcasts by the HH approach with (D) corresponding scores and prediction errors. For nowcasting, the number of newly-reported cases each week (blue line) are the only data available in real-time for that week, and help inform the estimate of the total number of cases that will be eventually reported (red line), shown with 95% prediction intervals (pink bands). For the HH approach, the 95% prediction intervals are very narrow and are thus difficult to see. The true number of cases eventually reported (black line) is known only in hindsight and is the nowcast target. Historical information on reporting is available within a 27-week moving window (grey shade) and used to make nowcasts. The score (brown line) and the difference between the true and mean estimated number of cases (grey line) are shown as a function of time.

https://doi.org/10.1371/journal.pcbi.1007735.g002

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Table 1. Performance measures for each nowcast approach by disease (mean % reported with no delay, computed across the complete time period^* of reports).

https://doi.org/10.1371/journal.pcbi.1007735.t001

The HH approach made predictions in only 55% of weeks in the dengue time series (Table 1). In this subset of weeks, the NobBS approach achieved relatively smooth and accurate tracking of the dengue time series (rRMSE = 0.464, average score = 0.274) despite low proportions of cases reported on the week of onset (Fig 1A and 1B). The 95% PI coverage was 0.85, indicating that the 95% PI included the true number of cases for 85% of the nowcasts. In comparison, the HH approach produced substantially less accurate point estimates and slightly broader uncertainty intervals (rRMSE = 1.24, average score = 0.161, 95% PI coverage = 0.90) with greater fluctuation in nowcasts from week-to-week (Fig 1C and 1D). On average, the HH approach led to more biased estimates and systematically overpredicted the number of cases for both ILI and dengue compared to NobBS. Because many weeks in the dengue data were low incidence, assigning a prediction of 0 to the HH approach’s missing nowcasts improved its rRMSE to 1.14 in the full time series compared to 1.24 for the subset over which nowcasts were generated from the model, though NobBS still surpassed the HH model’s accuracy on this and all other metrics (Table 1).

Nowcast point estimates tracked the ILI time series well for both approaches, though point estimates had greater error by all measures for the NobBS approach (NobBS rRMSE = 0.074 vs. HH rRMSE = 0.062; Table 1). However, the NobBS approach produced considerably wider prediction intervals (Figs 1C and 2C) resulting in both higher scores (NobBS average score = 0.218 vs. HH average score = 0.017) and 100% coverage by the 95% prediction intervals compared to 0% coverage for the HH (Table 1). We observed that at multiple deciles of the prediction interval, the interval provided conservative coverage, but that the calibration of narrower intervals was better (S1 Fig).

To assess the degree of autocorrelation and related smoothness in the NobBS predictions, we calculated the 1-week lagged autocorrelation of predictions (ρ_a) and compared this to the 1-week lagged autocorrelation of cases (ρ_c). In addition, we computed metrics reflecting the accuracy of the approaches in capturing the change in cases from week-to-week: the mean absolute error of the change (MAEΔ) and the RMSE of the change (RMSEΔ) (Table 2). The magnitude of change was much larger for the ILI data than dengue data, with average absolute value change of 1,312.6 cases/week versus 9.8 cases/week, yet both showed high autocorrelation (ρ_c = 0.958 for dengue and ρ_c = 0.972 for ILI). Comparing the full time series, the nowcasts produced by NobBS exhibited high autocorrelation for both diseases (ρ_a = 0.876 for dengue, 0.973 for ILI) while the HH approach yielded lower autocorrelation for dengue nowcasts, comparatively (ρ_a = 0.631 for dengue, 0.970 for ILI). For dengue, over the weeks in which at least 1 case was initially reported, the NobBS approach achieved both lower mean absolute difference between predicted and observed changes in cases (NobBS MAEΔ = 23 vs. HH MAEΔ = 50) and lower RMSE of the change (NobBS RMSEΔ = 35.8 vs. HH RMSEΔ = 64.6). In addition, NobBS outperformed the HH approach over the full time series of dengue cases (Table 2). For ILI, however, the metrics for the weekly change were similar for the two approaches (Table 2).

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Table 2. Performance measures for estimates of the change in disease incidence from the previous week.

https://doi.org/10.1371/journal.pcbi.1007735.t002

Reporting delays impact nowcast performance

The delay distributions between the reporting systems are strikingly different (Figs 1 and 2, S2 Fig). In the case of the dengue surveillance system, which includes specimen collection and laboratory testing, only approximately 4% of cases were processed during the week of onset, on average. In contrast, the U.S. Outpatient Influenza-like Illness (ILI) Surveillance Network (ILINet) captures only syndromic data reported electronically, with over 80% of ILI cases reported, on average, the same week they present (i.e. with no delay). Overall, we observed that the accuracy of nowcast point estimates (rRMSE) was higher for the ILI data compared to dengue, which may be related to the high proportion of cases reported with 0-weeks delay in these data. Large weekly absolute changes in the number of initial case reports also appeared to be related to increased error, particularly for dengue, which had high fluctuations in the number of initial reports over time (S1 Table, S3 Fig). Note that because of the difference in predictive distribution bin widths based on the number of cases that accrue for influenza vs. dengue (Materials & Methods), average scores are not comparable across diseases.

NobBS improves nowcasting with varying reporting delays

Dengue and ILI also exhibit differences in the trends of reporting delay probabilities over time, and we observed strikingly different comparative performance of the two nowcast approaches in these disease settings, namely that the HH approach performed comparably to NobBS for ILI but not for dengue. In the dengue data, we observe a noisier, more time-varying probability of reporting for cases, with more extreme fluctuations in the proportion of initial reports compared to ILI cases, which show more constant (tighter ranges of) reporting probabilities from week-to-week (S4 Fig). Independent of the initial proportion of cases reported (high vs. low), we hypothesized that these trends (relatively constant vs. time-varying) are particularly impactful on the performance of the nowcast, and that relatively constant reporting probabilities, as seen in the ILI data, may be linked to the higher accuracy of these predictions.

To test the robustness of the model, we simulated ILI data using the final counts from the true dataset, but imposing a time-varying delay distribution; specifically, with faster initial reporting during weeks of high incidence to reflect heightened awareness of an outbreak and increased urgency of reporting (described in Materials and Methods). Using these simulated data, we found that NobBS was relatively robust to changes in reporting delays (S5 Fig, S2 Table). In the context of stable reporting delays (original ILI data), NobBS performed comparably to the HH model (Fig 2, Table 1). However, in the presence of simulated time-varying reporting delays, NobBS outperformed the HH in terms of confidence (NobBS average score = 0.06 vs. HH average score ≈ 0), point estimates (NobBS rRMSE = 0.302 vs. HH rRMSE = 0.621), and accuracy of the predicted change (S3 Table). Such variations in the delay distribution are a reality in many epidemics[17].

Time-varying delays may be explicitly modeled in the HH approach, by pre-specifying the temporal change-points at which the delay distribution is expected to change–information that is rarely known in real-time. We therefore compared NobBS to the HH approach without explicitly modeling a time-varying delay distribution in either method. In the case of ILI and dengue data, we had no reason to believe or anticipate a systematic change in reporting over the time period covered.

Performance by year

The performance of both approaches in nowcasting ILI was relatively consistent by year across accuracy measures, but there were fluctuations in the year-to-year performance of dengue nowcasts (Table 3). Average scores of the dengue nowcasts (from both approaches) tended to be high in years that experienced a very low number of dengue cases (e.g. 2000, 2002, 2004, 2006), and therefore the performance of both approaches declined during higher incidence years. For dengue, the NobBS model was particularly effective at identifying periods of low incidence, with high probabilities assigned to the lowest outcome bin (0–25 cases, details in Materials and Methods) when the number of cases eventually reported was low (S6 Fig). On the other hand, during periods of high dengue activity, lower probabilities were assigned to the correct bin, reflecting greater uncertainty. Looking specifically at higher incidence time periods (Epidemiologic Weeks 40 through 20 of the following year for ILI, and all weeks in which the number of cases exceeded 50 for dengue), both nowcast approaches were less accurate and more uncertain than during other time periods (S4 Table). Overall, NobBS outperformed the benchmark approach on all performance measures for each year and most higher incidence time periods (Table 3; S4 Table).

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Table 3. Annual performance measures for each nowcast model, by disease.

All predicted weeks for each model are compared.

https://doi.org/10.1371/journal.pcbi.1007735.t003

Both approaches had their lowest accuracy on three high incidence dengue seasons: 1994, 2007, and 2010 (Table 3; S4 Table; Fig 1). The average annual scores for these years range between 0.041 and 0.170 across the NobBS and benchmark approaches, and between 0.000 and 0.081 for the high-incidence time periods specifically (Table 3; S4 Table), falling clearly below the rest of the years and higher incidence time periods in performance. These scores not only reflect unusually poor point estimate predictions as judged by rRMSE, but also the finding that the predictive distribution for weeks in these years for both approaches rarely included the true value of interest (a consequence of dramatic over- or underestimates during times of high disease activity).

Moving window sizes

We initially used moving windows of 104 weeks for dengue (a longer time series) and 27 weeks for ILI (a shorter time series) to leverage a large number of historical training weeks to train nowcast estimates. Moving windows allow for stable estimation of the recent delay distribution as information from very old and potentially less relevant weeks are forgotten. The size of the moving window reflects how quickly and smoothly changes in the data should be realized by the model: longer moving windows tend to produce smoother estimates, but the model may be less sensitive to abrupt changes in the data (e.g. changes in how quickly cases are reported during an outbreak) or shorter-interval secular trends, e.g. seasonality.

While we chose long moving windows for both diseases to capitalize on data availability, these considerations may affect the choice of moving window size and nowcast performance, depending on the data. In light of this, we experimented with moving windows of different lengths on the dengue data to assess the impact on nowcast performance. We tested moving windows of 5, 12, and 27 weeks (approx. 6 months) on the dengue data and, for each moving window, generated a set of nowcasts for December 1991-November 2010 in order to compare performance measures across all moving window sizes (5, 12, 27, and 104 weeks). A 5-week moving window produced substantially lower accuracy nowcasts (rRMSE = 7.381) with several steep case overestimates in 2007–08 and 2010 (S7A Fig). However, accuracy metrics for moving windows of 12 weeks or longer were similar to those using the full 104 week window (range in rRMSE: 0.6–0.655; average score: 0.35–0.37) (S5 Table; S7 Fig). While shorter moving windows often produced more accurate estimates of the reporting delay probability (S8A Fig), the estimated variance of the random walk process varied dramatically from week-to-week (S8B Fig) resulting in more dramatic overestimates at certain periods of extreme time-varying delays, suggesting a trade-off between delay estimation accuracy and more stable estimates of weekly cases.

Surveillance data

We collected data on approximately 53,000 cases of dengue in Puerto Rico and 2.77 million cases of ILI in the United States over a 21-year (1092 weeks) and 3.75-year (196 weeks) period, respectively. Time-stamped weekly dengue data for laboratory-confirmed cases of dengue in Puerto Rico were collected by the Puerto Rico Department of Health and Centers for Disease Control and Prevention. The times used for the analysis were the time of onset as reported by the reporting clinician and the time of laboratory report completion. ILI data originated from the U.S. Outpatient Influenza-like Illness Surveillance Network (ILINet), which consolidates information from over 2000 outpatient healthcare providers in the United States who report to the CDC on the number of patients with ILI. The times used for the analysis were the week of ILI-related care seeking and the week when those cases were posted online in FluView (https://gis.cdc.gov/grasp/fluview/fluportaldashboard.html) as collected in the DELPHI epidemiological data API (https://github.com/cmu-delphi/delphi-epidata). ILI data with delays of more than 6 months occasionally had irregularities, so we restricted the analyses to delays of up to 6 months.

Reporting triangle

Delays in reporting are often structurally decomposed into a (T x D) dimensional “reporting triangle,” where T is the most recent week (“now”) and D is the maximum reporting delay, in weeks, observed in the data. The data are right-truncated, since at any given week t, delays longer than T–t cannot be observed. For example, at week t = T, only the cases reported with delay d = 0 are observable; cases reported with longer delays (i.e. 1- or 2-week delays, d = 1 or d = 2) will be known in future weeks. In S6 Table, we present an example of the reporting triangle using ILI data.

For each week t, the goal of nowcasting is to produce estimates for the total number of cases eventually reported, N_t, based on an incomplete set of observed cases with delay d, n_t,d. Since not every n_t,d is observed for a delay d, but will be observed at some unknown time point in the future, N_t = sum(n_t,d).

Our approach is motivated by modeling the marginal cell counts of the reporting triangle, n_t,d in an adaptation of the loglinear chain ladder method developed in actuarial literature[13,14].

Bayesian nowcast model

Let n_t,d be the number of cases reported for week t with delay d. We assume that the underlying cases occur in a Poisson process such that

We apply this default assumption in the case of the dengue fever data.

We also allow for extra-Poisson variation, that is, when the variance is larger than the mean and a negative binomial process (of which the Poisson is a special case) is more appropriate. We apply this in the case of the influenza (ILI) data: and p(t,d) is the failure probability at week t and delay d, and (r) is the number of failures before stopping.

We then model the mean, λ_t,d, as a simple log-linear equation where α_t represents the true epidemiologic signal for week t and β_d as the probability of reporting with delay = d. In other words, NobBS contains random effects for week t and the reporting delay d. Exponentiating both sides of the equation, . This model is an adaptation of the general chain ladder model, proposed in the actuarial literature as an approach to nowcast the number of claims at time t with delay d with the form log (λ_t,d) = μ+α_t+β_d [15].

We place prior distributions on α_t and β_d reflecting properties of each parameter. Since β_d represents a probability vector containing delays = 0, …, D, we place on it a Dirichlet prior of length D:

The maximum delay D can be identified as the maximum observable delay in the data, which may change as the time series extends, or can be fixed at some value D thought to represent a very long delay. In the latter case, θ_D can be modeled as the probability of delay ≥ D. For dengue, we choose to fix D at 10 weeks, since over 99% of the cases observed in the first two years (prior to producing out-of-sample nowcasts) were reported within 10 weeks. For influenza, we chose D to be the longest possible delay within the 27-week moving window, or D = 26. The implications of choosing a maximum delay D within a moving window of W weeks means that the nowcast will include all cases arising with delays greater than or equal to D but less than or equal to W, thus excluding all cases with delays greater than W (see the reporting triangle in S6 Table).

We place weakly informative priors on θ representing a small number of hypothetical total cases (n = 10) distributed across delay bins, loosely representing the empirical probability of reporting delays for each delay d observed in the first two years of data for dengue and the first 6 months of data for ILI (training periods). As a sensitivity, we also placed weak priors on θ treating all delays with equal probability, but there was no material difference in the results.

We allow a dependency between successive α_t’s to capture the time evolution and autocorrelation of cases from week-to-week, commonly exhibited by epidemic curves. We therefore model α_t as a first-order random walk:

Because α_t is in natural log form, this constitutes a geometric random walk.

We place weakly informative priors on the precisions of the Normal distribution, . For the negative binomial stopping-time parameter, r, we place an informative Ga(60,20) prior to reflect belief that the process deviates moderately from the Poisson.

Models were compiled in JAGS on R (v 3.3.2) producing 10,000 posterior samples. Trace plots were visually reviewed for convergence.

NobBS R package implementation and recommendations

In the R package “NobBS” (cran.r-project.org/web/packages/NobBS/index.html), the nowcast model may be implemented automatically by specifying a data frame with one record (row) per case, with a variable specifying the date of case onset and a variable specifying the date of case report. Dates are assumed to be in a sequence of weeks or months, with no gaps in reporting (i.e. no weeks for which reporting did not happen). By default, the model runs a Poisson log-linear model with no moving window (i.e. all historical data is taken into account), a maximum delay D equal to 1 minus the length of the reporting time series (i.e. if 12 consecutive weeks of report data are provided, the longest possible delay is equal to 11 weeks), and ignoring any delays longer than D (because they are unobserved in the default implementation). By default, a weakly informative Ga(0.01,0.01) prior is placed on the precision of the first-order random walk, , indicating equal weight to a standard deviation of 1 as to a standard deviation of 10,000: in other words, little knowledge about the extent to which the number of cases in the present week differ from the previous week. A weakly informative Dirichlet(θ_d…D = 0.1) prior is placed on the vector of delay probabilities, β_d, indicating the weak prior belief that the probability of delay d is equal at all delays.

However, in “NobBS” all of these features are customizable and it is possible to leverage existing knowledge or historical information to impose more informative priors on the data. For example, it may be already known in the surveillance field that cases of dengue fever are rarely reported the same week they occur (i.e. that the probability of delay d = 0 is very low), but that most cases are reported within 4 weeks. In this example, it would be possible to design a stronger prior on β_d such that θ = (3,30,50,10,4,3), indicating that out of 100 cases distributed across delays, 3% of cases are expected to be reported with no delay, 30% of cases with a 1-week delay, 50% of cases with a 2-week delay, 10% of cases with a 3-week delay, 4% of cases with a 4-week delay, and 3% of cases with a 5-week delay. In other words, the user may directly parameterize the Dirichlet prior using knowledge of the delay distribution, and the sum of θ_d…D is related to the strength of the prior. Historical information (e.g. reporting data from the last several months or years) may also be used to inform the Dirichlet prior, such as through plotting the empirical distribution of delays in case reports. In the present study, for example, the observed delay distribution over the initial model training period was used to inform weakly informative Dirichlet priors for dengue and ILI, with the sum of θ_d…D being small.

The shape and rate hyperpriors of the Gamma(shape, rate) prior on may also be modified to reflect stronger prior knowledge of the relationship between cases from week-to-week, which controls the smoothness of the epidemic curve. Increasing the shape parameter with a more informative prior, e.g. Gamma(10,0.1), can force the differences in cases from week-to-week to be small, resulting in a smoother epidemic curve and mitigating large fluctuations in estimates.

In all cases, keep in mind that the posterior is a compromise between the prior and the likelihood, so with enough weeks of reporting, the observed data can dominate over the prior.

Specifying a moving window size (e.g. a moving window of 27 weeks) may be leveraged for computational advantage (faster runtime) or to reflect the belief that the delay distribution of very old reports is no longer relevant to inform current reports. Alternatively, the user may prefer to let the moving window expand each week, especially to leverage as much prior data as possible to inform the nowcasts in the absence of available historical information. As discussed in Results, the estimation of the parameters is sensitive to how much historical data is used to fit the model, and this decision is up to the user and could depend on the relevant disease reporting system. The moving window acts to consider (or “weight”) all past observations equally, and shortening the moving window reflects the extent to which the user wishes to consider only more recent data as informative–noting also that shorter moving windows produce more volatile estimates of delay probabilities (which generally increases the accuracy of) as well as more volatile estimates of the variance of the random walk (which generally decreases the accuracy of). Still, these parameters may be constrained through their priors as discussed above.

Nowcast estimates

We produced weekly nowcasts beginning with the 27^th week (influenza) and 104^th week (dengue) and through the final week of the series. This resulted in 989 weekly out-of-sample estimates of dengue cases and 170 weekly out-of-sample estimates of ILI. The time series of nowcast estimates for both diseases are shown in Figs 1 and 2.

We used a two-year moving window to estimate a stable delay distribution within the window. As a sensitivity, and to gauge the minimum amount of historical information required to produce accurate nowcasts, we also applied moving windows of 5, 12, and 27 weeks (approximately 6 months).

We used as a benchmark for comparison the “nowcast” function of the R package “surveillance” by Höhle and colleagues (described in ref.[9]) designed to produce Bayesian nowcasts for epidemics using a hierarchical model for n_{t, d ≤ T-t} | n_t,d, or the observed cases conditional on the expected total number of cases. We applied the function assuming a time-homogenous delay distribution and recommended parameterization described by the authors in http://staff.math.su.se/hoehle/blog/2016/07/19/nowCast.html, and for comparability, used the same moving window sizes (27 and 104 weeks) to produce nowcasts over the same time periods.

Model performance metrics

The mean absolute error (MAE), root mean square error (RMSE) and relative root mean square error (rRMSE) are defined, respectively, as: and were used to quantify the accuracy of point estimates, x_i, compared to true case numbers, y_i, across the different models at each week i.

To quantify the accuracy of the point estimates in capturing the change in cases from week i-1 to week i, we computed the mean absolute error of the change (MAEΔ) and the RMSE of the change (RMSEΔ):

To capture smoothness in predictions from week-to-week, we also calculated the lag-1 autocorrelation of predictions (ρ_a) and cases (ρ_c) between week i and week i-1. where x = the predicted or true cases at each week i.

The logarithmic scoring rule was used to quantify the accuracy of the posterior predictive distribution of the nowcast. Predictive distributions were assigned to a series of bins categorized across possible values of true case counts. We used bin widths of 25 cases for dengue and 1000 cases for influenza, allowing for a larger number of bins for ILI cases based on case ranges of approx. 0–400 for dengue and 4,000–40,000 for ILI. For a predictive distribution with binned probability pi for a given nowcast target, the logarithmic score was calculated as ln(pi). For example, there were 115 cases eventually observed for the week of January 20, 1992. The NobBS nowcast for this week, which assigned a probability of 0.4 to the bin [100,125), thus received a log score of ln(0.4) = -0.92. As in[22,23], a very low log score of -10 was assigned for weeks in which the predictive distribution did not include the true case value, for weeks in which the bin probability ≤ e-10. This rule provides a lower limit (-10) to the score of highly inaccurate predictions.

The average log score across all prediction weeks was computed for all models to assess nowcast performance. The exponentiated average log score yields a nowcast score that can be interpreted as the average probability assigned to the bin corresponding to the true number of cases, and is a metric for model comparison purposes used in several other forecast contexts[22,23]. In this paper, we present the exponentiated average log score and refer to this as the average score.

Simulated ILI data

To simulate ILI data with a time-varying probability of reporting delay d = 0, we drew, for each week, Pr(d = 0) from Unif(0.2, 0.9) for all weeks in which the total number of eventually-observed cases exceeded the mean of the ILI series (14,000 cases), and from Unif(0, 0.65) for all weeks in which the total observed case count was less than or equal to 14,000. This probability was used to calculate the simulated number of cases that would be observed with d = 0, out of the total number of cases that would be eventually observed for that week. The remaining cases were distributed to other delays ranging from 1–52 weeks using NB(0.9,0.4). This produced a rough approximation for a hypothetical scenario in which cases are reported faster (higher probability of d = 0) during weeks with higher disease activity (more cases), representing heightened awareness of an outbreak and increased urgency of reporting. However, it would also be possible to simulate the reverse scenario (slower reporting at weeks of higher incidence, to demonstrate the problem of reporting backfill during an outbreak) by swapping the two uniform priors.