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Modelling reporting delays for disease surveillance data
Leonardo Bastos, Theodoros Economou, Marcelo Gomes, Daniel Villela, Flavio Coelho, Oswaldo Cruz, Oliver Stoner, Trevor Bailey, Claudia Codeço
TL;DR
Reporting delays can distort disease surveillance, while under-reporting is a distinct issue from cases that have not yet been reported. The paper proposes a Bayesian hierarchical model for correcting delays and quantifying uncertainty; its applications include dengue and SARI surveillance, including an SARI outbreak anticipated two weeks earlier.
Problem
Reporting delays distort reported incidence and can misinform surveillance and warning systems, while correcting never-reported cases is a separate problem.
Source: §1 · p. 2Method
A Bayesian hierarchical model represents delayed counts with Negative Binomial marginals and uses INLA to obtain posterior samples for nowcasting.
Source: §3 · p. 6, §5 · p. 24Results
The proposed method anticipated the 2017 SARI outbreak in Paraná two weeks earlier, and its dengue nowcasts are used in Brazil's Info-Dengue warning system.
Source: §5 · p. 24Takeaways & Limitations
The framework was illustrated on dengue and SARI data and is presented as applicable to reporting systems represented by a run-off triangle.
Source: §5 · p. 23The correction targets reporting delays in observable cases, not under-reporting of cases that were never detected or reported.
Source: §1 · p. 2
Abstract
from arXiv · showhide
One difficulty for real-time tracking of epidemics is related to reporting delay. The reporting delay may be due to laboratory confirmation, logistic problems, infrastructure difficulties and so on. The ability to correct the available information as quickly as possible is crucial, in terms of decision making such as issuing warnings to the public and local authorities. A Bayesian hierarchical modelling approach is proposed as a flexible way of correcting the reporting delays and to quantify the associated uncertainty. Implementation of the model is fast, due to the use of the integrated nested Laplace approximation (INLA). The approach is illustrated on dengue fever incidence data in Rio de Janeiro, and Severe Acute Respiratory Illness (SARI) data in Paran\'a state, Brazil.
1 Introduction
Reporting delays can vary across places and over time, distorting reported incidence and potentially misinforming surveillance and warning systems. The paper proposes a flexible Bayesian framework to estimate missing observable cases and quantify uncertainty, with dengue and SARI applications in Brazil.
1 Introduction
- Delays vary across localities and with disease incidence, so reported incidence can misrepresent true incidence and misinform surveillance and warning systems.During high transmission, delays may either decrease with greater awareness or increase as health systems become saturated.Source: §1 · p. 2
- Reporting delay is treated as censoring of observable cases that will eventually be reported; the paper does not correct for never-reported cases.Addressing under-reporting generally requires additional information, such as prior rates or fully observed data.Source: §1 · p. 2
- The framework targets missing observable cases for nowcasting and potential forecasting, with reliable corrections and uncertainty quantification.The stated goals also include decision support and flexibility across complex data structures.Source: §1 · p. 2
- The framework is intended to accommodate covariates, residual spatio-temporal variability, and practical computation for potential real-time warning updates.Temporal dependence is identified as particularly important for outbreak detection.Source: §1 · pp. 2–3
- A Bayesian formulation is desirable because predictive distributions quantify uncertainty in correcting missing values; the applications use dengue and SARI data from Brazil.The stated aim is an online warning system for infectious diseases in Brazil.Source: §1 · p. 3
2 Background
The background defines reporting-delay data as a time-by-delay table with an unobserved run-off triangle that must be estimated for timely risk assessment. It reviews conditional-count and direct cell-count approaches, then positions the paper's extension of chain-ladder models to include spatial, temporal, and covariate structure.
2.1 The run-off triangle
- At time T, the grey run-off triangle contains occurred but unreported counts, so estimating it supports nowcasts and reliable risk assessment.Rows index event time and columns index reporting delay; the triangle's totals are also missing.Source: §2.1 · pp. 3–4
2.2 Current approaches
- One existing approach models counts conditional on totals using a hierarchical Poisson or Negative Binomial model followed by a Multinomial delay distribution.The delay probabilities can vary with time.Source: §2.2 · pp. 4–5
- The alternative chain-ladder framework models delayed cell counts directly; its Negative Binomial marginals can be motivated from a conditional Multinomial model, though delay probabilities and totals cannot be separated.The chain-ladder method was developed as a distribution-free estimator of missing delayed counts and later linked to a GLM.Source: §2.2 · p. 5
- The paper extends chain-ladder Negative Binomial marginals to allow spatial and temporal variation and covariate effects in reporting-delay counts.Spatial variation allows delay mechanisms to differ across regions and supports borrowing information across them.Source: §2.2 · p. 5
- Dependence across time and delay can represent temporal variation in both disease occurrence and the delay mechanism.Source: §2.2 · p. 6
- The approaches discussed are statistical rather than mechanistic, fitting the paper's focus on observational surveillance data and efficient real-time decision support.Mechanistic disease components may improve predictions but can require well-documented data and be computationally expensive.Source: §2.2 · p. 6
3 Model specification
The framework models delayed disease counts with Bayesian hierarchical components for temporal, delay, seasonal, and spatial variation. It uses posterior prediction to estimate missing counts and nowcast totals, with INLA as a faster approximate inference approach.
3 Model specification
- A conditional Negative Binomial model combines temporal, delay, interaction, seasonal, and covariate effects to describe counts by occurrence time and reporting delay.The temporal and delay effects can vary over time; seasonal effects capture recurring annual variation, and the model constrains its components to sum to zero for intercept identifiability.Source: §3 · pp. 6–8
3.1 Model implementation
- INLA provides approximate posterior samples for the latent Gaussian model, offering a faster alternative to computationally intensive MCMC.The authors note that this speed comes with approximations that must be considered.Source: §3.1 · p. 9, §3.3 · p. 11
3.2 Spatial variation
- Spatial random effects allow both disease counts and reporting delays to vary across areas, while pooling information to aid estimation in locations with fewer data.The spatial delay component captures area-specific delay structure; the SARI application uses an area-based formulation with an IAR spatial effect.Source: §3.2 · pp. 9–10
3.3 Nowcasting
- Posterior predictive samples estimate missing counts in the run-off triangle and yield nowcasts of the latest total, with earlier totals available for hindcasting.Predictions proceed sequentially from the top-right of the run-off triangle; simulated missing counts are summed to obtain marginal totals.Source: §3.3 · p. 11
4 Model application
The model is applied to dengue in Rio de Janeiro and SARI in Paraná; both implementations are used by local and national authorities with online warning systems.
4.1.1 Results and model checking
- At the 15th epidemic week of 2012, dengue predictions identified an outbreak despite currently reported counts that could suggest a decline.The model corrected the current total and the preceding nine weeks, with 95% prediction intervals.Source: §4.1 · p. 12, Figure 2 · p. 13, §4.1.1 · p. 13
- Predicted dengue totals captured observed ranks, while predictive checks also represented the totals’ mean, median, variance, and first eight autocorrelation lags.Ten of 68 totals were based on data the model had not seen.Source: §4.1.1 · pp. 13–14, Figure 3 · p. 14, Figure 4 · p. 14
- Only 3% of covariance tail-area probabilities were extreme, indicating that the model captured covariance among delay-column counts.Table 1 reports the lower tail-area probabilities for the sample covariance of each data column.Source: Table 1 · p. 15, §4.1.1 · p. 15
4.1.2 Sensitivity analysis of using INLA for prediction
- R-INLA predictive distributions matched MCMC comparisons broadly, with occasional slight underestimation of upper tails, while taking seconds rather than hours per chain.The comparison covered total counts for ten weeks and selected individual delayed counts.Source: §4.1.2 · p. 16, Figure 6 · p. 17
4.1.3 Estimates and rolling predictions
- Dengue estimates showed a plateauing temporal trend, a declining delay structure, and early-year seasonal increases; rolling predictions captured outbreak rises and declines.Most eventually reported counts were within the 95% prediction intervals, particularly for the current time point.Source: §4.1.3 · pp. 16–17, Figure 7 · p. 18, Figure 8 · p. 19
4.3 Data
- The SARI analysis used weekly reports from 22 health regions and allowed delay patterns to vary by region while sharing the overall temporal trend and time-delay interaction.The model also included spatially structured dependence between neighboring health regions.Source: §4.3 · pp. 19–20
4.3.1 Results
- Two weeks earlier than official data, the model detected the qualitative start of SARI epidemic activity in Paraná, potentially enabling earlier mitigation.The model identified the pre-epidemic threshold at week 14; official data crossed it only by the end of week 16.Source: Figure 9 · p. 21, §4.3.1 · p. 21
- SARI delay estimates declined on average but varied substantially across regions, especially during the first two weeks; the spatial-effect ratio was centered at 1.The temporal effects reflected seasonal reporting patterns, and the structured spatial effect captured most of the variance with minimal unstructured contribution.Source: §4.3.1 · pp. 21–23, Figure 10 · p. 22, Figure 11 · p. 23
5 Discussion
The framework is presented as applicable to any disease and to reporting systems represented by a run-off triangle, with dengue and SARI case studies illustrating different forms of temporal and spatial variation. Its fast Bayesian implementation is already used for surveillance warnings in Brazil.
5 Discussion
- The framework can apply to any disease and to reporting systems whose data have a run-off triangle structure, and is illustrated with dengue and SARI data in Brazil.The dengue model uses temporal and delay dependence; the SARI model incorporates spatial variability and dependence to borrow information across regions.Source: §5 · p. 23
- Although neither case-study model included covariates, the authors describe adding them in the R-INLA implementation as fairly trivial.Source: §5 · p. 24
- The dengue model's nowcasts anticipated the 2017 SARI outbreak in Paraná by 2 weeks, and predictive-count samples can support decision-theoretic warning decisions.The models are also used to produce influenza and SARI warnings across Brazil.Source: §5 · p. 24
- The Bayesian implementation is fast because it uses INLA to compute samples from marginal posteriors.The paper reports the dengue model is used in Info-Dengue, and its nowcasts are used by Brazil's Ministry of Health for influenza and SARI warnings.Source: §5 · p. 24