An approximate Bayesian approach for estimation of the instantaneous reproduction number under misreported epidemic data

In epidemic models, the effective reproduction number is of central importance to assess the transmission dynamics of an infectious disease and to orient health intervention strategies. Publicly shared data during an outbreak often suffers from two sources of misreporting (underreporting and delay in reporting) that should not be overlooked when estimating epidemiological parameters. The main statistical challenge in models that intrinsically account for a misreporting process lies in the joint estimation of the time-varying reproduction number and the delay/underreporting parameters. Existing Bayesian approaches typically rely on Markov chain Monte Carlo algorithms that are extremely costly from a computational perspective. We propose a much faster alternative based on Laplacian-P-splines (LPS) that combines Bayesian penalized B-splines for flexible and smooth estimation of the instantaneous reproduction number and Laplace approximations to selected posterior distributions for fast computation. Assuming a known generation interval distribution, the incidence at a given calendar time is governed by the epidemic renewal equation and the delay structure is specified through a composite link framework. Laplace approximations to the conditional posterior of the spline vector are obtained from analytical versions of the gradient and Hessian of the log-likelihood, implying a drastic speed-up in the computation of posterior estimates. Furthermore, the proposed LPS approach can be used to obtain point estimates and approximate credible intervals for the delay and reporting probabilities. Simulation of epidemics with different combinations for the underreporting rate and delay structure (one-day, two-day, and weekend delays) show that the proposed LPS methodology delivers fast and accurate estimates outperforming existing methods that do not take into account underreporting and delay patterns. Finally, LPS is illustrated in two real case studies of epidemic outbreaks.     

Near real-time surveillance of the SARS-CoV-2 epidemic with incomplete data

When responding to infectious disease outbreaks, rapid and accurate estimation of the epidemic trajectory is critical. However, two common data collection problems affect the reliability of the epidemiological data in real time: missing information on the time of first symptoms, and retrospective revision of historical information, including right censoring. Here, we propose an approach to construct epidemic curves in near real time that addresses these two challenges by 1) imputation of dates of symptom onset for reported cases using a dynamically-estimated “backward” reporting delay conditional distribution, and 2) adjustment for right censoring using the NobBS software package to nowcast cases by date of symptom onset. This process allows us to obtain an approximation of the time-varying reproduction number (R_t) in real time. We apply this approach to characterize the early SARS-CoV-2 outbreak in two Spanish regions between March and April 2020. We evaluate how these real-time estimates compare with more complete epidemiological data that became available later. We explore the impact of the different assumptions on the estimates, and compare our estimates with those obtained from commonly used surveillance approaches. Our framework can help improve accuracy, quantify uncertainty, and evaluate frequently unstated assumptions when recovering the epidemic curves from limited data obtained from public health systems in other locations.     

An exploration of fixed and random effects selection for longitudinal binary outcomes in the presence of nonignorable dropout

We explore a Bayesian approach to selection of variables that represent fixed and random effects in modeling of longitudinal binary outcomes with missing data caused by dropouts. We show via analytic results for a simple example that nonignorable missing data lead to biased parameter estimates. This bias results in selection of wrong effects asymptotically, which we can confirm via simulations for more complex settings. By jointly modeling the longitudinal binary data with the dropout process that possibly leads to nonignorable missing data, we are able to correct the bias in estimation and selection. Mixture priors with a point mass at zero are used to facilitate variable selection. We illustrate the proposed approach using a clinical trial for acute ischemic stroke.     

OutbreakFlow: Model-based Bayesian inference of disease outbreak dynamics with invertible neural networks and its application to the COVID-19 pandemics in Germany

Mathematical models in epidemiology are an indispensable tool to determine the dynamics and important characteristics of infectious diseases. Apart from their scientific merit, these models are often used to inform political decisions and interventional measures during an ongoing outbreak. However, reliably inferring the epidemical dynamics by connecting complex models to real data is still hard and requires either laborious manual parameter fitting or expensive optimization methods which have to be repeated from scratch for every application of a given model. In this work, we address this problem with a novel combination of epidemiological modeling with specialized neural networks. Our approach entails two computational phases: In an initial training phase, a mathematical model describing the epidemic is used as a coach for a neural network, which acquires global knowledge about the full range of possible disease dynamics. In the subsequent inference phase, the trained neural network processes the observed data of an actual outbreak and infers the parameters of the model in order to realistically reproduce the observed dynamics and reliably predict future progression. With its flexible framework, our simulation-based approach is applicable to a variety of epidemiological models. Moreover, since our method is fully Bayesian, it is designed to incorporate all available prior knowledge about plausible parameter values and returns complete joint posterior distributions over these parameters. Application of our method to the early Covid-19 outbreak phase in Germany demonstrates that we are able to obtain reliable probabilistic estimates for important disease characteristics, such as generation time, fraction of undetected infections, likelihood of transmission before symptom onset, and reporting delays using a very moderate amount of real-world observations.     

Faster indicators of chikungunya incidence using Google searches

Chikungunya, a mosquito-borne disease, is a growing threat in Brazil, where over 640,000 cases have been reported since 2017. However, there are often long delays between diagnoses of chikungunya cases and their entry in the national monitoring system, leaving policymakers without the up-to-date case count statistics they need. In contrast, weekly data on Google searches for chikungunya is available with no delay. Here, we analyse whether Google search data can help improve rapid estimates of chikungunya case counts in Rio de Janeiro, Brazil. We build on a Bayesian approach suitable for data that is subject to long and varied delays, and find that including Google search data reduces both model error and uncertainty. These improvements are largest during epidemics, which are particularly important periods for policymakers. Including Google search data in chikungunya surveillance systems may therefore help policymakers respond to future epidemics more quickly.     

QUANTITATIVE GENETIC MODELING AND INFERENCE IN THE PRESENCE OF NONIGNORABLE MISSING DATA

Natural selection is typically exerted at some specific life stages. If natural selection takes place before a trait can be measured, using conventional models can cause wrong inference about population parameters. When the missing data process relates to the trait of interest, a valid inference requires explicit modeling of the missing process. We propose a joint modeling approach, a shared parameter model, to account for nonrandom missing data. It consists of an animal model for the phenotypic data and a logistic model for the missing process, linked by the additive genetic effects. A Bayesian approach is taken and inference is made using integrated nested Laplace approximations. From a simulation study we find that wrongly assuming that missing data are missing at random can result in severely biased estimates of additive genetic variance. Using real data from a wild population of Swiss barn owls Tyto alba, our model indicates that the missing individuals would display large black spots; and we conclude that genes affecting this trait are already under selection before it is expressed. Our model is a tool to correctly estimate the magnitude of both natural selection and additive genetic variance.     

Birth dates vary with fixed and dynamic maternal features,offspring sex,and extreme climatic events in a high‐latitude marine mammal

Reproductive synchrony tends to be widespread in diverse species of plants and animals, especially at higher latitudes. However, for long‐lived mammals, birth dates for different individuals can vary by weeks within a population. A mother's birth timing can reveal useful information about her reproductive abilities and have important implications for the characteristics and survival of her offspring. Despite this, our current knowledge of factors associated with variation in birth dates is modest. We used long‐term data for known‐age Weddell seals in Antarctica and a Bayesian hierarchical modeling approach to study how birth dates varied with fixed and temporally varying features of mothers, whether sex allocation varied with birth timing, and annual variation in birth dates. Based on birth dates for 4465 pups born to 1117 mothers aged 4–31, we found that diverse features of mothers were associated with variation in birth dates. Maternal identity was the most important among these. Unlike most studies, which have reported that birth dates occur earlier as mothers age, we found that birth dates progressively occurred earlier in the year in the early part of a mother's reproductive life, reached a minimum at age 16, and then occurred later at later ages. Birth dates were positively related to a mother's age at primiparity and recent reproductive effort. The earliest birth dates were for pups born to prime‐age mothers who did not reproduce in the previous year but began reproduction early in life, suggesting that females in the best condition gave birth earlier than others. If so, our finding that male pups tended to be born earlier than females provides support for the Trivers–Willard sex‐allocation model. Average birth dates were quite consistent across years, except for 2 years that had notable delays and occurred during the period when massive icebergs were present and disrupted the ecosystem.     

A pseudo-bayesian shrinkage approach to regression with missing covariates

Summary We consider the linear regression of outcome Y on regressors W and Z with some values of W missing, when our main interest is the effect of Z on Y, controlling for W. Three common approaches to regression with missing covariates are (i) complete‐case analysis (CC), which discards the incomplete cases, and (ii) ignorable likelihood methods, which base inference on the likelihood based on the observed data, assuming the missing data are missing at random ( Rubin, 1976b ), and (iii) nonignorable modeling, which posits a joint distribution of the variables and missing data indicators. Another simple practical approach that has not received much theoretical attention is to drop the regressor variables containing missing values from the regression modeling (DV, for drop variables). DV does not lead to bias when either (i) the regression coefficient of W is zero or (ii) W and Z are uncorrelated. We propose a pseudo‐Bayesian approach for regression with missing covariates that compromises between the CC and DV estimates, exploiting information in the incomplete cases when the data support DV assumptions. We illustrate favorable properties of the method by simulation, and apply the proposed method to a liver cancer study. Extension of the method to more than one missing covariate is also discussed.     

Meta-analysis of diagnostic test accuracy assessment studies with varying number of thresholds

Current meta-analytic methods for diagnostic test accuracy are generally applicable to a selection of studies reporting only estimates of sensitivity and specificity, or at most, to studies whose results are reported using an equal number of ordered categories. In this article, we propose a new meta-analytic method to evaluate test accuracy and arrive at a summary receiver operating characteristic (ROC) curve for a collection of studies evaluating diagnostic tests, even when test results are reported in an unequal number of nonnested ordered categories. We discuss both non-Bayesian and Bayesian formulations of the approach. In the Bayesian setting, we propose several ways to construct summary ROC curves and their credible bands. We illustrate our approach with data from a recently published meta-analysis evaluating a single serum progesterone test for diagnosing pregnancy failure.     

On the probability model for asthma attacks

In environmental epidemiology, the impact of environmental agents on symptoms or health status is of interest. This influence is described quantitatively in the theory of Whittemore & Keller (1979). They formulated a logistic model for individuals that is useful in evaluation of panel studies in which each participant protocols whether he does or does not have a certain symptom each day. In the present paper an equation for the prevalence of symptoms in the study population that is defined as the fraction of symptomatic subjects is deduced from the model for individuals. The model for the aggregated quantity depends on the individuals' parameters in a nonlinear manner. The relationship between the individual-based model and the corresponding population-based model is illustrated by means of a simulated panel. Bayesian estimates of the parameters are calculated and compared for both approaches. Bayesian inference enables to apply the prevalence model to a population of non-identical individuals. For such a heterogeneous population, we observe an attenuation of environmental effects on the aggregated symptom prevalence in comparison to the individual-based approach. The presented theory is applicable not only to panel studies but also in time-series analysis of prevalences and incidences.     

Statistical analysis of relative labeled mass spectrometry data from complex samples using ANOVA

Statistical tools enable unified analysis of data from multiple global proteomic experiments, producing unbiased estimates of normalization terms despite the missing data problem inherent in these studies. The modeling approach, implementation, and useful visualization tools are demonstrated via a case study of complex biological samples assessed using the iTRAQ relative labeling protocol.     

The hierarchical metaregression approach and learning from clinical evidence

The hierarchical metaregression (HMR) approach is a multiparameter Bayesian approach for meta‐analysis, which generalizes the standard mixed effects models by explicitly modeling the data collection process in the meta‐analysis. The HMR allows to investigate the potential external validity of experimental results as well as to assess the internal validity of the studies included in a systematic review. The HMR automatically identifies studies presenting conflicting evidence and it downweights their influence in the meta‐analysis. In addition, the HMR allows to perform cross‐evidence synthesis, which combines aggregated results from randomized controlled trials to predict effectiveness in a single‐arm observational study with individual participant data (IPD). In this paper, we evaluate the HMR approach using simulated data examples. We present a new real case study in diabetes research, along with a new R package called jarbes (just a rather Bayesian evidence synthesis), which automatizes the complex computations involved in the HMR.     

Variational Phylodynamic Inference Using Pandemic-scale Data

The ongoing global pandemic has sharply increased the amount of data available to researchers in epidemiology and public health. Unfortunately, few existing analysis tools are capable of exploiting all of the information contained in a pandemic-scale data set, resulting in missed opportunities for improved surveillance and contact tracing. In this paper, we develop the variational Bayesian skyline (VBSKY), a method for fitting Bayesian phylodynamic models to very large pathogen genetic data sets. By combining recent advances in phylodynamic modeling, scalable Bayesian inference and differentiable programming, along with a few tailored heuristics, VBSKY is capable of analyzing thousands of genomes in a few minutes, providing accurate estimates of epidemiologically relevant quantities such as the effective reproduction number and overall sampling effort through time. We illustrate the utility of our method by performing a rapid analysis of a large number of SARS-CoV-2 genomes, and demonstrate that the resulting estimates closely track those derived from alternative sources of public health data.     

Inference for discretely observed stochastic kinetic networks with applications to epidemic modeling

We present a new method for Bayesian Markov Chain Monte Carlo-based inference in certain types of stochastic models, suitable for modeling noisy epidemic data. We apply the so-called uniformization representation of a Markov process, in order to efficiently generate appropriate conditional distributions in the Gibbs sampler algorithm. The approach is shown to work well in various data-poor settings, that is, when only partial information about the epidemic process is available, as illustrated on the synthetic data from SIR-type epidemics and the Center for Disease Control and Prevention data from the onset of the H1N1 pandemic in the United States.     

Addressing delayed case reporting in infectious disease forecast modeling

Infectious disease forecasting is of great interest to the public health community and policymakers, since forecasts can provide insight into disease dynamics in the near future and inform interventions. Due to delays in case reporting, however, forecasting models may often underestimate the current and future disease burden.In this paper, we propose a general framework for addressing reporting delay in disease forecasting efforts with the goal of improving forecasts. We propose strategies for leveraging either historical data on case reporting or external internet-based data to estimate the amount of reporting error. We then describe several approaches for adapting general forecasting pipelines to account for under- or over-reporting of cases. We apply these methods to address reporting delay in data on dengue fever cases in Puerto Rico from 1990 to 2009 and to reports of influenza-like illness (ILI) in the United States between 2010 and 2019. Through a simulation study, we compare method performance and evaluate robustness to assumption violations. Our results show that forecasting accuracy and prediction coverage almost always increase when correction methods are implemented to address reporting delay. Some of these methods required knowledge about the reporting error or high quality external data, which may not always be available. Provided alternatives include excluding recently-reported data and performing sensitivity analysis. This work provides intuition and guidance for handling delay in disease case reporting and may serve as a useful resource to inform practical infectious disease forecasting efforts.     

Estimating a Markovian Epidemic Model Using Household Serial Interval Data from the Early Phase of an Epidemic

The clinical serial interval of an infectious disease is the time between date of symptom onset in an index case and the date of symptom onset in one of its secondary cases. It is a quantity which is commonly collected during a pandemic and is of fundamental importance to public health policy and mathematical modelling. In this paper we present a novel method for calculating the serial interval distribution for a Markovian model of household transmission dynamics. This allows the use of Bayesian MCMC methods, with explicit evaluation of the likelihood, to fit to serial interval data and infer parameters of the underlying model. We use simulated and real data to verify the accuracy of our methodology and illustrate the importance of accounting for household size. The output of our approach can be used to produce posterior distributions of population level epidemic characteristics.     

Modelling under-reporting in epidemics

Under-reporting of infected cases is crucial for many diseases because of the bias it can introduce when making inference for the model parameters. The objective of this paper is to study the effect of under-reporting in epidemics by considering the stochastic Markovian SIR epidemic in which various reporting processes are incorporated. In particular, we first investigate the effect on the estimation process of ignoring under-reporting when it is present in an epidemic outbreak. We show that such an approach leads to under-estimation of the infection rate and the reproduction number. Secondly, by allowing for the fact that under-reporting is occurring, we develop suitable models for estimation of the epidemic parameters and explore how well the reporting rate and other model parameters can be estimated. We consider the case of a constant reporting probability and also more realistic assumptions which involve the reporting probability depending on time or the source of infection for each infected individual. Due to the incomplete nature of the data and reporting process, the Bayesian approach provides a natural modelling framework and we perform inference using data augmentation and reversible jump Markov chain Monte Carlo techniques.     

A bayesian approach to the multistate Jolly-Seber capture-recapture model

This article considers a Bayesian approach to the multistate extension of the Jolly-Seber model commonly used to estimate population abundance in capture-recapture studies. It extends the work of George and Robert (1992, Biometrika79, 677-683), which dealt with the Bayesian estimation of a closed population with only a single state for all animals. A super-population is introduced to model new entrants in the population. Bayesian estimates of abundance are obtained by implementing a Gibbs sampling algorithm based on data augmentation of the missing data in the capture histories when the state of the animal is unknown. Moreover, a partitioning of the missing data is adopted to ensure the convergence of the Gibbs sampling algorithm even in the presence of impossible transitions between some states. Lastly, we apply our methodology to a population of fish to estimate abundance and movement.     

A comparison of methods to estimate seasonal phenological development from BBCH scale recording

The BBCH scale is a two-digit key of growth stages in plants that is based on standardised definitions of plant development stages. The extended BBCH scale, used in this paper, enables the coding of the entire development cycle of all mono- and dicotyledonous plants. Using this key, the frequency distribution of phenological stages was recorded which required a less intense sampling frequency. The onset dates of single events were later estimated from the frequency distribution of BBCH codes. The purpose of this study was to present four different methods from which those onset dates can be estimated. Furthermore, the effects of (1) a less detailed observation key and (2) changes in the sampling frequency on estimates of onset dates were assessed. For all analyses, phenological data from the entire development cycle of four grass species were used. Estimates of onset dates determined by Weighted Plant Development (WPD), Pooled pre-/post-Stage Development (PSD), Cumulative Stage Development (CSD) and Ordinal Logistic Regression (OLR) methods can all be used to determine the phenological progression of plants. Moreover, results show that a less detailed observation key still resulted in similar onset dates, unless more than two consecutive stages were omitted. Further results reveal that the simulation of a less intense sampling frequency had only small impacts on estimates of onset dates. Thus, especially in remote areas where an observation interval of a week is not feasible, estimates derived from the frequency distribution of BBCH codes appear to be appropriate.     

Ambulance diagnosis of ST elevation myocardial infarction eligible for primary PCI

Reperfusion therapy for ST-elevation acute coronary syndromes aims at early and complete recanalisation of the infarct-related artery in order to salvage myocardium and improve both early and late clinical outcomes. The benefit rises exponentially the earlier therapy is initiated. The greatest number of lives saved is within the first hour after symptom onset: the golden hour. The exponential form of the curve relating mortality to time-to-reperfusion has major implications for the timing of treatment. The impact of delay in time-to-treatment lessens as the duration of ischaemia lengthens. Consequently, reducing delays will have a much more positive return in patients presenting early than for those presenting late.