Predicting case numbers during infectious disease outbreaks when some cases are undiagnosed

Sampling bias minimization in disease frequency estimates

An accurate estimate of the number of infected individuals in any disease is crucial. Current estimates are mainly based on the fraction of positive samples or the total number of positive samples. However, both methods are biased and sensitive to the sampling depth. We here propose an alternative method to use the attributes of each sample to estimate the change in the total number of positive patients in the total population. We present a Bayesian estimator assuming a combination of condition and time-dependent probability of being positive, and mixed implicit-explicit solution for the probability of a person with conditions i at time t of being positive. We use this estimate to predict the total probability of being positive at a given day t. We show that these estimate results are smooth and not sensitive to the properties of the samples. Moreover, these results are a better predictor of future mortality.

Evaluation of the number of undiagnosed infected in an outbreak using source of infection measurements

In times of outbreaks, an essential requirement for better monitoring is the evaluation of the number of undiagnosed infected individuals. An accurate estimate of this fraction is crucial for the assessment of the situation and the establishment of protective measures. In most current studies using epidemics models, the total number of infected is either approximated by the number of diagnosed individuals or is dependent on the model parameters and assumptions, which are often debated. We here study the relationship between the fraction of diagnosed infected out of all infected, and the fraction of infected with known contaminator out of all diagnosed infected. We show that those two are approximately the same in exponential models and across most models currently used in the study of epidemics, independently of the model parameters. As an application, we compute an estimate of the effective number of infected by the SARS-CoV-2 virus in various countries.

The R0 package: a toolbox to estimate reproduction numbers for epidemic outbreaks

Background

Several generic methods have been proposed to estimate transmission parameters during an outbreak, especially the reproduction number. However, as of today, no dedicated software exists that implements these methods and allow comparisons.

Results

A review of generic methods used to estimate transmissibility parameters during outbreaks was carried out. Most methods used the epidemic curve and the generation time distribution. Two categories of methods were available: those estimating the initial reproduction number, and those estimating a time dependent reproduction number. We implemented five methods as an R library, developed sensitivity analysis tools for each method and provided numerical illustrations of their use. A comparison of the performance of the different methods on simulated datasets is reported.

Conclusions

This software package allows a standardized and extensible approach to the estimation of the reproduction number and generation interval distribution from epidemic curves.

Modeling the variations in pediatric respiratory syncytial virus seasonal epidemics

Background

Seasonal respiratory syncytial virus (RSV) epidemics occur annually in temperate climates and result in significant pediatric morbidity and increased health care costs. Although RSV epidemics generally occur between October and April, the size and timing vary across epidemic seasons and are difficult to predict accurately. Prediction of epidemic characteristics would support management of resources and treatment.

Methods

The goals of this research were to examine the empirical relationships among early exponential growth rate, total epidemic size, and timing, and the utility of specific parameters in compartmental models of transmission in accounting for variation among seasonal RSV epidemic curves. RSV testing data from Primary Children's Medical Center were collected on children under two years of age (July 2001-June 2008). Simple linear regression was used explore the relationship between three epidemic characteristics (final epidemic size, days to peak, and epidemic length) and exponential growth calculated from four weeks of daily case data. A compartmental model of transmission was fit to the data and parameter estimated used to help describe the variation among seasonal RSV epidemic curves.

Results

The regression results indicated that exponential growth was correlated to epidemic characteristics. The transmission modeling results indicated that start time for the epidemic and the transmission parameter co-varied with the epidemic season.

Conclusions

The conclusions were that exponential growth was somewhat empirically related to seasonal epidemic characteristics and that variation in epidemic start date as well as the transmission parameter over epidemic years could explain variation in seasonal epidemic size. These relationships are useful for public health, health care providers, and infectious disease researchers.

An approach to estimate the number of SARS cases imported by international air travel

The worldwide spread of severe acute respiratory syndrome (SARS) raised questions about the risk of importation of such infection, in particular by air travel. Entry screening was implemented in some countries although poor evidence on its effectiveness is reported. We developed a model to estimate the number of imported SARS cases between regions, using the 2003 SARS epidemic data to apply this model for two scenarios: from Beijing to Frankfurt and from Hong Kong to London. We back-calculated the data to estimate individuals' time of infection and built a model where every individual has a probability of being isolated, of traveling, and of being undetected at arrival. The findings, consistent with what was observed in 2003, suggest that entry screening does not affect the predicted number of imported cases. Inversely, importation depends on the transmission dynamic in the country of origin (including control measures in place) and on the intensity of air travel between regions.

Quantifying the transmission potential of pandemic influenza

This article reviews quantitative methods to estimate the basic reproduction number of pandemic influenza, a key threshold quantity to help determine the intensity of interventions required to control the disease. Although it is difficult to assess the transmission potential of a probable future pandemic, historical epidemiologic data is readily available from previous pandemics, and as a reference quantity for future pandemic planning, mathematical and statistical analyses of historical data are crucial. In particular, because many historical records tend to document only the temporal distribution of cases or deaths (i.e. epidemic curve), our review focuses on methods to maximize the utility of time-evolution data and to clarify the detailed mechanisms of the spread of influenza.

First, we highlight structured epidemic models and their parameter estimation method which can quantify the detailed disease dynamics including those we cannot observe directly. Duration-structured epidemic systems are subsequently presented, offering firm understanding of the definition of the basic and effective reproduction numbers. When the initial growth phase of an epidemic is investigated, the distribution of the generation time is key statistical information to appropriately estimate the transmission potential using the intrinsic growth rate. Applications of stochastic processes are also highlighted to estimate the transmission potential using similar data. Critically important characteristics of influenza data are subsequently summarized, followed by our conclusions to suggest potential future methodological improvements.

Comparative estimation of the reproduction number for pandemic influenza from daily case notification data

The reproduction number, R, defined as the average number of secondary cases generated by a primary case, is a crucial quantity for identifying the intensity of interventions required to control an epidemic. Current estimates of the reproduction number for seasonal influenza show wide variation and, in particular, uncertainty bounds for R for the pandemic strain from 1918 to 1919 have been obtained only in a few recent studies and are yet to be fully clarified. Here, we estimate R using daily case notifications during the autumn wave of the influenza pandemic (Spanish flu) in the city of San Francisco, California, from 1918 to 1919. In order to elucidate the effects from adopting different estimation approaches, four different methods are used: estimation of R using the early exponential-growth rate (Method 1), a simple susceptible-exposed-infectious-recovered (SEIR) model (Method 2), a more complex SEIR-type model that accounts for asymptomatic and hospitalized cases (Method 3), and a stochastic susceptible-infectious-removed (SIR) with Bayesian estimation (Method 4) that determines the effective reproduction number Rt at a given time t. The first three methods fit the initial exponential-growth phase of the epidemic, which was explicitly determined by the goodness-of-fit test. Moreover, Method 3 was also fitted to the whole epidemic curve. Whereas the values of R obtained using the first three methods based on the initial growth phase were estimated to be 2.98 (95% confidence interval (CI): 2.73, 3.25), 2.38 (2.16, 2.60) and 2.20 (1.55, 2.84), the third method with the entire epidemic curve yielded a value of 3.53 (3.45, 3.62). This larger value could be an overestimate since the goodness-of-fit to the initial exponential phase worsened when we fitted the model to the entire epidemic curve, and because the model is established as an autonomous system without time-varying assumptions. These estimates were shown to be robust to parameter uncertainties, but the theoretical exponential-growth approximation (Method 1) shows wide uncertainty. Method 4 provided a maximum-likelihood effective reproduction number 2.10 (1.21, 2.95) using the first 17 epidemic days, which is consistent with estimates obtained from the other methods and an estimate of 2.36 (2.07, 2.65) for the entire autumn wave. We conclude that the reproduction number for pandemic influenza (Spanish flu) at the city level can be robustly assessed to lie in the range of 2.0-3.0, in broad agreement with previous estimates using distinct data.