Estimating the Basic Reproductive Number in the General Epidemic Model with an Unknown Initial Number of Susceptible Individuals

Abstract. In any epidemic, there may exist an unidentified subpopulation which might be naturally immune or isolated and who will not be involved in the transmission of the disease. Estimation of key parameters, for example, the basic reproductive number, without accounting for this possibility would underestimate the severity of the epidemics. Here, we propose a procedure to estimate the basic reproductive number ( R ₀ ) in an epidemic model with an unknown initial number of susceptibles. The infection process is usually not completely observed, but is reconstructed by a kernel‐smoothing method under a counting process framework. Simulation is used to evaluate the performance of the estimators for major epidemics. We illustrate the procedure using the Abakaliki smallpox data.     

Inference in disease transmission experiments by using stochastic epidemic models

Summary.  The paper extends the susceptible–exposed–infective–removed model to handle heterogeneity introduced by spatially arranged populations, biologically plausible distributional assumptions and incorporation of observations from additional diagnostic tests. These extensions are motivated by a desire to analyse disease transmission experiments in a more detailed fashion than before. Such experiments are performed by veterinarians to gain knowledge about the dynamics of an infectious disease. By fitting our spatial susceptible–exposed–infective–removed with diagnostic testing model to data for a specific disease and production environment a valuable decision support tool is obtained, e.g. when evaluating on-farm control measures. Partial observability of the epidemic process is an inherent problem when trying to estimate model parameters from experimental data. We therefore extend existing work on Markov chain Monte Carlo estimation in partially observable epidemics to the multitype epidemic set-up of our model. Throughout the paper, data from a Belgian classical swine fever virus transmission experiment are used as a motivating example.     

Efficient likelihood-free Bayesian Computation for?household epidemics

Considerable progress has been made in applying Markov chain Monte Carlo (MCMC) methods to the analysis of epidemic data. However, this likelihood based method can be inefficient due to the limited data available concerning an epidemic outbreak. This paper considers an alternative approach to studying epidemic data using Approximate Bayesian Computation (ABC) methodology. ABC is a simulation-based technique for obtaining an approximate sample from the posterior distribution of the parameters of the model and in an epidemic context is very easy to implement. A new approach to ABC is introduced which generates a set of values from the (approximate) posterior distribution of the parameters during each simulation rather than a single value. This is based upon coupling simulations with different sets of parameters and we call the resulting algorithm coupled ABC. The new methodology is used to analyse final size data for epidemics amongst communities partitioned into households. It is shown that for the epidemic data sets coupled ABC is more efficient than ABC and MCMC-ABC.     

Influenza vaccine efficacy trials: a simulation approach to understand failures from the past

The success of a seasonal influenza vaccine efficacy trial depends not only upon the design but also upon the annual epidemic characteristics. In this context, simulation methods are an essential tool in evaluating the performances of study designs under various circumstances. However, traditional methods for simulating time‐to‐event data are not suitable for the simulation of influenza vaccine efficacy trials because of the seasonality and heterogeneity of influenza epidemics. Instead, we propose a mathematical model parameterized with historical surveillance data, heterogeneous frailty among the subjects, survey‐based heterogeneous number of daily contact, and a mixed vaccine protection mechanism. We illustrate our methodology by generating multiple‐trial data similar to a large phase III trial that failed to show additional relative vaccine efficacy of an experimental adjuvanted vaccine compared with the reference vaccine. We show that small departures from the designing assumptions, such as a smaller range of strain protection for the experimental vaccine or the chosen endpoint, could lead to smaller probabilities of success in showing significant relative vaccine efficacy. Copyright © 2015 John Wiley & Sons, Ltd.     

Computation of final outcome probabilities for the generalised stochastic epidemic

This paper is concerned with methods for the numerical calculation of the final outcome distribution for a well-known stochastic epidemic model in a closed population. The model is of the SIR (Susceptible→Infected→ Removed) type, and the infectious period can have any specified distribution. The final outcome distribution is specified by the solution of a triangular system of linear equations, but the form of the distribution leads to inherent numerical problems in the solution. Here we employ multiple precision arithmetic to surmount these problems. As applications of our methodology, we assess the accuracy of two approximations that are frequently used in practice, namely an approximation for the probability of an epidemic occurring, and a Gaussian approximation to the final number infected in the event of an outbreak. We also present an example of Bayesian inference for the epidemic threshold parameter.     

Exact Bayesian Inference and Model Selection for Stochastic Models of Epidemics Among a Community of Households

Abstract.  Much recent methodological progress in the analysis of infectious disease data has been due to Markov chain Monte Carlo (MCMC) methodology. In this paper, it is illustrated that rejection sampling can also be applied to a family of inference problems in the context of epidemic models, avoiding the issues of convergence associated with MCMC methods. Specifically, we consider models for epidemic data arising from a population divided into households. The models allow individuals to be potentially infected both from outside and from within the household. We develop methodology for selection between competing models via the computation of Bayes factors. We also demonstrate how an initial sample can be used to adjust the algorithm and improve efficiency. The data are assumed to consist of the final numbers ultimately infected within a sample of households in some community. The methods are applied to data taken from outbreaks of influenza.     

Data and projections of HIV and AIDS in Portugal

Projections of AIDS incidence are critical for assessing future healthcare needs. This paper focuses on the method of back-calculation for obtaining forecasts. The first problem faced was the need to account for delays and underreporting in reporting of cases and to adjust the incidence data. The method used to estimate the reporting delay distribution is based on Poisson regression and involves cross-classifying each reported case by calendar time of diagnosis and reporting delay. The adjusted AIDS incidence data are then used to obtain short-term projections and lower bounds on the size of the AIDS epidemic. The estimation procedure 'back-calculates' from AIDS incidence data using the incubation period distribution to obtain estimates of the numbers previously infected. These numbers are then projected forward. The problem can be shown to reduce to estimating the size of a multinomial population. The expectation-maximization (EM) algorithm is used to obtain maximum-likelihood estimates when the density of infection times is parametrized as a step function. The methodology is applied to AIDS incidence data in Portugal for four different transmission categories: injecting drug users, sexual transmission (homosexual/bisexual and heterosexual contact) and other, mainly haemophilia and blood transfusion related, to obtain short-term projections and an estimate of the minimum size of the epidemic.     

Bayesian Portfolio Selection: An Empirical Analysis of the S&P 500 Index 1970–1996

In this article we present a technique for implementing large-scale optimal portfolio selection. We use high-frequency daily data to capture valuable statistical information in asset returns. We describe several statistical issues involved in quantitative approaches to portfolio selection. Our methodology applies to large-scale portfolio-selection problems in which the number of possible holdings is large relative to the estimation period provided by historical data. We illustrate our approach on an equity database that consists of stocks from the Standard and Poor's index, and we compare our portfolios to this benchmark index. Our methodology differs from the usual quadratic programming approach to portfolio selection in three ways: (1) We employ informative priors on the expected returns and variance-covariance matrices, (2) we use daily data for estimation purposes, with upper and lower holding limits for individual securities, and (3) we use a dynamic asset-allocation approach that is based on reestimating and then rebalancing the portfolio weights on a prespecified time window. The key inputs to the optimization process are the predictive distributions of expected returns and the predictive variance-covariance matrix. We describe the statistical issues involved in modeling these inputs for high-dimensional portfolio problems in which our data frequency is daily. In our application, we find that our optimal portfolio outperforms the underlying benchmark.     

Semiparametric estimation of the duration of immunity from infectious disease time series: influenza as a case-study

Summary.  An important epidemiological problem is to estimate the decay through time of immunity following infection. For this purpose, we propose a semiparametric time series epidemic model that is based on the mechanism of the susceptible–infected–recovered–susceptible system to analyse complex time series data. We develop an estimation method for the model. Simulations show that the approach proposed can capture the non-linearity of epidemics as well as estimate the decay of immunity. We apply our approach to influenza in France and the Netherlands and show a rapid decline in immunity following infection, which agrees with recent spatiotemporal analyses.     

Bayesian inference for nonlinear stochastic SIR epidemic model

ABSTRACT

Inference for epidemic parameters can be challenging, in part due to data that are intrinsically stochastic and tend to be observed by means of discrete-time sampling, which are limited in their completeness. The problem is particularly acute when the likelihood of the data is computationally intractable. Consequently, standard statistical techniques can become too complicated to implement effectively. In this work, we develop a powerful method for Bayesian paradigm for susceptible–infected–removed stochastic epidemic models via data-augmented Markov Chain Monte Carlo. This technique samples all missing values as well as the model parameters, where the missing values and parameters are treated as random variables. These routines are based on the approximation of the discrete-time epidemic by diffusion process. We illustrate our techniques using simulated epidemics and finally we apply them to the real data of Eyam plague.     

A note on robustness of D-optimal block designs for two-colour microarray experiments

Two-colour microarray experiments form an important tool in gene expression analysis. Due to the high risk of missing observations in microarray experiments, it is fundamental to concentrate not only on optimal designs but also on designs which are robust against missing observations. As an extension of Latif et al. (2009), we define the optimal breakdown number for a collection of designs to describe the robustness, and we calculate the breakdown number for various D-optimal block designs. We show that, for certain values of the numbers of treatments and arrays, the designs which are D-optimal have the highest breakdown number. Our calculations use methods from graph theory.     

ESTIMATING WITHIN-HOUSEHOLD CONTACT NETWORKS FROM EGOCENTRIC DATA

Acute respiratory diseases are transmitted over networks of social contacts. Large-scale simulation models are used to predict epidemic dynamics and evaluate the impact of various interventions, but the contact behavior in these models is based on simplistic and strong assumptions which are not informed by survey data. These assumptions are also used for estimating transmission measures such as the basic reproductive number and secondary attack rates. Development of methodology to infer contact networks from survey data could improve these models and estimation methods. We contribute to this area by developing a model of within-household social contacts and using it to analyze the Belgian POLYMOD data set, which contains detailed diaries of social contacts in a 24-hour period. We model dependency in contact behavior through a latent variable indicating which household members are at home. We estimate age-specific probabilities of being at home and age-specific probabilities of contact conditional on two members being at home. Our results differ from the standard random mixing assumption. In addition, we find that the probability that all members contact each other on a given day is fairly low: 0.49 for households with two 0-5 year olds and two 19-35 year olds, and 0.36 for households with two 12-18 year olds and two 36+ year olds. We find higher contact rates in households with 2-3 members, helping explain the higher influenza secondary attack rates found in households of this size.     

Bayesian inference for epidemics with two levels of mixing

Abstract.  Methodology for Bayesian inference is considered for a stochastic epidemic model which permits mixing on both local and global scales. Interest focuses on estimation of the within- and between-group transmission rates given data on the final outcome. The model is sufficiently complex that the likelihood of the data is numerically intractable. To overcome this difficulty, an appropriate latent variable is introduced, about which asymptotic information is known as the population size tends to infinity. This yields a method for approximate inference for the true model. The methods are applied to real data, tested with simulated data, and also applied to a simple epidemic model for which exact results are available for comparison.     

Modeling Competing Infectious Pathogens from a Bayesian Perspective: Application to Influenza Studies with Incomplete Laboratory Results

In seasonal influenza epidemics, pathogens such as respiratory syncytial virus (RSV) often co-circulate with influenza and cause influenza-like illness (ILI) in human hosts. However, it is often impractical to test for each potential pathogen or to collect specimens for each observed ILI episode, making inference about influenza transmission difficult. In the setting of infectious diseases, missing outcomes impose a particular challenge because of the dependence among individuals. We propose a Bayesian competing-risk model for multiple co-circulating pathogens for inference on transmissibility and intervention efficacies under the assumption that missingness in the biological confirmation of the pathogen is ignorable. Simulation studies indicate a reasonable performance of the proposed model even if the number of potential pathogens is misspecified. They also show that a moderate amount of missing laboratory test results has only a small impact on inference about key parameters in the setting of close contact groups. Using the proposed model, we found that a non-pharmaceutical intervention is marginally protective against transmission of influenza A in a study conducted in elementary schools.     

Optimal weighted two-sample t-test with partially paired data in a unified framework

In this paper, we provide a unified framework for two-sample t-test with partially paired data. We show that many existing two-sample t-tests with partially paired data can be viewed as special members in our unified framework. Some shortcomings of these t-tests are discussed. We also propose the asymptotically optimal weighted linear combination of the test statistics comparing all four paired and unpaired data sets. Simulation studies are used to illustrate the performance of our proposed asymptotically optimal weighted combinations of test statistics and compare with some existing methods. It is found that our proposed test statistic is generally more powerful. Three real data sets about CD4 count, DNA extraction concentrations, and the quality of sleep are also analyzed by using our newly introduced test statistic.     

TWO-PASS CUSUM TO IDENTIFY AGE-CLUSTER OUTBREAKS

The paper introduces a two-pass adaptive cumulative sum (CUSUM) statistic to identify age clusters (age grouping) that significantly contribute to epidemics or unusually high counts. If epidemiologists know that an epidemic is confined to a narrow age group, then this information not only makes it clear where to target the epidemiological effort but also helps them decide whether to respond. It is much easier to control an epidemic that starts in a narrow age range of the population, such as pre-school children, than an epidemic that is not confined demographically or geographically.     

Inference for Epidemics with Three Levels of Mixing: Methodology and Application to a Measles Outbreak

Abstract. A stochastic epidemic model is defined in which each individual belongs to a household, a secondary grouping (typically school or workplace) and also the community as a whole. Moreover, infectious contacts take place in these three settings according to potentially different rates. For this model, we consider how different kinds of data can be used to estimate the infection rate parameters with a view to understanding what can and cannot be inferred. Among other things we find that temporal data can be of considerable inferential benefit compared with final size data, that the degree of heterogeneity in the data can have a considerable effect on inference for non‐household transmission, and that inferences can be materially different from those obtained from a model with only two levels of mixing. We illustrate our findings by analysing a highly detailed dataset concerning a measles outbreak in Hagelloch, Germany.     

On the Optimality of Estimating the Tail Index and a Naive Estimator

Using a straightforward estimator for estimating the tail index of a distribution we illustrate the inherent difficulties of this problem. We prove strong and weak consistencies and central limit theorems for our naive estimator, and discuss its various rates of convergence under different conditions. We argue that, while optimal rates of convergence do exist under various conditions for a number of estimators of the tail index, the notion of an optimal sequence for this problem is bound to run into unsurmountable difficulties.     

Optimal U-type design for Bayesian nonparametric multiresponse prediction

This paper presents an extension of the work of Yue and Chatterjee (2010) about U-type designs for Bayesian nonparametric response prediction. We consider nonparametric Bayesian regression model with p responses. We use U-type designs with n runs, m factors and q levels for the nonparametric multiresponse prediction based on the asymptotic Bayesian criterion. A lower bound for the proposed criterion is established, and some optimal and nearly optimal designs for the illustrative models are given.     

ESTIMATING WITHIN-SCHOOL CONTACT NETWORKS TO UNDERSTAND INFLUENZA TRANSMISSION

Many epidemic models approximate social contact behavior by assuming random mixing within mixing groups (e.g., homes, schools, and workplaces). The effect of more realistic social network structure on estimates of epidemic parameters is an open area of exploration. We develop a detailed statistical model to estimate the social contact network within a high school using friendship network data and a survey of contact behavior. Our contact network model includes classroom structure, longer durations of contacts to friends than non-friends and more frequent contacts with friends, based on reports in the contact survey. We performed simulation studies to explore which network structures are relevant to influenza transmission. These studies yield two key findings. First, we found that the friendship network structure important to the transmission process can be adequately represented by a dyad-independent exponential random graph model (ERGM). This means that individual-level sampled data is sufficient to characterize the entire friendship network. Second, we found that contact behavior was adequately represented by a static rather than dynamic contact network. We then compare a targeted antiviral prophylaxis intervention strategy and a grade closure intervention strategy under random mixing and network-based mixing. We find that random mixing overestimates the effect of targeted antiviral prophylaxis on the probability of an epidemic when the probability of transmission in 10 minutes of contact is less than 0.004 and underestimates it when this transmission probability is greater than 0.004. We found the same pattern for the final size of an epidemic, with a threshold transmission probability of 0.005. We also find random mixing overestimates the effect of a grade closure intervention on the probability of an epidemic and final size for all transmission probabilities. Our findings have implications for policy recommendations based on models assuming random mixing, and can inform further development of network-based models.