The analysis of hospital infection data using hidden Markov models

Surveillance data for communicable nosocomial pathogens usually consist of short time series of low-numbered counts of infected patients. These often show overdispersion and autocorrelation. To date, almost all analyses of such data have ignored the communicable nature of the organisms and have used methods appropriate only for independent outcomes. Inferences that depend on such analyses cannot be considered reliable when patient-to-patient transmission is important. We propose a new method for analysing these data based on a mechanistic model of the epidemic process. Since important nosocomial pathogens are often carried asymptomatically with overt infection developing in only a proportion of patients, the epidemic process is usually only partially observed by routine surveillance data. We therefore develop a 'structured' hidden Markov model where the underlying Markov chain is generated by a simple transmission model. We apply both structured and standard (unstructured) hidden Markov models to time series for three important pathogens. We find that both methods can offer marked improvements over currently used approaches when nosocomial spread is important. Compared to the standard hidden Markov model, the new approach is more parsimonious, is more biologically plausible, and allows key epidemiological parameters to be estimated.     

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 Network-based Analysis of the 1861 Hagelloch Measles Data

Summary In this article, we demonstrate a statistical method for fitting the parameters of a sophisticated network and epidemic model to disease data. The pattern of contacts between hosts is described by a class of dyadic independence exponential-family random graph models (ERGMs), whereas the transmission process that runs over the network is modeled as a stochastic susceptible-exposed-infectious-removed (SEIR) epidemic. We fit these models to very detailed data from the 1861 measles outbreak in Hagelloch, Germany. The network models include parameters for all recorded host covariates including age, sex, household, and classroom membership and household location whereas the SEIR epidemic model has exponentially distributed transmission times with gamma-distributed latent and infective periods. This approach allows us to make meaningful statements about the structure of the population-separate from the transmission process-as well as to provide estimates of various biological quantities of interest, such as the effective reproductive number, R. Using reversible jump Markov chain Monte Carlo, we produce samples from the joint posterior distribution of all the parameters of this model-the network, transmission tree, network parameters, and SEIR parameters-and perform Bayesian model selection to find the best-fitting network model. We compare our results with those of previous analyses and show that the ERGM network model better fits the data than a Bernoulli network model previously used. We also provide a software package, written in R, that performs this type of analysis.     

Bayesian inference of hospital-acquired infectious diseases and control measures given imperfect surveillance data

This paper describes a stochastic epidemic model developed to infer transmission rates of asymptomatic communicable pathogens within a hospital ward. Inference is complicated by partial observation of the epidemic process and dependencies within the data. The epidemic process of nosocomial communicable pathogens can be partially observed by routine swabs testing for the presence of the pathogen. False-negative swab results must be accounted for and make it difficult to ascertain the number of patients who were colonized. Reversible jump Markov chain Monte Carlo methods are used within a Bayesian framework to make inferences about the colonization rates and unknown colonization times. The methods are applied to routinely collected data concerning methicillin-resistant Staphylococcus Aureus in an intensive care unit to estimate the effectiveness of isolation on reducing transmission of the bacterium.     

Using Combined Diagnostic Test Results to Hindcast Trends of Infection from Cross-Sectional Data

Infectious disease surveillance is key to limiting the consequences from infectious pathogens and maintaining animal and public health. Following the detection of a disease outbreak, a response in proportion to the severity of the outbreak is required. It is thus critical to obtain accurate information concerning the origin of the outbreak and its forward trajectory. However, there is often a lack of situational awareness that may lead to over- or under-reaction. There is a widening range of tests available for detecting pathogens, with typically different temporal characteristics, e.g. in terms of when peak test response occurs relative to time of exposure. We have developed a statistical framework that combines response level data from multiple diagnostic tests and is able to ‘hindcast’ (infer the historical trend of) an infectious disease epidemic. Assuming diagnostic test data from a cross-sectional sample of individuals infected with a pathogen during an outbreak, we use a Bayesian Markov Chain Monte Carlo (MCMC) approach to estimate time of exposure, and the overall epidemic trend in the population prior to the time of sampling. We evaluate the performance of this statistical framework on simulated data from epidemic trend curves and show that we can recover the parameter values of those trends. We also apply the framework to epidemic trend curves taken from two historical outbreaks: a bluetongue outbreak in cattle, and a whooping cough outbreak in humans. Together, these results show that hindcasting can estimate the time since infection for individuals and provide accurate estimates of epidemic trends, and can be used to distinguish whether an outbreak is increasing or past its peak. We conclude that if temporal characteristics of diagnostics are known, it is possible to recover epidemic trends of both human and animal pathogens from cross-sectional data collected at a single point in time.     

Bayesian selection of markov models for symbol sequences: application to microsaccadic eye movements

Complex biological dynamics often generate sequences of discrete events which can be described as a Markov process. The order of the underlying Markovian stochastic process is fundamental for characterizing statistical dependencies within sequences. As an example for this class of biological systems, we investigate the Markov order of sequences of microsaccadic eye movements from human observers. We calculate the integrated likelihood of a given sequence for various orders of the Markov process and use this in a Bayesian framework for statistical inference on the Markov order. Our analysis shows that data from most participants are best explained by a first-order Markov process. This is compatible with recent findings of a statistical coupling of subsequent microsaccade orientations. Our method might prove to be useful for a broad class of biological systems.     

Bayesian analysis of experimental epidemics of foot-and-mouth disease

We investigate the transmission dynamics of a certain type of foot-and-mouth disease (FMD) virus under experimental conditions. Previous analyses of experimental data from FMD outbreaks in non-homogeneously mixing populations of sheep have suggested a decline in viraemic level through serial passage of the virus, but these do not take into account possible variation in the length of the chain of viral transmission for each animal, which is implicit in the non-observed transmission process. We consider a susceptible-exposed-infectious-removed non-Markovian compartmental model for partially observed epidemic processes, and we employ powerful methodology (Markov chain Monte Carlo) for statistical inference, to address epidemiological issues under a Bayesian framework that accounts for all available information and associated uncertainty in a coherent approach. The analysis allows us to investigate the posterior distribution of the hidden transmission history of the epidemic, and thus to determine the effect of the length of the infection chain on the recorded viraemic levels, based on the posterior distribution of a p-value. Parameter estimates of the epidemiological characteristics of the disease are also obtained. The results reveal a possible decline in viraemia in one of the two experimental outbreaks. Our model also suggests that individual infectivity is related to the level of viraemia.     

Estimating Temporal Transmission Parameters from Infectious Disease Household Data, with Application to Taiwan SARS Data

Taking households having at least one infective as standard units and considering both a within-household infection rate and a global infection rate, we propose a Bayesian two level mixing S-I-R (susceptible-infective-removed) counting process model in which the transmission parameters may change over time and the parameters of interest are the within-household infection rate and the removal rate. Customized Markov chain Monte Carlo methods are developed for generating samples from the posterior distribution for inference purpose, based only on the removal times. The numerical performance of this method is examined in a simulation study. Applying this method to 2003 Taiwan SARS data, we find that the within-household infection rate decreases, the removal rate increases and their ratio is less than one and decreases significantly during the epidemic. This method allows the estimation of these parameters during the epidemic. For a rapidly transmitted disease, it provides a method to nearly real-time tracking of infection measures.     

Modeling and real-time prediction of classical swine fever epidemics

We propose a new method to analyze outbreak data of an infectious disease such as classical swine fever. The underlying model is a two-type branching process. It is used to deduce information concerning the epidemic from detected cases. In particular, the method leads to prediction of the future course of the epidemic and hence can be used as a basis for control policy decisions. We test the model with data from the large 1997-1998 classical swine fever epidemic in The Netherlands. It turns out that our results are in good agreement with the data.     

A Markov Chain Model of Population Growth with an Application in Epidemiology

This paper is concerned with a class of population growth processes in discrete time; the simple epidemic process is considered as a specific example. A Markov chain model is constructed and standard Markov methods are used to study the main biological concepts. A simple and explicit formula is obtained for the transient distribution of the population size. Then, the cost of the process is defined and the joint probability generating function of its components is derived. Finally, the results are extended to the case where the inter-transition periods are bounded i.i.d. random variables.     

Evaluating Subcriticality during the Ebola Epidemic in West Africa

The 2014–2015 Ebola outbreak is the largest and most widespread to date. In order to estimate ongoing transmission in the affected countries, we estimated the weekly average number of secondary cases caused by one individual infected with Ebola throughout the infectious period for each affected West African country using a stochastic hidden Markov model fitted to case data from the World Health Organization. If the average number of infections caused by one Ebola infection is less than 1.0, the epidemic is subcritical and cannot sustain itself. The epidemics in Liberia and Sierra Leone have approached subcriticality at some point during the epidemic; the epidemic in Guinea is ongoing with no evidence that it is subcritical. Response efforts to control the epidemic should continue in order to eliminate Ebola cases in West Africa.     

Analysis of Non-stationary Data for Heart-rate Fluctuations in Terms of Drift and Diffusion Coefficients

We describe a method for analyzing the stochasticity in non-stationary data for the beat-to-beat fluctuations in the heart rates of healthy subjects, as well as those with congestive heart failure. The method analyzes the return time series of the data as a Markov process, and computes the Markov time scale, i.e., the time scale over which the data are a Markov process. We also construct an effective stochastic continuum equation for the return series. We show that the drift and diffusion coefficients, as well as the amplitude of the return time series for healthy subjects are distinct from those with CHF. Thus, the method may potentially provide a diagnostic tool for distinguishing healthy subjects from those with congestive heart failure, as it can distinguish small differences between the data for the two classes of subjects in terms of well-defined and physically-motivated quantities. PACS: 05.10.Gg 05.40.-a, 05.45.Tp, 87.19.Hh     

Bayesian analysis of severe acute respiratory syndrome: the 2003 Hong Kong epidemic

This paper analyzes data arising from a Severe Acute Respiratory Syndrome (SARS) epidemic in Hong Kong in 2003 involving 1755 cases. A discrete time stochastic model that uses a back-projection approach is proposed. Markov Chain Monte Carlo (MCMC) methods are developed for estimation of model parameters. The algorithm is further extended to integrate numerically over unobserved variables of the model. Applying the method to SARS data from Hong Kong, a value of 3.88 with a posterior standard deviation of 0.09 was estimated for the basic reproduction number. An estimate of the transmission parameter at the beginning of the epidemic was also obtained as 0.149 with a posterior standard deviation of 0.003. A reduction in the transmission parameter during the course of the epidemic forced the effective reproduction number to cross the threshold value of one, seven days after control interventions were introduced. At the end of the epidemic, the effective reproduction number was as low as 0.001 suggesting that the epidemic was brought under control by the intervention measures introduced.     

A two-component model for counts of infectious diseases

We propose a stochastic model for the analysis of time series of disease counts as collected in typical surveillance systems on notifiable infectious diseases. The model is based on a Poisson or negative binomial observation model with two components: a parameter-driven component relates the disease incidence to latent parameters describing endemic seasonal patterns, which are typical for infectious disease surveillance data. An observation-driven or epidemic component is modeled with an autoregression on the number of cases at the previous time points. The autoregressive parameter is allowed to change over time according to a Bayesian changepoint model with unknown number of changepoints. Parameter estimates are obtained through the Bayesian model averaging using Markov chain Monte Carlo techniques. We illustrate our approach through analysis of simulated data and real notification data obtained from the German infectious disease surveillance system, administered by the Robert Koch Institute in Berlin. Software to fit the proposed model can be obtained from http://www.statistik.lmu.de/ approximately mhofmann/twins.     

A stochastic SIR model with contact-tracing: large population limits and statistical inference

This paper is devoted to the presentation and study of a specific stochastic epidemic model accounting for the effect of contact-tracing on the spread of an infectious disease. Precisely, one considers here the situation in which individuals identified as infected by the public health detection system may contribute to detecting other infectious individuals by providing information related to persons with whom they have had possibly infectious contacts. The control strategy, which consists of examining each individual who has been able to be identified on the basis of the information collected within a certain time period, is expected to efficiently reinforce the standard random-screening-based detection and considerably ease the epidemic. In the novel modelling of the spread of a communicable infectious disease considered here, the population of interest evolves through demographic, infection and detection processes, in a way that its temporal evolution is described by a stochastic Markov process, of which the component accounting for the contact-tracing feature is assumed to be valued in a space of point measures. For adequate scalings of the demographic, infection and detection rates, it is shown to converge to the weak deterministic solution of a PDE system, as a parameter n, interpreted as the population size, roughly speaking, becomes larger. From the perspective of the analysis of infectious disease data, this approximation result may serve as a key tool for exploring the asymptotic properties of standard inference methods such as maximum likelihood estimation. We state preliminary statistical results in this context. Eventually, relations of the model with the available data of the HIV epidemic in Cuba, in which country a contact-tracing detection system has been set up since 1986, is investigated and numerical applications are carried out.     

Three-stage stochastic epidemic model: an application to AIDS.

A three-stage stochastic epidemic model extending the so-called classical epidemic process to one that includes time-dependent transition probabilities is described, and a solution to the appropriate set of forward differential-difference equations is given. When an individual can move from being a susceptible to one infected with the HIV virus to one diagnosed as having AIDS, we can use this general model to describe an AIDS epidemic process. We obtain expressions for the mean and variance of the number of AIDS cases for some special cases. By comparing these with actual data, it is suggested that, for some categories of cases (in particular, children), this model might be a plausible model to describe the underlying mechanism of the AIDS epidemic.     

Parameterizing Spatial Models of Infectious Disease Transmission that Incorporate Infection Time Uncertainty Using Sampling-Based Likelihood Approximations

A class of discrete-time models of infectious disease spread, referred to as individual-level models (ILMs), are typically fitted in a Bayesian Markov chain Monte Carlo (MCMC) framework. These models quantify probabilistic outcomes regarding the risk of infection of susceptible individuals due to various susceptibility and transmissibility factors, including their spatial distance from infectious individuals. The infectious pressure from infected individuals exerted on susceptible individuals is intrinsic to these ILMs. Unfortunately, quantifying this infectious pressure for data sets containing many individuals can be computationally burdensome, leading to a time-consuming likelihood calculation and, thus, computationally prohibitive MCMC-based analysis. This problem worsens when using data augmentation to allow for uncertainty in infection times. In this paper, we develop sampling methods that can be used to calculate a fast, approximate likelihood when fitting such disease models. A simple random sampling approach is initially considered followed by various spatially-stratified schemes. We test and compare the performance of our methods with both simulated data and data from the 2001 foot-and-mouth disease (FMD) epidemic in the U.K. Our results indicate that substantial computation savings can be obtained—albeit, of course, with some information loss—suggesting that such techniques may be of use in the analysis of very large epidemic data sets.     

Estimation of Parameters for Macroparasite Population Evolution Using Approximate Bayesian Computation

Summary We estimate the parameters of a stochastic process model for a macroparasite population within a host using approximate Bayesian computation (ABC). The immunity of the host is an unobserved model variable and only mature macroparasites at sacrifice of the host are counted. With very limited data, process rates are inferred reasonably precisely. Modeling involves a three variable Markov process for which the observed data likelihood is computationally intractable. ABC methods are particularly useful when the likelihood is analytically or computationally intractable. The ABC algorithm we present is based on sequential Monte Carlo, is adaptive in nature, and overcomes some drawbacks of previous approaches to ABC. The algorithm is validated on a test example involving simulated data from an autologistic model before being used to infer parameters of the Markov process model for experimental data. The fitted model explains the observed extra‐binomial variation in terms of a zero‐one immunity variable, which has a short‐lived presence in the host.