Bayesian nonparametric inference for heterogeneously mixing infectious disease models.

SignificanceMathematical models of infectious disease transmission continue to play a vital role in understanding, mitigating, and preventing outbreaks. The vast majority of epidemic models in the literature are parametric, meaning that they contain inherent assumptions about how transmission occurs in a population. However, such assumptions can be lacking in appropriate biological or epidemiological justification and in consequence lead to erroneous scientific conclusions and misleading predictions. We propose a flexible Bayesian nonparametric framework that avoids the need to make strict model assumptions about the infection process and enables a far more data-driven modeling approach for inferring the mechanisms governing transmission. We use our methods to enhance our understanding of the transmission mechanisms of the 2001 UK foot and mouth disease outbreak.

Pair-based likelihood approximations for stochastic epidemic models.

Fitting stochastic epidemic models to data is a non-standard problem because data on the infection processes defined in such models are rarely observed directly. This in turn means that the likelihood of the observed data is intractable in the sense that it is very computationally expensive to obtain. Although data-augmented Markov chain Monte Carlo (MCMC) methods provide a solution to this problem, employing a tractable augmented likelihood, such methods typically deteriorate in large populations due to poor mixing and increased computation time. Here, we describe a new approach that seeks to approximate the likelihood by exploiting the underlying structure of the epidemic model. Simulation study results show that this approach can be a serious competitor to data-augmented MCMC methods. Our approach can be applied to a wide variety of disease transmission models, and we provide examples with applications to the common cold, Ebola, and foot-and-mouth disease.

Key questions for modelling COVID-19 exit strategies.

Combinations of intense non-pharmaceutical interventions (lockdowns) were introduced worldwide to reduce SARS-CoV-2 transmission. Many governments have begun to implement exit strategies that relax restrictions while attempting to control the risk of a surge in cases. Mathematical modelling has played a central role in guiding interventions, but the challenge of designing optimal exit strategies in the face of ongoing transmission is unprecedented. Here, we report discussions from the Isaac Newton Institute 'Models for an exit strategy' workshop (11-15 May 2020). A diverse community of modellers who are providing evidence to governments worldwide were asked to identify the main questions that, if answered, would allow for more accurate predictions of the effects of different exit strategies. Based on these questions, we propose a roadmap to facilitate the development of reliable models to guide exit strategies. This roadmap requires a global collaborative effort from the scientific community and policymakers, and has three parts: (i) improve estimation of key epidemiological parameters; (ii) understand sources of heterogeneity in populations; and (iii) focus on requirements for data collection, particularly in low-to-middle-income countries. This will provide important information for planning exit strategies that balance socio-economic benefits with public health.

Modelling, Bayesian inference, and model assessment for nosocomial pathogens using whole-genome-sequence data.

Whole-genome sequencing of pathogens in outbreaks of infectious disease provides the potential to reconstruct transmission pathways and enhance the information contained in conventional epidemiological data. In recent years, there have been numerous new methods and models developed to exploit such high-resolution genetic data. However, corresponding methods for model assessment have been largely overlooked. In this article, we develop both new modelling methods and new model assessment methods, specifically by building on the work of Worby et al. Although the methods are generic in nature, we focus specifically on nosocomial pathogens and analyze a dataset collected during an outbreak of MRSA in a hospital setting.

Evaluating hospital infection control measures for antimicrobial-resistant pathogens using stochastic transmission models: Application to vancomycin-resistant enterococci in intensive care units.

Nosocomial pathogens such as methicillin-resistant Staphylococcus aureus (MRSA) and vancomycin-resistant Enterococci (VRE) are the cause of significant morbidity and mortality among hospital patients. It is important to be able to assess the efficacy of control measures using data on patient outcomes. In this paper, we describe methods for analysing such data using patient-level stochastic models which seek to describe the underlying unobserved process of transmission. The methods are applied to detailed longitudinal patient-level data on vancomycin-resistant Enterococci from a study in a US hospital with eight intensive care units (ICUs). The data comprise admission and discharge dates, dates and results of screening tests, and dates during which precautionary measures were in place for each patient during the study period. Results include estimates of the efficacy of the control measures, the proportion of unobserved patients colonized with vancomycin-resistant Enterococci, and the proportion of patients colonized on admission.

Modelling and Bayesian analysis of the Abakaliki smallpox data.

The celebrated Abakaliki smallpox data have appeared numerous times in the epidemic modelling literature, but in almost all cases only a specific subset of the data is considered. The only previous analysis of the full data set relied on approximation methods to derive a likelihood and did not assess model adequacy. The data themselves continue to be of interest due to concerns about the possible re-emergence of smallpox as a bioterrorism weapon. We present the first full Bayesian statistical analysis using data-augmentation Markov chain Monte Carlo methods which avoid the need for likelihood approximations and which yield a wider range of results than previous analyses. We also carry out model assessment using simulation-based methods. Our findings suggest that the outbreak was largely driven by the interaction structure of the population, and that the introduction of control measures was not the sole reason for the end of the epidemic. We also obtain quantitative estimates of key quantities including reproduction numbers.

Reconstructing transmission trees for communicable diseases using densely sampled genetic data.

Whole genome sequencing of pathogens from multiple hosts in an epidemic offers the potential to investigate who infected whom with unparalleled resolution, potentially yielding important insights into disease dynamics and the impact of control measures. We considered disease outbreaks in a setting with dense genomic sampling, and formulated stochastic epidemic models to investigate person-to-person transmission, based on observed genomic and epidemiological data. We constructed models in which the genetic distance between sampled genotypes depends on the epidemiological relationship between the hosts. A data augmented Markov chain Monte Carlo algorithm was used to sample over the transmission trees, providing a posterior probability for any given transmission route. We investigated the predictive performance of our methodology using simulated data, demonstrating high sensitivity and specificity, particularly for rapidly mutating pathogens with low transmissibility. We then analyzed data collected during an outbreak of methicillin-resistant Staphylococcus aureus in a hospital, identifying probable transmission routes and estimating epidemiological parameters. Our approach overcomes limitations of previous methods, providing a framework with the flexibility to allow for unobserved infection times, multiple independent introductions of the pathogen, and within-host genetic diversity, as well as allowing forward simulation.

Bayesian non-parametric inference for stochastic epidemic models using Gaussian Processes.

This paper considers novel Bayesian non-parametric methods for stochastic epidemic models. Many standard modeling and data analysis methods use underlying assumptions (e.g. concerning the rate at which new cases of disease will occur) which are rarely challenged or tested in practice. To relax these assumptions, we develop a Bayesian non-parametric approach using Gaussian Processes, specifically to estimate the infection process. The methods are illustrated with both simulated and real data sets, the former illustrating that the methods can recover the true infection process quite well in practice, and the latter illustrating that the methods can be successfully applied in different settings.

Stochastic epidemic models featuring contact tracing with delays.

This paper is concerned with a stochastic model for the spread of an SEIR (susceptible → exposed (=latent) → infective → removed) epidemic with a contact tracing scheme, in which removed individuals may name some of their infectious contacts, who are then removed if they have not been already after some tracing delay. The epidemic is analysed via an approximating, modified birth-death process, for which a type-reproduction number is derived in terms of unnamed individuals, that is shown to be infinite when the contact rate is sufficiently large. We obtain explicit results under the assumption of either constant or exponentially distributed infectious periods, including the epidemic extinction probability in the former case. Numerical illustrations show that, while the distributions of latent periods and delays have an effect on the spread of the epidemic, the assumption of whether the delays experienced by individuals infected by the same individual are of the same or independent length makes little difference.

Bayesian model choice for epidemic models with two levels of mixing.

This paper considers the problem of choosing between competing models for infectious disease final outcome data in a population that is partitioned into households. The epidemic models are stochastic individual-based transmission models of the susceptible-infective-removed type. The main focus is on various algorithms for the estimation of Bayes factors, of which a path sampling-based algorithm is seen to give the best results. We also explore theoretical properties in the case where the within-model prior distributions become increasingly uninformative, which show the need for caution when using Bayes factors as a model choice tool. A suitable form of deviance information criterion is also considered for comparison. The theory and methods are illustrated with both artificial data, and influenza data from the Tecumseh study of illness.

Estimation of vaccine efficacy and critical vaccination coverage in partially observed outbreaks.

Classical approaches to estimate vaccine efficacy are based on the assumption that a person's risk of infection does not depend on the infection status of others. This assumption is untenable for infectious disease data where such dependencies abound. We present a novel approach to estimating vaccine efficacy in a Bayesian framework using disease transmission models. The methodology is applied to outbreaks of mumps in primary schools in the Netherlands. The total study population consisted of 2,493 children in ten primary schools, of which 510 (20%) were known to have been infected, and 832 (33%) had unknown infection status. The apparent vaccination coverage ranged from 12% to 93%, and the apparent infection attack rate varied from 1% to 76%. Our analyses show that vaccination reduces the probability of infection per contact substantially but not perfectly ([Formula: see text] = 0.933; 95CrI: 0.908-0.954). Mumps virus appears to be moderately transmissible in the school setting, with each case yielding an estimated 2.5 secondary cases in an unvaccinated population ([Formula: see text] = 2.49; 95%CrI: 2.36-2.63), resulting in moderate estimates of the critical vaccination coverage (64.2%; 95%CrI: 61.7-66.7%). The indirect benefits of vaccination are highest in populations with vaccination coverage just below the critical vaccination coverage. In these populations, it is estimated that almost two infections can be prevented per vaccination. We discuss the implications for the optimal control of mumps in heterogeneously vaccinated populations.

Estimating the effectiveness of isolation and decolonization measures in reducing transmission of methicillin-resistant Staphylococcus aureus in hospital general wards.

Infection control for hospital pathogens such as methicillin-resistant Staphylococcus aureus (MRSA) often takes the form of a package of interventions, including the use of patient isolation and decolonization treatment. Such interventions, though widely used, have generated controversy because of their significant resource implications and the lack of robust evidence with regard to their effectiveness at reducing transmission. The aim of this study was to estimate the effectiveness of isolation and decolonization measures in reducing MRSA transmission in hospital general wards. Prospectively collected MRSA surveillance data from 10 general wards at Guy's and St. Thomas' hospitals, London, United Kingdom, in 2006-2007 were used, comprising 14,035 patient episodes. Data were analyzed with a Markov chain Monte Carlo algorithm to model transmission dynamics. The combined effect of isolation and decolonization was estimated to reduce transmission by 64% (95% confidence interval: 37, 79). Undetected MRSA-positive patients were estimated to be the source of 75% (95% confidence interval: 67, 86) of total transmission events. Isolation measures combined with decolonization treatment were strongly associated with a reduction in MRSA transmission in hospital general wards. These findings provide support for active methods of MRSA control, but further research is needed to determine the relative importance of isolation and decolonization in preventing transmission.

Introduction and snapshot review: relating infectious disease transmission models to data.

Disease transmission models are becoming increasingly important both to public health policy makers and to scientists across many disciplines. We review some of the key aspects of how and why such models are related to data from infectious disease outbreaks, and identify a number of future challenges in the field.

Estimation of measles vaccine efficacy and critical vaccination coverage in a highly vaccinated population.

Measles is a highly infectious disease that has been targeted for elimination from four WHO regions. Whether and under which conditions this goal is feasible is, however, uncertain since outbreaks have been documented in populations with high vaccination coverage (more than 90%). Here, we use the example of a large outbreak in a German public school to show how estimates of key epidemiological parameters such as the basic reproduction number (R(0)), vaccine efficacy (VE(S)) and critical vaccination coverage (p(c)) can be obtained from partially observed outbreaks in highly vaccinated populations. Our analyses rely on Bayesian methods of inference based on the final size distribution of outbreak size, and use data which are easily collected. For the German public school the analyses indicate that the basic reproduction number of measles is higher than previously thought (R(0) = 30.8, 95% credible interval: 23.6-40.4), that the vaccine is highly effective in preventing infection (VE(S) = 0.997, 95% credible interval: 0.993-0.999), and that a vaccination coverage in excess of 95 per cent may be necessary to achieve herd immunity (p(c) = 0.971, 95% credible interval: 0.961-0.978). We discuss the implications for measles elimination from highly vaccinated populations.

Assessing the role of undetected colonization and isolation precautions in reducing methicillin-resistant Staphylococcus aureus transmission in intensive care units.

BACKGROUND: Screening and isolation are central components of hospital methicillin-resistant Staphylococcus aureus (MRSA) control policies. Their prevention of patient-to-patient spread depends on minimizing undetected and unisolated MRSA-positive patient days. Estimating these MRSA-positive patient days and the reduction in transmission due to isolation presents a major methodological challenge, but is essential for assessing both the value of existing control policies and the potential benefit of new rapid MRSA detection technologies. Recent methodological developments have made it possible to estimate these quantities using routine surveillance data. METHODS: Colonization data from admission and weekly nares cultures were collected from eight single-bed adult intensive care units (ICUs) over 17 months. Detected MRSA-positive patients were isolated using single rooms and barrier precautions. Data were analyzed using stochastic transmission models and model fitting was performed within a Bayesian framework using a Markov chain Monte Carlo algorithm, imputing unobserved MRSA carriage events. RESULTS: Models estimated the mean percent of colonized-patient-days attributed to undetected carriers as 14.1% (95% CI (11.7, 16.5)) averaged across ICUs. The percent of colonized-patient-days attributed to patients awaiting results averaged 7.8% (6.2, 9.2). Overall, the ratio of estimated transmission rates from unisolated MRSA-positive patients and those under barrier precautions was 1.34 (0.45, 3.97), but varied widely across ICUs. CONCLUSIONS: Screening consistently detected >80% of colonized-patient-days. Estimates of the effectiveness of barrier precautions showed considerable uncertainty, but in all units except burns/general surgery and one cardiac surgery ICU, the best estimates were consistent with reductions in transmission associated with barrier precautions.

Bayesian inference for stochastic multitype epidemics in structured populations using sample data.

This paper is concerned with the development of new methods for Bayesian statistical inference for structured-population stochastic epidemic models, given data in the form of a sample from a population with known structure. Specifically, the data are assumed to consist of final outcome information, so that it is known whether or not each individual in the sample ever became a clinical case during the epidemic outbreak. The objective is to make inference for the infection rate parameters in the underlying model of disease transmission. The principal challenge is that the required likelihood of the data is intractable in all but the simplest cases. Demiris and O'Neill (2005b) used data augmentation methods involving a certain random graph in a Markov chain Monte Carlo setting to address this situation in the special case where the sample is the same as the entire population. Here, we take an approach relying on broadly similar principles, but for which the implementation details are markedly different. Specifically, to cover the general case of sample data, we use an alternative data augmentation scheme and employ noncentering methods. The methods are illustrated using data from an influenza outbreak.

Control of emerging infectious diseases using responsive imperfect vaccination and isolation.

This paper is concerned with a stochastic model for the spread of an SEIR (susceptible --> exposed (= latent) --> infective --> removed) epidemic among a population partitioned into households, featuring different rates of infection for within and between households. The model incorporates responsive vaccination and isolation policies, based upon the appearance of diagnosed cases in households. Different models for imperfect vaccine response are considered. A threshold parameter R*, which determines whether or not a major epidemic can occur, and the probability of a major epidemic are obtained for different infectious and latent period distributions. Simpler expressions for these quantities are obtained in the limiting case of infinite within-household infection rate. Numerical studies suggest that the choice of infectious period distribution and whether or not latent individuals are vaccine-sensitive have a material influence on the spread of the epidemic, while, for given vaccine efficacy, the choice of vaccine action model is less influential. They also suggest that an effective isolation policy has a more significant impact than vaccination. The results show that R* alone is not sufficient to summarise the potential for an epidemic.

Estimating vaccine effects from studies of outbreaks in household pairs.

The traditional way to measure efficacy of a vaccine, with respect to reduced susceptibility and reduced infectivity once infected, is to look at relative attack rates. Although straightforward to apply, such measures do not take disease transmission into account, with the consequence that they can depend strongly on the community setting, the duration of the study period, the way participants are recruited into the study and the virulence of the infection. Sometimes they give a very misleading assessment of the vaccine, as we illustrate by examples. Here measures of vaccine efficacy are considered that avoid these defects, and estimation procedures are presented for studies based on outbreaks in household pairs. Such studies enable estimation of vaccine effects on susceptibility, infectivity and transmission. We propose that the vaccine efficacy measures be estimated, without making any assumptions about the nature of the vaccine response, by consistent estimates of bounds for the measures.

Bayesian model choice and infection route modelling in an outbreak of Norovirus.

A data set from an outbreak of gastroenteritis in a school is analysed using a stochastic transmission model. The causative agent of the outbreak is believed to be a Norovirus, spread through person-to-person contact. Particular attention is given to the question of whether or not vomiting episodes enhance the spread of the virus via aerosol transmission. The methodology developed uses Bayesian model choice, implemented with reversible-jump Markov chain Monte Carlo methods. The methodology appears to be highly sensitive to assumptions made concerning the data, which provides some assurance that the conclusions are driven by observations rather than the underlying model and methodology.

Estimating vaccine effects on transmission of infection from household outbreak data.

This article is concerned with a method for making inferences about various measures of vaccine efficacy. These measures describe reductions in susceptibility and in the potential to transmit infection. The method uses data on household outbreaks; it is based on a model that allows for transmission of infection both from within a household and from the outside. The use of household data is motivated by the hope that these are informative about vaccine-induced reduction of the potential to transmit infection, as household outbreaks contain some information about the possible source of infection. For illustration, the method is applied to observed data on household outbreaks of smallpox. These data are of the form needed and the number of households is of a size that can be managed in a vaccine trial. It is found that vaccine effects, such as the mean reduction in susceptibility and the mean reduction in the potential to infect others, per infectious contact, can be estimated with precision. However, a more specific parameter reflecting the reduction in infectivity for individuals partially responding to vaccination is not estimated well in the application. An evaluation of the method using artificial data shows that this parameter can be estimated with greater precision when we have outbreak data on a large number of small households.