Using model-based proposals for fast parameter inference on discrete state space, continuous-time Markov processes

Semi-parametric modeling of SARS-CoV-2 transmission using tests, cases, deaths, and seroprevalence data

Mechanistic models fit to streaming surveillance data are critical to understanding the transmission dynamics of an outbreak as it unfolds in real-time. However, transmission model parameter estimation can be imprecise, and sometimes even impossible, because surveillance data are noisy and not informative about all aspects of the mechanistic model. To partially overcome this obstacle, Bayesian models have been proposed to integrate multiple surveillance data streams. We devised a modeling framework for integrating SARS-CoV-2 diagnostics test and mortality time series data, as well as seroprevalence data from cross-sectional studies, and tested the importance of individual data streams for both inference and forecasting. Importantly, our model for incidence data accounts for changes in the total number of tests performed. We model the transmission rate, infection-to-fatality ratio, and a parameter controlling a functional relationship between the true case incidence and the fraction of positive tests as time-varying quantities and estimate changes of these parameters nonparametrically. We compare our base model against modified versions which do not use diagnostics test counts or seroprevalence data to demonstrate the utility of including these often unused data streams. We apply our Bayesian data integration method to COVID-19 surveillance data collected in Orange County, California between March 2020 and February 2021 and find that 32-72% of the Orange County residents experienced SARS-CoV-2 infection by mid-January, 2021. Despite this high number of infections, our results suggest that the abrupt end of the winter surge in January 2021 was due to both behavioral changes and a high level of accumulated natural immunity.

A linear noise approximation for stochastic epidemic models fit to partially observed incidence counts

Stochastic epidemic models (SEMs) fit to incidence data are critical to elucidating outbreak dynamics, shaping response strategies, and preparing for future epidemics. SEMs typically represent counts of individuals in discrete infection states using Markov jump processes (MJPs), but are computationally challenging as imperfect surveillance, lack of subject-level information, and temporal coarseness of the data obscure the true epidemic. Analytic integration over the latent epidemic process is impossible, and integration via Markov chain Monte Carlo (MCMC) is cumbersome due to the dimensionality and discreteness of the latent state space. Simulation-based computational approaches can address the intractability of the MJP likelihood, but are numerically fragile and prohibitively expensive for complex models. A linear noise approximation (LNA) that approximates the MJP transition density with a Gaussian density has been explored for analyzing prevalence data in large-population settings, but requires modification for analyzing incidence counts without assuming that the data are normally distributed. We demonstrate how to reparameterize SEMs to appropriately analyze incidence data, and fold the LNA into a data augmentation MCMC framework that outperforms deterministic methods, statistically, and simulation-based methods, computationally. Our framework is computationally robust when the model dynamics are complex and applies to a broad class of SEMs. We evaluate our method in simulations that reflect Ebola, influenza, and SARS-CoV-2 dynamics, and apply our method to national surveillance counts from the 2013-2015 West Africa Ebola outbreak.

Estimation of age-stratified contact rates during the COVID-19 pandemic using a novel inference algorithm

Well parameterized epidemiological models including accurate representation of contacts are fundamental to controlling epidemics. However, age-stratified contacts are typically estimated from pre-pandemic/peace-time surveys, even though interventions and public response likely alter contacts. Here, we fit age-stratified models, including re-estimation of relative contact rates between age classes, to public data describing the 2020-2021 COVID-19 outbreak in England. This data includes age-stratified population size, cases, deaths, hospital admissions and results from the Coronavirus Infection Survey (almost 9000 observations in all). Fitting stochastic compartmental models to such detailed data is extremely challenging, especially considering the large number of model parameters being estimated (over 150). An efficient new inference algorithm ABC-MBP combining existing approximate Bayesian computation (ABC) methodology with model-based proposals (MBPs) is applied. Modified contact rates are inferred alongside time-varying reproduction numbers that quantify changes in overall transmission due to pandemic response, and age-stratified proportions of asymptomatic cases, hospitalization rates and deaths. These inferences are robust to a range of assumptions including the values of parameters that cannot be estimated from available data. ABC-MBP is shown to enable reliable joint analysis of complex epidemiological data yielding consistent parametrization of dynamic transmission models that can inform data-driven public health policy and interventions. This article is part of the theme issue 'Technical challenges of modelling real-life epidemics and examples of overcoming these'.

Challenges in estimation, uncertainty quantification and elicitation for pandemic modelling

The estimation of parameters and model structure for informing infectious disease response has become a focal point of the recent pandemic. However, it has also highlighted a plethora of challenges remaining in the fast and robust extraction of information using data and models to help inform policy. In this paper, we identify and discuss four broad challenges in the estimation paradigm relating to infectious disease modelling, namely the Uncertainty Quantification framework, data challenges in estimation, model-based inference and prediction, and expert judgement. We also postulate priorities in estimation methodology to facilitate preparation for future pandemics.

Stochastic Epidemic Models inference and diagnosis with Poisson Random Measure Data Augmentation

We present a new Bayesian inference method for compartmental models that takes into account the intrinsic stochasticity of the process. We show how to formulate a SIR-type Markov jump process as the solution of a stochastic differential equation with respect to a Poisson Random Measure (PRM), and how to simulate the process trajectory deterministically from a parameter value and a PRM realization. This forms the basis of our Data Augmented MCMC, which consists of augmenting parameter space with the unobserved PRM value. The resulting simple Metropolis-Hastings sampler acts as an efficient simulation-based inference method, that can easily be transferred from model to model. Compared with a recent Data Augmentation method based on Gibbs sampling of individual infection histories, PRM-augmented MCMC scales much better with epidemic size and is far more flexible. It is also found to be competitive with Particle MCMC for moderate epidemics when using approximate simulations. PRM-augmented MCMC also yields a posteriori estimates of the PRM, that represent process stochasticity, and which can be used to validate the model. A pattern of deviation from the PRM prior distribution will indicate that the model underfits the data and help to understand the cause. We illustrate this by fitting a non-seasonal model to some simulated seasonal case count data. Applied to the Zika epidemic of 2013 in French Polynesia, our approach shows that a simple SEIR model cannot correctly reproduce both the initial sharp increase in the number of cases as well as the final proportion of seropositive. PRM augmentation thus provides a coherent story for Stochastic Epidemic Model inference, where explicitly inferring process stochasticity helps with model validation.

Identifiability analysis for stochastic differential equation models in systems biology

Mathematical models are routinely calibrated to experimental data, with goals ranging from building predictive models to quantifying parameters that cannot be measured. Whether or not reliable parameter estimates are obtainable from the available data can easily be overlooked. Such issues of parameter identifiability have important ramifications for both the predictive power of a model, and the mechanistic insight that can be obtained. Identifiability analysis is well-established for deterministic, ordinary differential equation (ODE) models, but there are no commonly adopted methods for analysing identifiability in stochastic models. We provide an accessible introduction to identifiability analysis and demonstrate how existing ideas for analysis of ODE models can be applied to stochastic differential equation (SDE) models through four practical case studies. To assess structural identifiability, we study ODEs that describe the statistical moments of the stochastic process using open-source software tools. Using practically motivated synthetic data and Markov chain Monte Carlo methods, we assess parameter identifiability in the context of available data. Our analysis shows that SDE models can often extract more information about parameters than deterministic descriptions. All code used to perform the analysis is available on Github.

A practical guide to pseudo-marginal methods for computational inference in systems biology

For many stochastic models of interest in systems biology, such as those describing biochemical reaction networks, exact quantification of parameter uncertainty through statistical inference is intractable. Likelihood-free computational inference techniques enable parameter inference when the likelihood function for the model is intractable but the generation of many sample paths is feasible through stochastic simulation of the forward problem. The most common likelihood-free method in systems biology is approximate Bayesian computation that accepts parameters that result in low discrepancy between stochastic simulations and measured data. However, it can be difficult to assess how the accuracy of the resulting inferences are affected by the choice of acceptance threshold and discrepancy function. The pseudo-marginal approach is an alternative likelihood-free inference method that utilises a Monte Carlo estimate of the likelihood function. This approach has several advantages, particularly in the context of noisy, partially observed, time-course data typical in biochemical reaction network studies. Specifically, the pseudo-marginal approach facilitates exact inference and uncertainty quantification, and may be efficiently combined with particle filters for low variance, high-accuracy likelihood estimation. In this review, we provide a practical introduction to the pseudo-marginal approach using inference for biochemical reaction networks as a series of case studies. Implementations of key algorithms and examples are provided using the Julia programming language; a high performance, open source programming language for scientific computing (https://github.com/davidwarne/Warne2019_GuideToPseudoMarginal).

Simulation and inference algorithms for stochastic biochemical reaction networks: from basic concepts to state-of-the-art

Stochasticity is a key characteristic of intracellular processes such as gene regulation and chemical signalling. Therefore, characterizing stochastic effects in biochemical systems is essential to understand the complex dynamics of living things. Mathematical idealizations of biochemically reacting systems must be able to capture stochastic phenomena. While robust theory exists to describe such stochastic models, the computational challenges in exploring these models can be a significant burden in practice since realistic models are analytically intractable. Determining the expected behaviour and variability of a stochastic biochemical reaction network requires many probabilistic simulations of its evolution. Using a biochemical reaction network model to assist in the interpretation of time-course data from a biological experiment is an even greater challenge due to the intractability of the likelihood function for determining observation probabilities. These computational challenges have been subjects of active research for over four decades. In this review, we present an accessible discussion of the major historical developments and state-of-the-art computational techniques relevant to simulation and inference problems for stochastic biochemical reaction network models. Detailed algorithms for particularly important methods are described and complemented with Matlab^® implementations. As a result, this review provides a practical and accessible introduction to computational methods for stochastic models within the life sciences community.

Posterior-based proposals for speeding up Markov chain Monte Carlo

Markov chain Monte Carlo (MCMC) is widely used for Bayesian inference in models of complex systems. Performance, however, is often unsatisfactory in models with many latent variables due to so-called poor mixing, necessitating the development of application-specific implementations. This paper introduces 'posterior-based proposals' (PBPs), a new type of MCMC update applicable to a huge class of statistical models (whose conditional dependence structures are represented by directed acyclic graphs). PBPs generate large joint updates in parameter and latent variable space, while retaining good acceptance rates (typically 33%). Evaluation against other approaches (from standard Gibbs/random walk updates to state-of-the-art Hamiltonian and particle MCMC methods) was carried out for widely varying model types: an individual-based model for disease diagnostic test data, a financial stochastic volatility model, a mixed model used in statistical genetics and a population model used in ecology. While different methods worked better or worse in different scenarios, PBPs were found to be either near to the fastest or significantly faster than the next best approach (by up to a factor of 10). PBPs, therefore, represent an additional general purpose technique that can be usefully applied in a wide variety of contexts.

Bayesian model discrimination for partially-observed epidemic models

An efficient method for Bayesian model selection is presented for a broad class of continuous-time Markov chain models and is subsequently applied to two important problems in epidemiology. The first problem is to identify the shape of the infectious period distribution; the second problem is to determine whether individuals display symptoms before, at the same time, or after they become infectious. In both cases we show that the correct model can be identified, in the majority of cases, from symptom onset data generated from multiple outbreaks in small populations. The method works by evaluating the likelihood using a particle filter that incorporates a novel importance sampling algorithm designed for partially-observed continuous-time Markov chains. This is combined with another importance sampling method to unbiasedly estimate the model evidence. These come with estimates of precision, which allow for stopping criterion to be employed. Our method is general and can be applied to a wide range of model selection problems in biological and epidemiological systems with intractable likelihood functions.

Inference of epidemiological parameters from household stratified data

We consider a continuous-time Markov chain model of SIR disease dynamics with two levels of mixing. For this so-called stochastic households model, we provide two methods for inferring the model parameters-governing within-household transmission, recovery, and between-household transmission-from data of the day upon which each individual became infectious and the household in which each infection occurred, as might be available from First Few Hundred studies. Each method is a form of Bayesian Markov Chain Monte Carlo that allows us to calculate a joint posterior distribution for all parameters and hence the household reproduction number and the early growth rate of the epidemic. The first method performs exact Bayesian inference using a standard data-augmentation approach; the second performs approximate Bayesian inference based on a likelihood approximation derived from branching processes. These methods are compared for computational efficiency and posteriors from each are compared. The branching process is shown to be a good approximation and remains computationally efficient as the amount of data is increased.

Efficient Data Augmentation for Fitting Stochastic Epidemic Models to Prevalence Data

Stochastic epidemic models describe the dynamics of an epidemic as a disease spreads through a population. Typically, only a fraction of cases are observed at a set of discrete times. The absence of complete information about the time evolution of an epidemic gives rise to a complicated latent variable problem in which the state space size of the epidemic grows large as the population size increases. This makes analytically integrating over the missing data infeasible for populations of even moderate size. We present a data augmentation Markov chain Monte Carlo (MCMC) framework for Bayesian estimation of stochastic epidemic model parameters, in which measurements are augmented with subject-level disease histories. In our MCMC algorithm, we propose each new subject-level path, conditional on the data, using a time-inhomogeneous continuous-time Markov process with rates determined by the infection histories of other individuals. The method is general, and may be applied to a broad class of epidemic models with only minimal modifications to the model dynamics and/or emission distribution. We present our algorithm in the context of multiple stochastic epidemic models in which the data are binomially sampled prevalence counts, and apply our method to data from an outbreak of influenza in a British boarding school.