Non-exponential tolerance to infection in epidemic systems--modeling, inference, and assessment

A combined neural ODE-Bayesian optimization approach to resolve dynamics and estimate parameters for a modified SIR model with immune memory

We propose a novel hybrid approach that integrates Neural Ordinary Differential Equations (NODEs) with Bayesian optimization to address the dynamics and parameter estimation of a modified time-delay-type Susceptible-Infected-Removed (SIR) model incorporating immune memory. This approach leverages a neural network to produce continuous multi-wave infection profiles by learning from both data and the model. The time-delay component of the SIR model, expressed through a convolutional integral, results in an integro-differential equation. To resolve these dynamics, we extend the NODE framework, employing a Runge-Kutta solver, to handle the challenging convolution integral, enabling us to fit the data and learn the parameters and dynamics of the model. Additionally, through Bayesian optimization, we enhance prediction accuracy while focusing on long-term dynamics. Our model, applied to COVID-19 data from Mexico, South Africa, and South Korea, effectively learns critical time-dependent parameters and provides accurate short- and long-term predictions. This combined methodology allows for early prediction of infection peaks, offering significant lead time for public health responses.

Modelling digital and manual contact tracing for COVID-19. Are low uptakes and missed contacts deal-breakers?

Comprehensive testing schemes, followed by adequate contact tracing and isolation, represent the best public health interventions we can employ to reduce the impact of an ongoing epidemic when no or limited vaccine supplies are available and the implications of a full lockdown are to be avoided. However, the process of tracing can prove feckless for highly-contagious viruses such as SARS-CoV-2. The interview-based approaches often miss contacts and involve significant delays, while digital solutions can suffer from insufficient adoption rates or inadequate usage patterns. Here we present a novel way of modelling different contact tracing strategies, using a generalized multi-site mean-field model, which can naturally assess the impact of manual and digital approaches alike. Our methodology can readily be applied to any compartmental formulation, thus enabling the study of more complex pathogen dynamics. We use this technique to simulate a newly-defined epidemiological model, SEIR-T, and show that, given the right conditions, tracing in a COVID-19 epidemic can be effective even when digital uptakes are sub-optimal or interviewers miss a fair proportion of the contacts.

Model selection and parameter estimation for dynamic epidemic models via iterated filtering: application to rotavirus in Germany

Despite the wide application of dynamic models in infectious disease epidemiology, the particular modeling of variability in the different model components is often subjective rather than the result of a thorough model selection process. This is in part because inference for a stochastic transmission model can be difficult since the likelihood is often intractable due to partial observability. In this work, we address the question of adequate inclusion of variability by demonstrating a systematic approach for model selection and parameter inference for dynamic epidemic models. For this, we perform inference for six partially observed Markov process models, which assume the same underlying transmission dynamics, but differ with respect to the amount of variability they allow for. The inference framework for the stochastic transmission models is provided by iterated filtering methods, which are readily implemented in the R package pomp by King and others (2016, Statistical inference for partially observed Markov processes via the R package pomp. Journal of Statistical Software69, 1–43). We illustrate our approach on German rotavirus surveillance data from 2001 to 2008, discuss practical difficulties of the methods used and calculate a model based estimate for the basic reproduction number \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$R_0$\end{document} using these data.

Latent likelihood ratio tests for assessing spatial kernels in epidemic models

One of the most important issues in the critical assessment of spatio-temporal stochastic models for epidemics is the selection of the transmission kernel used to represent the relationship between infectious challenge and spatial separation of infected and susceptible hosts. As the design of control strategies is often based on an assessment of the distance over which transmission can realistically occur and estimation of this distance is very sensitive to the choice of kernel function, it is important that models used to inform control strategies can be scrutinised in the light of observation in order to elicit possible evidence against the selected kernel function. While a range of approaches to model criticism is in existence, the field remains one in which the need for further research is recognised. In this paper, building on earlier contributions by the authors, we introduce a new approach to assessing the validity of spatial kernels—the latent likelihood ratio tests—which use likelihood-based discrepancy variables that can be used to compare the fit of competing models, and compare the capacity of this approach to detect model mis-specification with that of tests based on the use of infection-link residuals. We demonstrate that the new approach can be used to formulate tests with greater power than infection-link residuals to detect kernel mis-specification particularly when the degree of mis-specification is modest. This new tests avoid the use of a fully Bayesian approach which may introduce undesirable complications related to computational complexity and prior sensitivity.

Trade-off between local transmission and long-range dispersal drives infectious disease outbreak size in spatially structured populations

Transmission of infectious diseases between immobile hosts (e.g., plants, farms) is strongly dependent on the spatial distribution of hosts and the distance-dependent probability of transmission. As the interplay between these factors is poorly understood, we use spatial process and transmission modelling to investigate how epidemic size is shaped by host clustering and spatial range of transmission. We find that for a given degree of clustering and individual-level infectivity, the probability that an epidemic occurs after an introduction is generally higher if transmission is predominantly local. However, local transmission also impedes transfer of the infection to new clusters. A consequence is that the total number of infections is maximal if the range of transmission is intermediate. In highly clustered populations, the infection dynamics is strongly determined by the probability of transmission between clusters of hosts, whereby local clusters act as multiplier of infection. We show that in such populations, a metapopulation model sometimes provides a good approximation of the total epidemic size, using probabilities of local extinction, the final size of infections in local clusters, and probabilities of cluster-to-cluster transmission. As a real-world example we analyse the case of avian influenza transmission between poultry farms in the Netherlands.

Transmission of infectious diseases between immobile hosts depends on the transmission characteristics of the infection and on the spatial distribution of hosts. Examples include infectious diseases of plants that are spread by wind or via vectors (e.g., Asiatic citrus canker spread between citrus trees), diseases that are transmitted between local host populations (e.g., sylvatic plague transmitted between rodents living in burrows), diseases of production animals that are spread between farms (e.g., avian influenza in poultry transmitted from farm to farm). We use spatial transmission modelling to investigate how the total number of infections over the course of an epidemic is determined by host clustering and spatial range of transmission. We find that for a given degree of clustering and infectivity of hosts, the number of infections is maximal if the spatial range of transmission is intermediate. In highly clustered populations we show that epidemic size can be approximated by a metapopulation model, illustrating that in such populations the transmission dynamics is dominated by transmission between clusters of hosts.

A stochastic model for MRSA transmission within a hospital ward incorporating environmental contamination

Methicillin-resistant Staphylococcus aureus (MRSA) transmission in hospital wards is associated with adverse outcomes for patients and increased costs for hospitals. The transmission process is inherently stochastic and the randomness emphasized by the small population sizes involved. As such, a stochastic model was proposed to describe the MRSA transmission process, taking into account the related contribution and modelling of the associated microbiological environmental contamination. The model was used to evaluate the performance of five common interventions and their combinations on six potential outcome measures of interest under two hypothetical disease burden settings. The model showed that the optimal intervention combination varied depending on the outcome measure and burden setting. In particular, it was found that certain outcomes only required a small subset of targeted interventions to control the outcome measure, while other outcomes still reported reduction in the outcome distribution with up to all five interventions included. This study describes a new stochastic model for MRSA transmission within a ward and highlights the use of the generalized Mann–Whitney statistic to compare the distribution of the outcome measures under different intervention combinations to assist in planning future interventions in hospital wards under different potential outcome measures and disease burden.

A Systematic Bayesian Integration of Epidemiological and Genetic Data

Genetic sequence data on pathogens have great potential to inform inference of their transmission dynamics ultimately leading to better disease control. Where genetic change and disease transmission occur on comparable timescales additional information can be inferred via the joint analysis of such genetic sequence data and epidemiological observations based on clinical symptoms and diagnostic tests. Although recently introduced approaches represent substantial progress, for computational reasons they approximate genuine joint inference of disease dynamics and genetic change in the pathogen population, capturing partially the joint epidemiological-evolutionary dynamics. Improved methods are needed to fully integrate such genetic data with epidemiological observations, for achieving a more robust inference of the transmission tree and other key epidemiological parameters such as latent periods. Here, building on current literature, a novel Bayesian framework is proposed that infers simultaneously and explicitly the transmission tree and unobserved transmitted pathogen sequences. Our framework facilitates the use of realistic likelihood functions and enables systematic and genuine joint inference of the epidemiological-evolutionary process from partially observed outbreaks. Using simulated data it is shown that this approach is able to infer accurately joint epidemiological-evolutionary dynamics, even when pathogen sequences and epidemiological data are incomplete, and when sequences are available for only a fraction of exposures. These results also characterise and quantify the value of incomplete and partial sequence data, which has important implications for sampling design, and demonstrate the abilities of the introduced method to identify multiple clusters within an outbreak. The framework is used to analyse an outbreak of foot-and-mouth disease in the UK, enhancing current understanding of its transmission dynamics and evolutionary process.

In the midst of increasingly available sequence data of pathogens, a key challenge is to better integrate these data with traditional epidemiological data, with the proximate goal of reliable prediction and the ultimate aim of effective management of disease outbreaks. Although substantial advances have been made for such an integration, and they have improved our understandings of many disease dynamics which are not available otherwise, current methods have relied on fast algorithms, rather than achieving a systematic integration and accurate inference of the joint epidemiological-evolutionary process. Building on methods in current literature, this paper describes a novel Bayesian approach for systematically integrating these two streams of data. We propose a computationally tractable Bayesian inferential algorithm which takes the full joint epidemiological-evolutionary process into account. Using this algorithm, we study systematically the value of genetic data, providing valuable insights into future sampling designs. The algorithm is subsequently applied to real-world dataset describing the spread of animal foot-and-mouth disease in the UK, demonstrating the importance of such a systematic integration achieved with our methodology.

Survival time of the susceptible-infected-susceptible infection process on a graph

The survival time T is the longest time that a virus, a meme, or a failure can propagate in a network. Using the hitting time of the absorbing state in an uniformized embedded Markov chain of the continuous-time susceptible-infected-susceptible (SIS) Markov process, we derive an exact expression for the average survival time E[T] of a virus in the complete graph K_{N} and the star graph K_{1,N-1}. By using the survival time, instead of the average fraction of infected nodes, we propose a new method to approximate the SIS epidemic threshold τ_{c} that, at least for K_{N} and K_{1,N-1}, correctly scales with the number of nodes N and that is superior to the epidemic threshold τ_{c}^{(1)}=1/λ_{1} of the N-intertwined mean-field approximation, where λ_{1} is the spectral radius of the adjacency matrix of the graph G. Although this new approximation of the epidemic threshold offers a more intuitive understanding of the SIS process, it remains difficult to compare outbreaks in different graph types. For example, the survival in an arbitrary graph seems upper bounded by the complete graph and lower bounded by the star graph as a function of the normalized effective infection rate τ/τ_{c}^{(1)}. However, when the average fraction of infected nodes is used as a basis for comparison, the virus will survive in the star graph longer than in any other graph, making the star graph the worst-case graph instead of the complete graph. Finally, in non-Markovian SIS, the distribution of the spreading attempts over the infectious period of a node influences the survival time, even if the expected number of spreading attempts during an infectious period (the non-Markovian equivalent of the effective infection rate) is kept constant. Both early and late infection attempts lead to shorter survival times. Interestingly, just as in Markovian SIS, the survival times appear to be exponentially distributed, regardless of the infection and curing time distributions.

New model diagnostics for spatio-temporal systems in epidemiology and ecology

A cardinal challenge in epidemiological and ecological modelling is to develop effective and easily deployed tools for model assessment. The availability of such methods would greatly improve understanding, prediction and management of disease and ecosystems. Conventional Bayesian model assessment tools such as Bayes factors and the deviance information criterion (DIC) are natural candidates but suffer from important limitations because of their sensitivity and complexity. Posterior predictive checks, which use summary statistics of the observed process simulated from competing models, can provide a measure of model fit but appropriate statistics can be difficult to identify. Here, we develop a novel approach for diagnosing mis-specifications of a general spatio-temporal transmission model by embedding classical ideas within a Bayesian analysis. Specifically, by proposing suitably designed non-centred parametrization schemes, we construct latent residuals whose sampling properties are known given the model specification and which can be used to measure overall fit and to elicit evidence of the nature of mis-specifications of spatial and temporal processes included in the model. This model assessment approach can readily be implemented as an addendum to standard estimation algorithms for sampling from the posterior distributions, for example Markov chain Monte Carlo. The proposed methodology is first tested using simulated data and subsequently applied to data describing the spread of Heracleum mantegazzianum (giant hogweed) across Great Britain over a 30-year period. The proposed methods are compared with alternative techniques including posterior predictive checking and the DIC. Results show that the proposed diagnostic tools are effective in assessing competing stochastic spatio-temporal transmission models and may offer improvements in power to detect model mis-specifications. Moreover, the latent-residual framework introduced here extends readily to a broad range of ecological and epidemiological models.

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.

Non-Markovian infection spread dramatically alters the susceptible-infected-susceptible epidemic threshold in networks

Most studies on susceptible-infected-susceptible epidemics in networks implicitly assume Markovian behavior: the time to infect a direct neighbor is exponentially distributed. Much effort so far has been devoted to characterize and precisely compute the epidemic threshold in susceptible-infected-susceptible Markovian epidemics on networks. Here, we report the rather dramatic effect of a nonexponential infection time (while still assuming an exponential curing time) on the epidemic threshold by considering Weibullean infection times with the same mean, but different power exponent α. For three basic classes of graphs, the Erdős-Rényi random graph, scale-free graphs and lattices, the average steady-state fraction of infected nodes is simulated from which the epidemic threshold is deduced. For all graph classes, the epidemic threshold significantly increases with the power exponents α. Hence, real epidemics that violate the exponential or Markovian assumption can behave seriously differently than anticipated based on Markov theory.