Outbreaks in care homes may lead to substantial disease burden if not mitigated.

The number of COVID-19 outbreaks reported in UK care homes rose rapidly in early March of 2020. Owing to the increased co-morbidities and therefore worse COVID-19 outcomes for care home residents, it is important that we understand this increase and its future implications. We demonstrate the use of an SIS model where each nursing home is an infective unit capable of either being susceptible to an outbreak (S) or in an active outbreak (I). We use a generalized additive model to approximate the trend in growth rate of outbreaks in care homes and find the fit to be improved in a model where the growth rate is proportional to the number of current care home outbreaks compared with a model with a constant growth rate. Using parameters found from the outbreak-dependent growth rate, we predict a 73% prevalence of outbreaks in UK care homes without intervention as a reasonable worst-case planning assumption. This article is part of the theme issue 'Modelling that shaped the early COVID-19 pandemic response in the UK'.

Scabies in residential care homes: Modelling, inference and interventions for well-connected population sub-units.

In the context of an ageing population, understanding the transmission of infectious diseases such as scabies through well-connected sub-units of the population, such as residential care homes, is particularly important for the design of efficient interventions to mitigate against the effects of those diseases. Here, we present a modelling methodology based on the efficient solution of a large-scale system of linear differential equations that allows statistical calibration of individual-based random models to real data on scabies in residential care homes. In particular, we review and benchmark different numerical methods for the integration of the differential equation system, and then select the most appropriate of these methods to perform inference using Markov chain Monte Carlo. We test the goodness-of-fit of this model using posterior predictive intervals and propagate forward the resulting parameter uncertainty in a Bayesian framework to consider the economic cost of delayed interventions against scabies, quantifying the benefits of prompt action in the event of detection. We also revisit the previous methodology used to assess the safety of treatments in small population sub-units-in this context ivermectin-and demonstrate that even a very slight relaxation of the implicit assumption of homogeneous death rates significantly increases the plausibility of the hypothesis that ivermectin does not cause excess mortality based upon the data of Barkwell and Shields.

Gaussian process approximations for fast inference from infectious disease data.

We present a flexible framework for deriving and quantifying the accuracy of Gaussian process approximations to non-linear stochastic individual-based models of epidemics. We develop this for the SIR and SEIR models, and we show how it can be used to perform quick maximum likelihood inference for the underlying parameters given population estimates of the number of infecteds or cases at given time points. We also show how the unobserved processes can be inferred at the same time as the underlying parameters.

Extinction times in the subcritical stochastic SIS logistic epidemic.

Many real epidemics of an infectious disease are not straightforwardly super- or sub-critical, and the understanding of epidemic models that exhibit such complexity has been identified as a priority for theoretical work. We provide insights into the near-critical regime by considering the stochastic SIS logistic epidemic, a well-known birth-and-death chain used to model the spread of an epidemic within a population of a given size N. We study the behaviour of the process as the population size N tends to infinity. Our results cover the entire subcritical regime, including the "barely subcritical" regime, where the recovery rate exceeds the infection rate by an amount that tends to 0 as [Formula: see text] but more slowly than [Formula: see text]. We derive precise asymptotics for the distribution of the extinction time and the total number of cases throughout the subcritical regime, give a detailed description of the course of the epidemic, and compare to numerical results for a range of parameter values. We hypothesise that features of the course of the epidemic will be seen in a wide class of other epidemic models, and we use real data to provide some tentative and preliminary support for this theory.

Heterogeneous network epidemics: real-time growth, variance and extinction of infection.

Recent years have seen a large amount of interest in epidemics on networks as a way of representing the complex structure of contacts capable of spreading infections through the modern human population. The configuration model is a popular choice in theoretical studies since it combines the ability to specify the distribution of the number of contacts (degree) with analytical tractability. Here we consider the early real-time behaviour of the Markovian SIR epidemic model on a configuration model network using a multitype branching process. We find closed-form analytic expressions for the mean and variance of the number of infectious individuals as a function of time and the degree of the initially infected individual(s), and write down a system of differential equations for the probability of extinction by time t that are numerically fast compared to Monte Carlo simulation. We show that these quantities are all sensitive to the degree distribution-in particular we confirm that the mean prevalence of infection depends on the first two moments of the degree distribution and the variance in prevalence depends on the first three moments of the degree distribution. In contrast to most existing analytic approaches, the accuracy of these results does not depend on having a large number of infectious individuals, meaning that in the large population limit they would be asymptotically exact even for one initial infectious individual.

Exact and approximate moment closures for non-Markovian network epidemics.

Moment-closure techniques are commonly used to generate low-dimensional deterministic models to approximate the average dynamics of stochastic systems on networks. The quality of such closures is usually difficult to asses and furthermore the relationship between model assumptions and closure accuracy are often difficult, if not impossible, to quantify. Here we carefully examine some commonly used moment closures, in particular a new one based on the concept of maximum entropy, for approximating the spread of epidemics on networks by reconstructing the probability distributions over triplets based on those over pairs. We consider various models (SI, SIR, SEIR and Reed-Frost-type) under Markovian and non-Markovian assumption characterising the latent and infectious periods. We initially study with care two special networks, namely the open triplet and closed triangle, for which we can obtain analytical results. We then explore numerically the exactness of moment closures for a wide range of larger motifs, thus gaining understanding of the factors that introduce errors in the approximations, in particular the presence of a random duration of the infectious period and the presence of overlapping triangles in a network. We also derive a simpler and more intuitive proof than previously available concerning the known result that pair-based moment closure is exact for the Markovian SIR model on tree-like networks under pure initial conditions. We also extend such a result to all infectious models, Markovian and non-Markovian, in which susceptibles escape infection independently from each infected neighbour and for which infectives cannot regain susceptible status, provided the network is tree-like and initial conditions are pure. This works represent a valuable step in enriching intuition and deepening understanding of the assumptions behind moment closure approximations and for putting them on a more rigorous mathematical footing.

For principled model fitting in mathematical biology.

I argue for a principled approach to model fitting in mathematical biology that combines statistical and mechanistic insights.

Dynamics of stochastic epidemics on heterogeneous networks.

Epidemic models currently play a central role in our attempts to understand and control infectious diseases. Here, we derive a model for the diffusion limit of stochastic susceptible-infectious-removed (SIR) epidemic dynamics on a heterogeneous network. Using this, we consider analytically the early asymptotic exponential growth phase of such epidemics, showing how the higher order moments of the network degree distribution enter into the stochastic behaviour of the epidemic. We find that the first three moments of the network degree distribution are needed to specify the variance in disease prevalence fully, meaning that the skewness of the degree distribution affects the variance of the prevalence of infection. We compare these asymptotic results to simulation and find a close agreement for city-sized populations.

From Markovian to pairwise epidemic models and the performance of moment closure approximations.

Many if not all models of disease transmission on networks can be linked to the exact state-based Markovian formulation. However the large number of equations for any system of realistic size limits their applicability to small populations. As a result, most modelling work relies on simulation and pairwise models. In this paper, for a simple SIS dynamics on an arbitrary network, we formalise the link between a well known pairwise model and the exact Markovian formulation. This involves the rigorous derivation of the exact ODE model at the level of pairs in terms of the expected number of pairs and triples. The exact system is then closed using two different closures, one well established and one that has been recently proposed. A new interpretation of both closures is presented, which explains several of their previously observed properties. The closed dynamical systems are solved numerically and the results are compared to output from individual-based stochastic simulations. This is done for a range of networks with the same average degree and clustering coefficient but generated using different algorithms. It is shown that the ability of the pairwise system to accurately model an epidemic is fundamentally dependent on the underlying large-scale network structure. We show that the existing pairwise models are a good fit for certain types of network but have to be used with caution as higher-order network structures may compromise their effectiveness.

Modelling the impact of local reactive school closures on critical care provision during an influenza pandemic.

Despite the fact that the 2009 H1N1 pandemic influenza strain was less severe than had been feared, both seasonal epidemics of influenza-like-illness and future influenza pandemics have the potential to place a serious burden on health services. The closure of schools has been postulated as a means of reducing transmission between children and hence reducing the number of cases at the peak of an epidemic; this is supported by the marked reduction in cases during school holidays observed across the world during the 2009 pandemic. However, a national policy of long-duration school closures could have severe economic costs. Reactive short-duration closure of schools in regions where health services are close to capacity offers a potential compromise, but it is unclear over what spatial scale and time frame closures would need to be made to be effective. Here, using detailed geographical information for England, we assess how localized school closures could alleviate the burden on hospital intensive care units (ICUs) that are reaching capacity. We show that, for a range of epidemiologically plausible assumptions, considerable local coordination of school closures is needed to achieve a substantial reduction in the number of hospitals where capacity is exceeded at the peak of the epidemic. The heterogeneity in demand per hospital ICU bed means that even widespread school closures are unlikely to have an impact on whether demand will exceed capacity for many hospitals. These results support the UK decision not to use localized school closures as a control mechanism, but have far wider international public-health implications. The spatial heterogeneities in both population density and hospital capacity that give rise to our results exist in many developed countries, while our model assumptions are sufficiently general to cover a wide range of pathogens. This leads us to believe that when a pandemic has severe implications for ICU capacity, only widespread school closures (with their associated costs and organizational challenges) are sufficient to mitigate the burden on the worst-affected hospitals.

Calculation of disease dynamics in a population of households.

Early mathematical representations of infectious disease dynamics assumed a single, large, homogeneously mixing population. Over the past decade there has been growing interest in models consisting of multiple smaller subpopulations (households, workplaces, schools, communities), with the natural assumption of strong homogeneous mixing within each subpopulation, and weaker transmission between subpopulations. Here we consider a model of SIRS (susceptible-infectious-recovered-susceptible) infection dynamics in a very large (assumed infinite) population of households, with the simplifying assumption that each household is of the same size (although all methods may be extended to a population with a heterogeneous distribution of household sizes). For this households model we present efficient methods for studying several quantities of epidemiological interest: (i) the threshold for invasion; (ii) the early growth rate; (iii) the household offspring distribution; (iv) the endemic prevalence of infection; and (v) the transient dynamics of the process. We utilize these methods to explore a wide region of parameter space appropriate for human infectious diseases. We then extend these results to consider the effects of more realistic gamma-distributed infectious periods. We discuss how all these results differ from standard homogeneous-mixing models and assess the implications for the invasion, transmission and persistence of infection. The computational efficiency of the methodology presented here will hopefully aid in the parameterisation of structured models and in the evaluation of appropriate responses for future disease outbreaks.

A motif-based approach to network epidemics.

Networks have become an indispensable tool in modelling infectious diseases, with the structure of epidemiologically relevant contacts known to affect both the dynamics of the infection process and the efficacy of intervention strategies. One of the key reasons for this is the presence of clustering in contact networks, which is typically analysed in terms of prevalence of triangles in the network. We present a more general approach, based on the prevalence of different four-motifs, in the context of ODE approximations to network dynamics. This is shown to outperform existing models for a range of small world networks.