Deterministic epidemic models overestimate the basic reproduction number of observed outbreaks.

The basic reproduction number, R0, is a well-known quantifier of epidemic spread. However, a class of existing methods for estimating R0 from incidence data early in the epidemic can lead to an over-estimation of this quantity. In particular, when fitting deterministic models to estimate the rate of spread, we do not account for the stochastic nature of epidemics and that, given the same system, some outbreaks may lead to epidemics and some may not. Typically, an observed epidemic that we wish to control is a major outbreak. This amounts to implicit selection for major outbreaks which leads to the over-estimation problem. We formally characterised the split between major and minor outbreaks by using Otsu's method which provides us with a working definition. We show that by conditioning a 'deterministic' model on major outbreaks, we can more reliably estimate the basic reproduction number from an observed epidemic trajectory.

Eco-evolutionary dynamics in finite network-structured populations with migration.

We consider the effect of network structure on the evolution of a population. Models of this kind typically consider a population of fixed size and distribution. Here we consider eco-evolutionary dynamics where population size and distribution can change through birth, death and migration, all of which are separate processes. This allows complex interaction and migration behaviours that are dependent on competition. For migration, we assume that the response of individuals to competition is governed by tolerance to their group members, such that less tolerant individuals are more likely to move away due to competition. We look at the success of a mutant in the rare mutation limit for the complete, cycle and star networks. Unlike models with fixed population size and distribution, the distribution of the individuals per site is explicitly modelled by considering the dynamics of the population. This in turn determines the mutant appearance distribution for each network. Where a mutant appears impacts its success as it determines the competition it faces. For low and high migration rates the complete and cycle networks have similar mutant appearance distributions resulting in similar success levels for an invading mutant. A higher migration rate in the star network is detrimental for mutant success because migration results in a crowded central site where a mutant is more likely to appear.

Aortic stenosis post-COVID-19: a mathematical model on waiting lists and mortality.

OBJECTIVES: To provide estimates for how different treatment pathways for the management of severe aortic stenosis (AS) may affect National Health Service (NHS) England waiting list duration and associated mortality. DESIGN: We constructed a mathematical model of the excess waiting list and found the closed-form analytic solution to that model. From published data, we calculated estimates for how the strategies listed under Interventions may affect the time to clear the backlog of patients waiting for treatment and the associated waiting list mortality. SETTING: The NHS in England. PARTICIPANTS: Estimated patients with AS in England. INTERVENTIONS: (1) Increasing the capacity for the treatment of severe AS, (2) converting proportions of cases from surgery to transcatheter aortic valve implantation and (3) a combination of these two. RESULTS: In a capacitated system, clearing the backlog by returning to pre-COVID-19 capacity is not possible. A conversion rate of 50% would clear the backlog within 666 (533-848) days with 1419 (597-2189) deaths while waiting during this time. A 20% capacity increase would require 535 (434-666) days, with an associated mortality of 1172 (466-1859). A combination of converting 40% cases and increasing capacity by 20% would clear the backlog within a year (343 (281-410) days) with 784 (292-1324) deaths while awaiting treatment. CONCLUSION: A strategy change to the management of severe AS is required to reduce the NHS backlog and waiting list deaths during the post-COVID-19 'recovery' period. However, plausible adaptations will still incur a substantial wait to treatment and many hundreds dying while waiting.

Effectiveness of the BNT162b2 (Pfizer-BioNTech) and the ChAdOx1 nCoV-19 (Oxford-AstraZeneca) vaccines for reducing susceptibility to infection with the Delta variant (B.1.617.2) of SARS-CoV-2.

BACKGROUND: From January to May 2021 the alpha variant (B.1.1.7) of SARS-CoV-2 was the most commonly detected variant in the UK. Following this, the Delta variant (B.1.617.2) then became the predominant variant. The UK COVID-19 vaccination programme started on 8th December 2020. Prior to the Delta variant, most vaccine effectiveness studies focused on the alpha variant. We therefore aimed to estimate the effectiveness of the BNT162b2 (Pfizer-BioNTech) and the ChAdOx1 nCoV-19 (Oxford-AstraZeneca) vaccines in preventing symptomatic and asymptomatic infection with respect to the Delta variant in a UK setting. METHODS: We used anonymised public health record data linked to infection data (PCR) using the Combined Intelligence for Population Health Action resource. We then constructed an SIR epidemic model to explain SARS-CoV-2 infection data across the Cheshire and Merseyside region of the UK. Vaccines were assumed to be effective after 21 days for 1 dose and 14 days for 2 doses. RESULTS: We determined that the effectiveness of the Oxford-AstraZeneca vaccine in reducing susceptibility to infection is 39% (95% credible interval [34, 43]) and 64% (95% credible interval [61, 67]) for a single dose and a double dose respectively. For the Pfizer-BioNTech vaccine, the effectiveness is 20% (95% credible interval [10, 28]) and 84% (95% credible interval [82, 86]) for a single-dose and a double dose respectively. CONCLUSION: Vaccine effectiveness for reducing susceptibility to SARS-CoV-2 infection shows noticeable improvement after receiving two doses of either vaccine. Findings also suggest that a full course of the Pfizer-BioNTech provides the optimal protection against infection with the Delta variant. This reinforces the need to complete the full course programme to maximise individual protection and reduce transmission.

Approximating Quasi-Stationary Behaviour in Network-Based SIS Dynamics.

Deterministic approximations to stochastic Susceptible-Infectious-Susceptible models typically predict a stable endemic steady-state when above threshold. This can be hard to relate to the underlying stochastic dynamics, which has no endemic steady-state but can exhibit approximately stable behaviour. Here, we relate the approximate models to the stochastic dynamics via the definition of the quasi-stationary distribution (QSD), which captures this approximately stable behaviour. We develop a system of ordinary differential equations that approximate the number of infected individuals in the QSD for arbitrary contact networks and parameter values. When the epidemic level is high, these QSD approximations coincide with the existing approximation methods. However, as we approach the epidemic threshold, the models deviate, with these models following the QSD and the existing methods approaching the all susceptible state. Through consistently approximating the QSD, the proposed methods provide a more robust link to the stochastic models.

Divide-and-conquer: machine-learning integrates mammalian and viral traits with network features to predict virus-mammal associations.

Our knowledge of viral host ranges remains limited. Completing this picture by identifying unknown hosts of known viruses is an important research aim that can help identify and mitigate zoonotic and animal-disease risks, such as spill-over from animal reservoirs into human populations. To address this knowledge-gap we apply a divide-and-conquer approach which separates viral, mammalian and network features into three unique perspectives, each predicting associations independently to enhance predictive power. Our approach predicts over 20,000 unknown associations between known viruses and susceptible mammalian species, suggesting that current knowledge underestimates the number of associations in wild and semi-domesticated mammals by a factor of 4.3, and the average potential mammalian host-range of viruses by a factor of 3.2. In particular, our results highlight a significant knowledge gap in the wild reservoirs of important zoonotic and domesticated mammals' viruses: specifically, lyssaviruses, bornaviruses and rotaviruses.

Evolutionary bet-hedging in structured populations.

As ecosystems evolve, species can become extinct due to fluctuations in the environment. This leads to the evolutionary adaption known as bet-hedging, where species hedge against these fluctuations to reduce their likelihood of extinction. Environmental variation can be either within or between generations. Previous work has shown that selection for bet-hedging against within-generational variation should not occur in large populations. However, this work has been limited by assumptions of well-mixed populations, whereas real populations usually have some degree of structure. Using the framework of evolutionary graph theory, we show that through adding competition structure to the population, within-generational variation can have a significant impact on the evolutionary process for any population size. This complements research using subdivided populations, which suggests that within-generational variation is important when local population sizes are small. Together, these conclusions provide evidence to support observations by some ecologists that are contrary to the widely held view that only between-generational environmental variation has an impact on natural selection. This provides theoretical justification for further empirical study into this largely unexplored area.

Impact of climatic, demographic and disease control factors on the transmission dynamics of COVID-19 in large cities worldwide.

Approximately a year into the COVID-19 pandemic caused by the SARS-CoV-2 virus, many countries have seen additional "waves" of infections, especially in the temperate northern hemisphere. Other vulnerable regions, such as South Africa and several parts of South America have also seen cases rise, further impacting local economies and livelihoods. Despite substantial research efforts to date, it remains unresolved as to whether COVID-19 transmission has the same sensitivity to climate observed for other common respiratory viruses such as seasonal influenza. Here, we look for empirical evidence of seasonality using a robust estimation framework. For 359 large cities across the world, we estimated the basic reproduction number (R0) using logistic growth curves fitted to cumulative case data. We then assess evidence for association with climatic variables through ordinary least squares (OLS) regression. We find evidence of seasonality, with lower R0 within cities experiencing greater surface radiation (coefficient = -0.005, p < 0.001), after adjusting for city-level variation in demographic and disease control factors. Additionally, we find association between R0 and temperature during the early phase of the epidemic in China. However, climatic variables had much weaker explanatory power compared to socioeconomic and disease control factors. Rates of transmission and health burden of the continuing pandemic will be ultimately determined by population factors and disease control policies.

Evolutionary graph theory derived from eco-evolutionary dynamics.

A biologically motivated individual-based framework for evolution in network-structured populations is developed that can accommodate eco-evolutionary dynamics. This framework is used to construct a network birth and death model. The evolutionary graph theory model, which considers evolutionary dynamics only, is derived as a special case, highlighting additional assumptions that diverge from real biological processes. This is achieved by introducing a negative ecological feedback loop that suppresses ecological dynamics by forcing births and deaths to be coupled. We also investigate how fitness, a measure of reproductive success used in evolutionary graph theory, is related to the life-history of individuals in terms of their birth and death rates. In simple networks, these ecologically motivated dynamics are used to provide new insight into the spread of adaptive mutations, both with and without clonal interference. For example, the star network, which is known to be an amplifier of selection in evolutionary graph theory, can inhibit the spread of adaptive mutations when individuals can die naturally.

Integration of shared-pathogen networks and machine learning reveals the key aspects of zoonoses and predicts mammalian reservoirs.

Diseases that spread to humans from animals, zoonoses, pose major threats to human health. Identifying animal reservoirs of zoonoses and predicting future outbreaks are increasingly important to human health and well-being and economic stability, particularly where research and resources are limited. Here, we integrate complex networks and machine learning approaches to develop a new approach to identifying reservoirs. An exhaustive dataset of mammal-pathogen interactions was transformed into networks where hosts are linked via their shared pathogens. We present a methodology for identifying important and influential hosts in these networks. Ensemble models linking network characteristics with phylogeny and life-history traits are then employed to predict those key hosts and quantify the roles they undertake in pathogen transmission. Our models reveal drivers explaining host importance and demonstrate how these drivers vary by pathogen taxa. Host importance is further integrated into ensemble models to predict reservoirs of zoonoses of various pathogen taxa and quantify the extent of pathogen sharing between humans and mammals. We establish predictors of reservoirs of zoonoses, showcasing host influence to be a key factor in determining these reservoirs. Finally, we provide new insight into the determinants of zoonosis-sharing, and contrast these determinants across major pathogen taxa.

Methods for approximating stochastic evolutionary dynamics on graphs.

Population structure can have a significant effect on evolution. For some systems with sufficient symmetry, analytic results can be derived within the mathematical framework of evolutionary graph theory which relate to the outcome of the evolutionary process. However, for more complicated heterogeneous structures, computationally intensive methods are required such as individual-based stochastic simulations. By adapting methods from statistical physics, including moment closure techniques, we first show how to derive existing homogenised pair approximation models and the exact neutral drift model. We then develop node-level approximations to stochastic evolutionary processes on arbitrarily complex structured populations represented by finite graphs, which can capture the different dynamics for individual nodes in the population. Using these approximations, we evaluate the fixation probability of invading mutants for given initial conditions, where the dynamics follow standard evolutionary processes such as the invasion process. Comparisons with the output of stochastic simulations reveal the effectiveness of our approximations in describing the stochastic processes and in predicting the probability of fixation of mutants on a wide range of graphs. Construction of these models facilitates a systematic analysis and is valuable for a greater understanding of the influence of population structure on evolutionary processes.

Localization of eigenvector centrality in networks with a cut vertex.

We show that eigenvector centrality exhibits localization phenomena on networks that can be easily partitioned by the removal of a vertex cut set, the most extreme example being networks with a cut vertex. Three distinct types of localization are identified in these structures. One is related to the well-established hub node localization phenomenon and the other two are introduced and characterized here. We gain insights into these problems by deriving the relationship between eigenvector centrality and Katz centrality. This leads to an interpretation of the principal eigenvector as an approximation to more robust centrality measures which exist in the full span of an eigenbasis of the adjacency matrix.

A Combined In Vitro/In Silico Approach to Identifying Off-Target Receptor Toxicity.

Many xenobiotics can bind to off-target receptors and cause toxicity via the dysregulation of downstream transcription factors. Identification of subsequent off-target toxicity in these chemicals has often required extensive chemical testing in animal models. An alternative, integrated in vitro/in silico approach for predicting toxic off-target functional responses is presented to refine in vitro receptor identification and reduce the burden on in vivo testing. As part of the methodology, mathematical modeling is used to mechanistically describe processes that regulate transcriptional activity following receptor-ligand binding informed by transcription factor signaling assays. Critical reactions in the signaling cascade are identified to highlight potential perturbation points in the biochemical network that can guide and optimize additional in vitro testing. A physiologically based pharmacokinetic model provides information on the timing and localization of different levels of receptor activation informing whole-body toxic potential resulting from off-target binding.

Impact of the infectious period on epidemics.

The duration of the infectious period is a crucial determinant of the ability of an infectious disease to spread. We consider an epidemic model that is network based and non-Markovian, containing classic Kermack-McKendrick, pairwise, message passing, and spatial models as special cases. For this model, we prove a monotonic relationship between the variability of the infectious period (with fixed mean) and the probability that the infection will reach any given subset of the population by any given time. For certain families of distributions, this result implies that epidemic severity is decreasing with respect to the variance of the infectious period. The striking importance of this relationship is demonstrated numerically. We then prove, with a fixed basic reproductive ratio (R_{0}), a monotonic relationship between the variability of the posterior transmission probability (which is a function of the infectious period) and the probability that the infection will reach any given subset of the population by any given time. Thus again, even when R_{0} is fixed, variability of the infectious period tends to dampen the epidemic. Numerical results illustrate this but indicate the relationship is weaker. We then show how our results apply to message passing, pairwise, and Kermack-McKendrick epidemic models, even when they are not exactly consistent with the stochastic dynamics. For Poissonian contact processes, and arbitrarily distributed infectious periods, we demonstrate how systems of delay differential equations and ordinary differential equations can provide upper and lower bounds, respectively, for the probability that any given individual has been infected by any given time.

A control analysis perspective on Katz centrality.

Methods for efficiently controlling dynamics propagated on networks are usually based on identifying the most influential nodes. Knowledge of these nodes can be used for the targeted control of dynamics such as epidemics, or for modifying biochemical pathways relating to diseases. Similarly they are valuable for identifying points of failure to increase network resilience in, for example, social support networks and logistics networks. Many measures, often termed 'centrality', have been constructed to achieve these aims. Here we consider Katz centrality and provide a new interpretation as a steady-state solution to continuous-time dynamics. This enables us to implement a sensitivity analysis which is similar to metabolic control analysis used in the analysis of biochemical pathways. The results yield a centrality which quantifies, for each node, the net impact of its absence from the network. It also has the desirable property of requiring a node with a high centrality to play a central role in propagating the dynamics of the system by having the capacity to both receive flux from others and then to pass it on. This new perspective on Katz centrality is important for a more comprehensive analysis of directed networks.

The relationships between message passing, pairwise, Kermack-McKendrick and stochastic SIR epidemic models.

We consider a very general stochastic model for an SIR epidemic on a network which allows an individual's infectious period, and the time it takes to contact each of its neighbours after becoming infected, to be correlated. We write down the message passing system of equations for this model and prove, for the first time, that it has a unique feasible solution. We also generalise an earlier result by proving that this solution provides a rigorous upper bound for the expected epidemic size (cumulative number of infection events) at any fixed time [Formula: see text]. We specialise these results to a homogeneous special case where the graph (network) is symmetric. The message passing system here reduces to just four equations. We prove that cycles in the network inhibit the spread of infection, and derive important epidemiological results concerning the final epidemic size and threshold behaviour for a major outbreak. For Poisson contact processes, this message passing system is equivalent to a non-Markovian pair approximation model, which we show has well-known pairwise models as special cases. We show further that a sequence of message passing systems, starting with the homogeneous one just described, converges to the deterministic Kermack-McKendrick equations for this stochastic model. For Poisson contact and recovery, we show that this convergence is monotone, from which it follows that the message passing system (and hence also the pairwise model) here provides a better approximation to the expected epidemic size at time [Formula: see text] than the Kermack-McKendrick model.

The Differential Warming Response of Britain's Rivers (1982-2011).

River water temperature is a hydrological feature primarily controlled by topographical, meteorological, climatological, and anthropogenic factors. For Britain, the study of freshwater temperatures has focussed mainly on observations made in England and Wales; similar comprehensive data sets for Scotland are currently unavailable. Here we present a model for the whole of mainland Britain over three recent decades (1982-2011) that incorporates geographical extrapolation to Scotland. The model estimates daily mean freshwater temperature for every river segment and for any day in the studied period, based upon physico-geographical features, daily mean air and sea temperatures, and available freshwater temperature measurements. We also extrapolate the model temporally to predict future warming of Britain's rivers given current observed trends. Our results highlight the spatial and temporal diversity of British freshwater temperatures and warming rates. Over the studied period, Britain's rivers had a mean temperature of 9.84°C and experienced a mean warming of +0.22°C per decade, with lower rates for segments near lakes and in coastal regions. Model results indicate April as the fastest-warming month (+0.63°C per decade on average), and show that most rivers spend on average ever more days of the year at temperatures exceeding 10°C, a critical threshold for several fish pathogens. Our results also identify exceptional warming in parts of the Scottish Highlands (in April and September) and pervasive cooling episodes, in December throughout Britain and in July in the southwest of England (in Wales, Cornwall, Devon, and Dorset). This regional heterogeneity in rates of change has ramifications for current and future water quality, aquatic ecosystems, as well as for the spread of waterborne diseases.

Discrete-time moment closure models for epidemic spreading in populations of interacting individuals.

Understanding the dynamics of spread of infectious diseases between individuals is essential for forecasting the evolution of an epidemic outbreak or for defining intervention policies. The problem is addressed by many approaches including stochastic and deterministic models formulated at diverse scales (individuals, populations) and different levels of detail. Here we consider discrete-time SIR (susceptible-infectious-removed) dynamics propagated on contact networks. We derive a novel set of 'discrete-time moment equations' for the probability of the system states at the level of individual nodes and pairs of nodes. These equations form a set which we close by introducing appropriate approximations of the joint probabilities appearing in them. For the example case of SIR processes, we formulate two types of model, one assuming statistical independence at the level of individuals and one at the level of pairs. From the pair-based model we then derive a model at the level of the population which captures the behavior of epidemics on homogeneous random networks. With respect to their continuous-time counterparts, the models include a larger number of possible transitions from one state to another and joint probabilities with a larger number of individuals. The approach is validated through numerical simulation over different network topologies.

Complete hierarchies of SIR models on arbitrary networks with exact and approximate moment closure.

We first generalise ideas discussed by Kiss et al. (2015) to prove a theorem for generating exact closures (here expressing joint probabilities in terms of their constituent marginal probabilities) for susceptible-infectious-removed (SIR) dynamics on arbitrary graphs (networks). For Poisson transmission and removal processes, this enables us to obtain a systematic reduction in the number of differential equations needed for an exact 'moment closure' representation of the underlying stochastic model. We define 'transmission blocks' as a possible extension of the block concept in graph theory and show that the order at which the exact moment closure representation is curtailed is the size of the largest transmission block. More generally, approximate closures of the hierarchy of moment equations for these dynamics are typically defined for the first and second order yielding mean-field and pairwise models respectively. It is frequently implied that, in principle, closed models can be written down at arbitrary order if only we had the time and patience to do this. However, for epidemic dynamics on networks, these higher-order models have not been defined explicitly. Here we unambiguously define hierarchies of approximate closed models that can utilise subsystem states of any order, and show how well-known models are special cases of these hierarchies.

Epidemic control analysis: designing targeted intervention strategies against epidemics propagated on contact networks.

In cases where there are limited resources for the eradication of an epidemic, or where we seek to minimise possible adverse impacts of interventions, it is essential to optimise the efficacy of control measures. We introduce a new approach, Epidemic Control Analysis (ECA), to design effective targeted intervention strategies to mitigate and control the propagation of infections across heterogeneous contact networks. We exemplify this methodology in the context of a newly developed individual-level deterministic Susceptible-Infectious-Susceptible (SIS) epidemiological model (we also briefly consider applications to Susceptible-Infectious-Removed (SIR) dynamics). This provides a flexible way to systematically determine the impact of interventions on endemic infections in the population. Individuals are ranked based on their influence on the level of infectivity. The highest-ranked individuals are prioritised for targeted intervention. Many previous intervention strategies have determined prioritisation based mainly on the position of individuals in the network, described by various local and global network centrality measures, and their chance of being infectious. Comparisons of the predictions of the proposed strategy with those of widely used targeted intervention programmes on various model and real-world networks reveal its efficiency and accuracy. It is demonstrated that targeting central individuals or individuals that have high infection probability is not always the best strategy. The importance of individuals is not determined by network structure alone, but can be highly dependent on the infection dynamics. This interplay between network structure and infection dynamics is effectively captured by ECA.