Matthew James Keeling

Networks and epidemic models

Networks and the epidemiology of directly transmitted infectious diseases are fundamentally linked. The foundations of epidemiology and early epidemiological models were based on population wide random-mixing, but in practice each individual has a finite set of contacts to whom they can pass infection; the ensemble of all such contacts forms a ‘mixing network’. Knowledge of the structure of the network allows models to compute the epidemic dynamics at the population scale from the individual-level behaviour of infections. Therefore, characteristics of mixing networks—and how these deviate from the random-mixing norm—have become important applied concerns that may enhance the understanding and prediction of epidemic patterns and intervention measures. Here, we review the basis of epidemiological theory (based on random-mixing models) and network theory (based on work from the social sciences and graph theory). We then describe a variety of methods that allow the mixing network, or an approximation to the network, to be ascertained. It is often the case that time and resources limit our ability to accurately find all connections within a network, and hence a generic understanding of the relationship between network structure and disease dynamics is needed. Therefore, we review some of the variety of idealized network types and approximation techniques that have been utilized to elucidate this link. Finally, we look to the future to suggest how the two fields of network theory and epidemiological modelling can deliver an improved understanding of disease dynamics and better public health through effective disease control.

The effects of local spatial structure on epidemiological invasions.

Predicting the likely success of invasions is vitally important in ecology and especially epidemiology. Whether an organism can successfully invade and persist in the short–term is highly dependent on the spatial correlations that develop in the early stages of invasion. By modelling the correlations between individuals, we are able to understand the role of spatial heterogeneity in invasion dynamics without the need for large–scale computer simulations. Here, a natural methodology is developed for modelling the behaviour of individuals in a fixed network. This formulation is applied to the spread of a disease through a structured network to determine invasion thresholds and some statistical properties of a single epidemic.

Modelling vaccination strategies against foot-and-mouth disease.

Vaccination has proved a powerful defence against a range of infectious diseases of humans and animals. However, its potential to control major epidemics of foot-and-mouth disease (FMD) in livestock is contentious. Using an individual farm-based model, we consider either national prophylactic vaccination campaigns in advance of an outbreak, or combinations of reactive vaccination and culling strategies during an epidemic. Consistent with standard epidemiological theory, mass prophylactic vaccination could reduce greatly the potential for a major epidemic, while the targeting of high-risk farms increases efficiency. Given sufficient resources and preparation, a combination of reactive vaccination and culling might control ongoing epidemics. We also explore a reactive strategy, ‘predictive’ vaccination, which targets key spatial transmission loci and can reduce markedly the long tail that characterizes many FMD epidemics. These analyses have broader implications for the control of human and livestock infectious diseases in heterogeneous spatial landscapes.

Modeling dynamic and network heterogeneities in the spread of sexually transmitted diseases

A wide range of communicable human diseases can be considered as spreading through a network of possible transmission routes. The implied network structure is vital in determining disease dynamics, especially when the average number of connections per individual is small as is the case for many sexually transmitted diseases (STDs). Here we develop an intuitive mathematical framework to deal with the heterogeneities implicit within contact networks and those that arise because of the infection process. These models are compared with full stochastic simulations and show excellent agreement across a wide range of parameters. We show how such models can be used to estimate parameters of epidemiological importance, and how they can be extended to examine the effectiveness of various control strategies, in particular screening programs and contact tracing.

Identification of 100 fundamental ecological questions

Summary 1. Fundamental ecological research is both intrinsically interesting and provides the basic knowledge required to answer applied questions of importance to the management of the natural world. The 100th anniversary of the British Ecological Society in 2013 is an opportune moment to reflect on the current status of ecology as a science and look forward to high-light priorities for future work.

Planning for smallpox outbreaks

Mathematical models of viral transmission and control are important tools for assessing the threat posed by deliberate release of the smallpox virus and the best means of containing an outbreak. Models must balance biological realism against limitations of knowledge, and uncertainties need to be accurately communicated to policy-makers. Smallpox poses the particular challenge that key biological, social and spatial factors affecting disease spread in contemporary populations must be elucidated largely from historical studies undertaken before disease eradication in 1979. We review the use of models in smallpox planning within the broader epidemiological context set by recent outbreaks of both novel and re-emerging pathogens.

Appropriate Models for the Management of Infectious Diseases

Background Mathematical models have become invaluable management tools for epidemiologists, both shedding light on the mechanisms underlying observed dynamics as well as making quantitative predictions on the effectiveness of different control measures. Here, we explain how substantial biases are introduced by two important, yet largely ignored, assumptions at the core of the vast majority of such models. Methods and Findings First, we use analytical methods to show that (i) ignoring the latent period or (ii) making the common assumption of exponentially distributed latent and infectious periods (when including the latent period) always results in underestimating the basic reproductive ratio of an infection from outbreak data. We then proceed to illustrate these points by fitting epidemic models to data from an influenza outbreak. Finally, we document how such unrealistic a priori assumptions concerning model structure give rise to systematically overoptimistic predictions on the outcome of potential management options. Conclusion This work aims to highlight that, when developing models for public health use, we need to pay careful attention to the intrinsic assumptions embedded within classical frameworks.

Disease evolution on networks: the role of contact structure

Owing to their rapid reproductive rate and the severe penalties for reduced fitness, diseases are under immense evolutionary pressure. Understanding the evolutionary response of diseases in new situations has clear public–health consequences, given the changes in social and movement patterns over recent decades and the increased use of antibiotics. This paper investigates how a disease may adapt in response to the routes of transmission available between infected and susceptible individuals. The potential transmission routes are defined by a computer–generated contact network, which we describe as either local (highly clustered networks where connected individuals are likely to share common contacts) or global (unclustered networks with a high proportion of long–range connections). Evolution towards stable strategies operates through the gradual random mutation of disease traits (transmission rate and infectious period) whenever new infections occur. In contrast to mean–field models, the use of contact networks greatly constrains the evolutionary dynamics. In the local networks, high transmission rates are selected for, as there is intense competition for susceptible hosts between disease progeny. By contrast, global networks select for moderate transmission rates because direct competition between progeny is minimal and a premium is placed upon persistence. All networks show a very slow but steady rise in the infectious period.

Seasonnally forced disease dynamics explored as switching between attractors

Biological phenomena offer a rich diversity of problems that can be understood using mathematical techniques. Three key features common to many biological systems are temporal forcing, stochasticity and nonlinearity. Here, using simple disease models compared to data, we examine how these three factors interact to produce a range of complicated dynamics. The study of disease dynamics has been amongst the most theoretically developed areas of mathematical biology; simple models have been highly successful in explaining the dynamics of a wide variety of diseases. Models of childhood diseases incorporate seasonal variation in contact rates due to the increased mixing during school terms compared to school holidays. This ‘binary’ nature of the seasonal forcing results in dynamics that can be explained as switching between two nonlinear spiral sinks. Finally, we consider the stability of the attractors to understand the interaction between the deterministic dynamics and demographic and environmental stochasticity. Throughout attention is focused on the behaviour of measles, whooping cough and rubella.

Optimal reactive vaccination strategies for a foot-and-mouth outbreak in the UK

Foot-and-mouth disease (FMD) in the UK provides an ideal opportunity to explore optimal control measures for an infectious disease. The presence of fine-scale spatio-temporal data for the 2001 epidemic has allowed the development of epidemiological models that are more accurate than those generally created for other epidemics and provide the opportunity to explore a variety of alternative control measures. Vaccination was not used during the 2001 epidemic; however, the recent DEFRA (Department for Environment Food and Rural Affairs) contingency plan details how reactive vaccination would be considered in future. Here, using the data from the 2001 epidemic, we consider the optimal deployment of limited vaccination capacity in a complex heterogeneous environment. We use a model of FMD spread to investigate the optimal deployment of reactive ring vaccination of cattle constrained by logistical resources. The predicted optimal ring size is highly dependent upon logistical constraints but is more robust to epidemiological parameters. Other ways of targeting reactive vaccination can significantly reduce the epidemic size; in particular, ignoring the order in which infections are reported and vaccinating those farms closest to any previously reported case can substantially reduce the epidemic. This strategy has the advantage that it rapidly targets new foci of infection and that determining an optimal ring size is unnecessary.