the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Using a novel interpretation of dynamic networks, we analyse the network of livestock movements in Great Britain in order to determine the risk of a large epidemic of foot-and-mouth disease (FMD). This network is exceptionally well characterized, as there are legal requirements that the date, source, destination and number of animals be recorded and held on central databases. We identify a percolation threshold in the structure of the livestock network, indicating that, while there is little possibility of a national epidemic of FMD in winter when the catastrophic 2001 epidemic began, there remains a risk in late summer or early autumn. These predictions are corroborated by a non-parametric simulation in which the movements of livestock in 2003 and 2004 are replayed as they occurred. Despite the risk, we show that the network displays small-world properties which can be exploited to target surveillance and control and drastically reduce this risk.

During the 2001 foot and mouth disease epidemic in the UK, initial dissemination of the disease to widespread geographical regions was attributed to livestock movement, especially of sheep. In response, recording schemes to provide accurate data describing the movement of large livestock in Great Britain (GB) were introduced. Using these data, we reconstruct directed contact networks within the sheep industry and identify key epidemiological properties of these networks. There is clear seasonality in sheep movements, with a peak of intense activity in August and September and an associated high risk of a large epidemic. The high correlation between the in and out degree of nodes favours disease transmission. However, the contact networks were largely dissasortative: highly connected nodes mostly connect to nodes with few contacts, effectively slowing the spread of disease. This is a result of bipartite-like network properties, with most links occurring between highly active markets and less active farms. When comparing sheep movement networks (SMNs) to randomly generated networks with the same number of nodes and node degrees, despite structural differences (such as disassortativity and higher frequency of even path lengths in the SMNs), the characteristic path lengths within the SMNs are close to values computed from the corresponding random networks, showing that SMNs have 'small-world'-like properties. Using the network properties, we show that targeted biosecurity or surveillance at highly connected nodes would be highly effective in preventing a large and widespread epidemic.

The dynamics of disease transmission strongly depends on the properties of the population contact network. Pair-approximation models and individual-based network simulation have been used extensively to model contact networks with non-trivial properties. In this paper, using a continuous time Markov chain, we start from the exact formulation of a simple epidemic model on an arbitrary contact network and rigorously derive and prove some known results that were previously mainly justified based on some biological hypotheses. The main result of the paper is the illustration of the link between graph automorphisms and the process of lumping whereby the number of equations in a system of linear differential equations can be significantly reduced. The main advantage of lumping is that the simplified lumped system is not an approximation of the original system but rather an exact version of this. For a special class of graphs, we show how the lumped system can be obtained by using graph automorphisms. Finally, we discuss the advantages and possible applications of exact epidemic models and lumping.

No abstract

Livestock movements in Great Britain (GB) are well recorded and are a unique record of the network of connections among livestock-holding locations. These connections can be critical for disease spread, as in the 2001 epidemic of foot-and-mouth disease (FMD) in the UK. Here, the movement data are used to construct an individual-farm-based model of the initial spread of FMD in GB and determine the susceptibility of the GB livestock industry to future outbreaks under the current legislative requirements. Transmission through movements is modelled, with additional local spread unrelated to the known movements. Simulations show that movements can result in a large nationwide epidemic, but only if cattle are heavily involved, or the epidemic occurs in late summer or early autumn. Inclusion of random local spread can considerably increase epidemic size, but has only a small impact on the spatial extent of the disease. There is a geographical bias in the epidemic size reached, with larger epidemics originating in Scotland and the north of England than elsewhere.

We analyse the relationship between the network of livestock movements in the UK and the dynamics of two diseases: foot-and-mouth disease (FMD), which has an incubation period of days, and scrapie, which incubates over years. For FMD, the time-scale of expected epidemics is similar to the time-scale of the evolution of the network. We argue that, under appropriate conditions, a static network analysis can be an appropriate tool for gaining insights into disease dynamics even when the relevant time-scales are similar, as with FMD. We show that a subclass of 'linkage moves' maintains the network structure, and so removing these links has a dramatic effect on the number of potentially infected farms, an effect corroborated by simulations. In contrast, because scrapie has a low probability of transmission per contact and a long incubation period, a static network representation is probably appropriate; however, the signature of the network in the pattern of transmission is likely to be faint. Scrapie-notifying farms were more likely to be associated with each other via trading at markets than were control farms; however, network community structure proves to be less representative of prevalence patterns than geographical region. These contradictory indicators emphasize that appropriate observation time frames and good discrimination among types of potentially infectious contacts are vital in order for network analysis to be a valuable epidemiological tool.

Combinations of intense non-pharmaceutical interventions (lockdowns) were introduced worldwide to reduce SARS-CoV-2 transmission. Many governments have begun to implement exit strategies that relax restrictions while attempting to control the risk of a surge in cases. Mathematical modelling has played a central role in guiding interventions, but the challenge of designing optimal exit strategies in the face of ongoing transmission is unprecedented. Here, we report discussions from the Isaac Newton Institute ‘Models for an exit strategy’ workshop (11–15 May 2020). A diverse community of modellers who are providing evidence to governments worldwide were asked to identify the main questions that, if answered, would allow for more accurate predictions of the effects of different exit strategies. Based on these questions, we propose a roadmap to facilitate the development of reliable models to guide exit strategies. This roadmap requires a global collaborative effort from the scientific community and policymakers, and has three parts: (i) improve estimation of key epidemiological parameters; (ii) understand sources of heterogeneity in populations; and (iii) focus on requirements for data collection, particularly in low-to-middle-income countries. This will provide important information for planning exit strategies that balance socio-economic benefits with public health.