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This chapter provides an overview of stochastic models for epidemics, focusing on topics that have preoccupied researchers for the last 15 years, and identifying continuing challenges. After some historical background and a brief account of basic deterministic models for transmission of infectious diseases, the principles of stochastic modeling of epidemics in homogeneous populations are outlined. The chapter then discusses the complications that arise owing to heterogeneity of host population, of mixing within the population, and of the network among the population, due for example to its social or spatial structure. The chapter concludes with a brief discussion of statistical issues.

This chapter discusses the spread of diseases over contact networks between individuals and the methods used to model this process. The chapter begins with an introduction to the classic models of mathematical epidemiology, including the SI model, the SIR model, and the SIS model. Models for coinfection and competition between diseases are also discussed, as well as “complex contagion” models used to represent the spread of information. The remainder of the chapter deals with the behavior of these models on networks, where the behavior of spreading diseases depends strongly on network structure. It is shown that the SIR model maps to a bond percolation process on networks, allowing us to solve for static properties such as the total number of individuals infected in a disease outbreak. The case of the configuration model is developed in detail and the calculations are extended to competing diseases, coinfection, and complex contagion. Time-dependent behavior of diseases on networks is also studied using various differential equation approximations, including pair approximations and degree-based approximations.

This chapter discusses ‘epidemiology’, which in medical literature refers to the field that seeks to identify key correlates for a particular disease. In this book, however, it is the study of host–parasite population dynamics as a branch of population biology and population genetics. Beginning with the Swiss mathematician Daniel Bernoulli, who used a mathematical model to analyse the dynamics of a small-pox epidemic in Paris, epidemiology provides a great help to the study of evolutionary parasitology. The classical Nicholson-Bailey model is one example of an epidemiology model that analyses host–parasitoid systems. The chapter examines other models, such as SIR-models, in order to further analyse the epidemiology of hosts and their microparasites. It also talks about the epidemiology of vectored microparasites, like malaria, which can also similarly be analysed with modified SIR-models such as the classical Ross-Macdonald model.

Epidemic processes are relevant to studying the propagation of infectious diseases, but their current use extends also to the study of propagation of ideas in the society or memes and news in online social media. In most of the relevant applications epidemic spreading does not actually take place on a single network but propagates in a multilayer network where different types of interaction play different roles. This chapter provides a comprehensive view of the effect that multilayer network structures have on epidemic processes. The Susceptible–Infected–Susceptible (SIS) Model and the Susceptible–Infected–Removed (SIR) Model are characterized on multilayer networks. Additionally, it is shown that the multilayer networks framework can also allow us to study interacting Awareness and epidemic spreading, competing networks and epidemics in temporal networks.

This chapter reviews graph generation techniques in the context of applications. The first case study is power grids, where proposed strategies to prevent blackouts have been tested on tailored random graphs. The second case study is in social networks. Applications of random graphs to social networks are extremely wide ranging – the particular aspect looked at here is modelling the spread of disease on a social network – and how a particular construction based on projecting from a bipartite graph successfully captures some of the clustering observed in real social networks. The third case study is on null models of food webs, discussing the specific constraints relevant to this application, and the topological features which may contribute to the stability of an ecosystem. The final case study is taken from molecular biology, discussing the importance of unbiased graph sampling when considering if motifs are over-represented in a protein–protein interaction network.