Introduction

Mathematical modeling has provided many significant insights concerning the epidemiology of infectious diseases. The most notable of these include threshold conditions (involving the so-called 'basic reproductive number') that describe when invasion and persistence of an infection is possible.1-3 The development of much of this theory has revolved around the use of extremely simple models, such as deterministic compartmental models. Typically, the population of interest is subdivided into a small number of compartments based on infection status (e.g. susceptible, infectious or recovered) and the flows between these compartments are described by a low dimensional set of ordinary differential equations. The derivation of these equations typically involves a number of simplifying assumptions, an important example of which is that the population is well-mixed, which will be discussed in detail below.

The simplicity of these models facilitates the use of analytic techniques to gain general understanding, but at the cost of oversimplifying the biology of real-world disease processes. The weaknesses of simple models have long been clear, particularly when model behavior has been compared to epidemiologic data, and this has lead to the development of increasingly complex models that attempt to account for more details of the underlying biology.1 Much of this complexity can be incorporated within the population-level framework provided by compartmental models.

Individual-level models offer a fundamentally different way of describing biological populations. In this approach, every individual in the population is accounted for as a separate entity. The complexity of such models makes analysis difficult, and numerical simulation computationally intensive. Furthermore, these models must include some description of the interactions between the individuals that make up the population. Unless a large number of simplifying assumptions are made, specifying these interactions is a major task whenever there are more than a handful of individuals to be considered.

Network models (also known as graph models) provide a natural way of describing a population and its interactions. Nodes (vertices) of the graph represent individuals and edges (links) depict interactions between individuals that could potentially lead to transmission of infection. It is interesting to note that similar network representations can be used in a number of contexts, such as transportation networks, communication networks (including the internet and World Wide Web) and social networks (including friendship, movie actor and scientific collaboration networks).4-6

In this chapter we shall discuss the development and use of network models in epidemiology. While network models have long been discussed in the theoretical epidemiology literature, they have recently received a large amount of attention amongst the statistical physics community. This has been fueled by the desire to better understand the structure of social and large-scale technological networks, and the increases in computational power that have made the simulation of reasonably-sized network models a feasible proposition. A main aim of this review is to bridge the epidemi-

ologic and statistical physics approaches to network models for infectious diseases, highlighting the important contributions made by both research communities.

This chapter is organized as follows. We shall first discuss some of the epidemiologic settings in which network models are employed. Our attention will then turn to ways in which networks are described, including measures that attempt to capture important properties of graphs. An important part of this discussion will include the feasibility of employing such methods to describe real-world networks, particularly when only incomplete information is available. We shall then describe some classes of networks that have received particular attention. Finally, we discuss the impact of network structure on the spread of infection and some of the ways in which control measures must account for this structure.

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