Many diseases spread through human populations via close physical interactions. The interpersonal contact patterns that underlie disease transmission can naturally be thought to form a network, where links join individuals who interact with each other. During an outbreak, disease then spreads along these links. All epidemiological models make assumptions about the underlying network of interactions, often without explicitly stating them. Contact network models, however, mathematically formalize this intuitive concept so that epidemiological calculations can explicitly consider complex patterns of interactions.
Formally, a contact (or social) network model explicitly represents host interactions that mediate disease spread. A node in a contact network represents an individual host, and an edge between two nodes represents an interaction that may allow disease transmission. A node's degree is the number of edges attached to it (i.e. its number of contacts) and the degree distribution of a network is the frequency distribution of degrees throughout the entire population.
An exact contact network model requires knowledge of every individual in a population and every disease-causing contact between individuals (e.g. sneezing in the case of airborne diseases or sexual contact in the case of sexually transmitted diseases). For even small populations, this is typically unfeasible, and thus researchers typically work with approximate networks. There are several techniques for gathering the information needed to build realistic contact network models. They include tracing all infected individuals and their contacts during or following an outbreak (e.g. Klovdahl et al. 1977
), surveying individuals in populations (e.g. Eubank et al. 2004
) and using census (e.g. Meyers et al. 2005
), social characteristic (e.g. Halloran et al. 2002
) or other collected data (e.g. Meyers et al. 2003
Characterizing network structure has become a multidisciplinary cottage industry, with researchers across epidemiology, sociology, biology, computer science and physics searching volumes of data for meaningful patterns. Researchers often look for global statistical properties in network data, and have paid special attention to small-world networks—characterized by high levels of both local clustering and global connectivity (Watts & Strogatz 1998
)—and scale-free networks—characterized by degree distributions that follow a power-law distribution with a small fraction of very highly connected hubs (Barabási & Albert 1999
). Scale-free networks have been reported in many technological (e.g. the Internet, the World Wide Web) and biological systems (e.g. metabolic, protein interactions, transcription regulation and protein domain; Albert et al. 1999
; Faloutsos et al. 1999
; Wuchty 2001
; Giot et al. 2003
; Bork et al. 2004
; Hatzimanikatis et al. 2004
; Luscombe et al. 2004
). These highly structured networks are often contrasted with three classes of ‘null’ networks: (i) lattices
in which all nodes have the same degree, and any given node is connected to physically proximate nodes, (ii) regular random networks
in which all nodes have the same degree, but any given node is connected to randomly chosen nodes throughout the network, and (iii) Poisson random networks
(also called Erdos–Renyi random graphs) in which some specified total number of edges are assigned to nodes completely at random, thus yielding a Poisson degree distribution across the network.
The field has focused particularly on scale-free random networks (May & Lloyd 2001
; Dezso & Barabasi 2002
; Pastor-Satorras & Vespignani 2002
), based on the apparent ubiquity of such networks in natural and human-made systems and a (limited) set of studies of epidemiologically relevant contact patterns (Liljeros et al. 2001
). These networks are characterized by the presence of hosts with anomalously high numbers of potential disease-causing contacts, called super-spreaders, which have important epidemiological implications (Shen et al. 2004
; Lloyd-Smith et al. 2005
). With large or infinite variance in degree, scale-free networks can have exceedingly low or non-existent epidemic thresholds; this means that even the sparsest networks are highly vulnerable to epidemics. Despite recent popularity, however, it is not clear that realistic epidemiological networks are generally scale-free, and thus whether they deserve so much attention in epidemiology. Here, we address this by characterizing the structures of several real-world networks.
We limit our discussion to random networks with arbitrary degree distributions, including random networks with regular, Poisson, exponential and scale-free degree distributions (). These classes of networks have been well studied with respect to the spread of epidemics and are representatives of a spectrum of network structures. We focus exclusively on the epidemiological impact of the degree distribution, although other network characteristics such as clustering (Watts & Strogatz 1998
; Keeling 1999
; Moore & Newman 2000
; Petermann & Rios 2004
) and degree correlations (Boguna et al. 2003
) are also important.
Figure 1 Examples of (a) a regular random network with 15 nodes and mean=5, (b) a Poisson random graph with 15 nodes and mean=5, (c) a scale-free random graph with 100 nodes and mean=5, (d) the Zachary Karate Club contact network (Zachary 1977) with 34 nodes and (more ...)
We will also focus exclusively on static networks, i.e. networks in which contacts are assumed to be fixed during the infectious period of an individual. The permanence of contacts captured by static networks offers a more realistic model of human contact behaviour than that by traditional epidemiological models. However, a recent study (Volz & Meyers submitted
) suggests that static networks may only be an approximate model for diseases that spread slowly relative to the rate at which individuals change the numbers and identities of their contacts.