Networks and epidemic models

Abstract

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.

Figure 1

For the same simple network (thin grey lines), the type of network information that is achieved using infection tracing (left), contact tracing (middle) and diary-based studies (right). For infection and contact tracing, circles represent infected individuals, while the square shows the primary infectious case; for the diary-based study, those taking part are shown with open circles. For infection tracing, only sources of infection are traced and some individuals (e.g. top left) have multiple potential sources of infection. For contact tracing, a proportion of all contacts from infectious individuals are traced. Finally, with a diary-based study, although almost all links can be traced, the lack of a unique identifier means that often links from different individuals cannot be connected.

Figure 2

Five distinct network types containing 100 individuals. These are from left to right: random, lattice, small world (top row), spatial and scale-free (bottom row). A network generated by an exponential random graph model is not shown, as this flexible framework can encompass a huge variety of network types. For the scale-free network, the bottom right-hand graph shows the power-law distribution of individuals with a given degree from 1000 replicate networks; for this example, the power-law exponent is −3.3. The random, spatial and scale-free networks all utilize the same position of individuals, although for the random and scale-free network, the position of the individuals is irrelevant for forming connections. In all five graphs, the average number of contacts per individual is approximately four. For the scale-free network, individuals with high numbers of contacts are represented by larger dots and are shaded grey.

Figure 3

Typical SIR epidemics on the five network types shown in figure 2. These are from left to right: random, lattice, small world (top row), spatial and scale-free (bottom row). Each graph shows 100 epidemic curves (grey) together with the average for all major epidemics (black) for a single example of each network type; therefore, all variability within each graph is a result of the stochastic nature of transmission and not variation in the network. All five networks contain 10000 individuals, although all individuals are not necessarily interconnected as part of a giant component. For the spatial and scale-free networks, approximately 88 and 74% are part of the giant component and can therefore potentially become infected. For these networks, the proportion of infectious individuals has been rescaled as a fraction of the giant component. In all networks, the average number of contacts per individual is approximately 4, although for the scale-free network, there is considerable heterogeneity with one individual having 85 contacts. For consistency, the small-world network is formed from a two-dimensional lattice (not a one-dimensional circle as shown in figure 2) with 10 additional random ‘long-range’ contacts. The dashed lines show the effect on the mean epidemic of increasing the number of long-range contacts to 20 and 100. (τ=1, g=0.5, b=d=0).