Pathogens, social networks, and the paradox of transmission scaling

Abstract

Understanding the scaling of transmission is critical to predicting how infectious diseases will affect populations of different sizes and densities. The two classic "mean-field" epidemic models-either assuming density-dependent or frequency-dependent transmission-make predictions that are discordant with patterns seen in either within-population dynamics or across-population comparisons. In this paper, we propose that the source of this inconsistency lies in the greatly simplifying "mean-field" assumption of transmission within a fully-mixed population. Mixing in real populations is more accurately represented by a network of contacts, with interactions and infectious contacts confined to the local social neighborhood. We use network models to show that density-dependent transmission on heterogeneous networks often leads to apparent frequency dependency in the scaling of transmission across populations of different sizes. Network-methodology allows us to reconcile seemingly conflicting patterns of within- and across-population epidemiology.

Figure 1

Scaling of measles transmission. The estimated mean transmission rate (β) of measles in England and Wales plotted against increasing city size in thousands. Reproduced from [19].

Figure 2

Classes of social networks. Two classes of social networks where the node represents an individual and the edge a social connection or epidemiological relevant contact according to edge distributions (i.e., contact networks) that are described by (a) Poisson networks and (b) power law networks.

Figure 3

Scaling of transmission on Poisson (a, b), exponential (c, d), and scale-free (e, f) networks. Left-hand panels are the mean realized per capita transmission rate, $\widehat{\beta }$, plotted against network size. Right-hand panels are the mean number of infections per edge between susceptible and infected nodes. Solid lines indicate a constant mean number of contacts for all population sizes. Dashed lines indicate a mean number of contacts that increase proportional to the square root of population size. Dotted lines indicate a mean number of contacts that increase linearly with the population size. Vertical bars give the standard deviation in observations from 30 simulated networks.