Spatial analyses of wildlife contact networks

Abstract

Datasets from which wildlife contact networks of epidemiological importance can be inferred are becoming increasingly common. A largely unexplored facet of these data is finding evidence of spatial constraints on who has contact with whom, despite theoretical epidemiologists having long realized spatial constraints can play a critical role in infectious disease dynamics. A graph dissimilarity measure is proposed to quantify how close an observed contact network is to being purely spatial whereby its edges are completely determined by the spatial arrangement of its nodes. Statistical techniques are also used to fit a series of mechanistic models for contact rates between individuals to the binary edge data representing presence or absence of observed contact. These are the basis for a second measure that quantifies the extent to which contacts are being mediated by distance. We apply these methods to a set of 128 contact networks of field voles (Microtus agrestis) inferred from mark-recapture data collected over 7 years and from four sites. Large fluctuations in vole abundance allow us to demonstrate that the networks become increasingly similar to spatial proximity graphs as vole density increases. The average number of contacts, 〈k〉, was (i) positively correlated with vole density across the range of observed densities and (ii) for two of the four sites a saturating function of density. The implications for pathogen persistence in wildlife may be that persistence is relatively unaffected by fluctuations in host density because at low density 〈k〉 is low but hosts move more freely, and at high density 〈k〉 is high but transmission is hampered by local build-up of infected or recovered animals.

Keywords: Microtus agrestis; dissimilarity measure; epidemiology; field vole; graph; mathematical model.

Figure 1.

Spatial and non-spatial plots of two of the 32 contact networks inferred from trapping sessions conducted at the Kielder Site (KSC) during (a) winter (from 13 November 2001 to 20 January 2002) and (b) summer (from 28 June 2002 to 26 July 2002). The non-spatial versions of these networks, (c) and (d), respectively, are produced in the software package R where there is an attempt to more clearly display the structure of the networks by minimizing the number of edge-crossings. (Online version in colour.)

Figure 2.

The mean degree of each of the 32 networks, from each of the four sites, plotted against estimated population density. A linear function and a power function were fitted to the data separately for each site. In (a,b), for BHP and ROB, a power function was a better fit (adjusted R² values were, respectively, 0.669 versus 0.427 and 0.778 versus 0.642), while in (c,d), for PLJ and KCS, a linear function was the better fit (adjusted R² values were, respectively, 0.783 versus 0.6689 and 0.7308 versus 0.6907). (Online version in colour.)

Figure 3.

The CV (the standard deviation of the degree sequence scaled by the average degree) of each of the 32 networks, from each of the four sites, plotted against estimated population density. The CV tends to drop as population density increases. Consistent with the plots of mean degree shown in figure 2, the results for sites BHP and ROB appear to show a similar pattern to each other, as does the pairing PLJ and KCS. (Online version in colour.)

Figure 4.

The results for the dissimilarity measure, D, quantifying the difference between an observed spatial graph and the family of proximity graphs based on the underlying point pattern. The values of D for the 32 networks from each site are plotted against estimated population density. The same pattern, that the observed graphs tend to become more similar to proximity graphs as network size increases, is replicated at each site. (Online version in colour.)

Figure 5.

The model parameter λ (see Material and methods) which represents the strength of spatial constraints on contact rates between voles, were estimated for each of the 32 networks from each of the four sites and plotted against estimated population density. The results for the four sites are consistent and show a positive correlation. Pearson product–moment correlation coefficients are given as insets. (Online version in colour.)