Space and contact networks: capturing the locality of disease transmission

Abstract

While an arbitrary level of complexity may be included in simulations of spatial epidemics, computational intensity and analytical intractability mean that such models often lack transparency into the determinants of epidemiological dynamics. Although numerous approaches attempt to resolve this complexity-tractability trade-off, moment closure methods arguably offer the most promising and robust frameworks for capturing the role of the locality of contact processes on global disease dynamics. While a close analogy may be made between full stochastic spatial transmission models and dynamic network models, we consider here the special case where the dynamics of the network topology change on time-scales much longer than the epidemiological processes imposed on them; in such cases, the use of static network models are justified. We show that in such cases, static network models may provide excellent approximations to the underlying spatial contact process through an appropriate choice of the effective neighbourhood size. We also demonstrate the robustness of this mapping by examining the equivalence of deterministic approximations to the full spatial and network models derived under third-order moment closure assumptions. For systems where deviation from homogeneous mixing is limited, we show that pair equations developed for network models are at least as good an approximation to the underlying stochastic spatial model as more complex spatial moment equations, with both classes of approximation becoming less accurate only for highly localized kernels.

Figure 1

Infectious individuals i in the spatial model pose an infection hazard βU(x_j−y_i) to susceptible neighbours j, with the dependence on distance being determined by the kernel U(r) (here schematically represented by the greyscale shading around each individual). Linking all (i, j) pairs according to this spatial kernel allows one to define an equivalent contact network with a fixed infection hazard between pairs of τ=(β/n), where the mean neighbourhood size $n=1/\int U{(r)}^2\text{d}\mathit{r}$ and the network connectivity $\varphi =2\pi n\iiint U(R)U(r^{\prime })U(r)r^{\prime }\text{d}{r}^{\prime }r\text{d}r\text{d}\theta$, so that the corresponding network may be parameterized directly from knowledge of U(r) (see §2).

Figure 2

Comparing the moment equations (3.2) and ODEs (3.7) for an offset power-law, exponential and Gaussian kernel for the mean infective densities $\overline{I}(t)$ and average S–I covariance ${\overline{c}}_{SI}(t)$ over time. The ODEs are plotted for best-fit k and with all other parameters as per table 1.

Figure 3

Comparing the SIR dynamics resulting from the average of 100 stochastic realizations of the full spatial model, the spatial model under power-3 closure and the pair equations derived from the mappings for an offset power-law kernel. Parameters as per figure 2 except $\overline{I}(t=0)=0.001$, the basic reproduction number R₀=4 and the infectious period 1/γ=7 days. The pair equations are plotted for the four relevant k values in table 1.