Space and contact networks: capturing the locality of disease transmission.

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.