4- Spatial ModelThe hierarchical model of the previous section allowed us to explore the consequences of localized program sharing. In this section, we shall learn more about the effects of locality by studying epidemics on a completely different topological structure -- a d-dimensional cartesian lattice. Each point in the lattice represents a node which can infect or be infected by all nodes within some local neighborhood. As in the hierarchical model, locality immediately implies the existence of cliques, but their form is somewhat different in the spatial model. For example, consider the neighborhood of infectible nodes surrounding node A. If we move one step to the right to node B, we find that B's neighborhood has many nodes in common with A's. Although it may be less realistic in some respects than the hierarchical model, the spatial model offers the advantage of being more amenable to deterministic analysis. Given our experience that random graphs with sufficient connectivity are reasonably well-described by a deterministic approximation, we expect this to hold for the spatial model as well, provided that the neighborhood is sufficiently large. First, we shall use a deterministic approximation to derive an equation for the spatio-temporal dynamics of an epidemic. Then, we shall confirm these results with simulations.
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