Dynamics for the spatial mean

Since the dynamics of the total number of infected depends on the number of neighboring pairs due to the non-linearity in the transition rates, e.g. W1-Iiji ~ Ii ■ I?, we need to examine clusters of sites. The methods we use here are in analogy with the methods used for the non-spatial master equations.

We consider statistics for the number of clusters with certain shapes, starting with the number of single sites that are infected. For the total number of infected sites we have [I] [S] := i=1 (1 — Ii). For pairs we have [II] and triples [III] := £^ £^

Ii and respectively

Sfc=1 Jij Jjk • IiIjIk or triangles [A] := 2fc=i Jij Jjk Jki • IiIjIk and so on. These spatial averages, e.g [I] := N=1 Ii, depend on the ensemble (I1,...,IN) which changes with j = 1 Jij Ii

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