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Limit theorems for the spread of epidemics and forest fires. (English) Zbl 0667.92016
In a certain two-dimensional epidemic model, each site of the square lattice may be in any of three states, 1 (healthy), i (infected), or 0 (immune). A healthy site may be infected by an infected neighbour at a certain rate, and an infected site remains infected for a random time before becoming immune. There is a critical value \(\alpha_ c\) of the infection rate \(\alpha\) which marks the onset of the regime in which an epidemic may take place from a single initial infective.
This model is related to a certain dependent percolation process. A shape theorem is proved for the epidemic process: if \(\alpha >\alpha_ c\), then the set of sites infected at time t from a single initial infective is approximately tC for some convex set C. Related results are discussed for two-dimensional percolation.
Reviewer: G.Grimmett

MSC:
92D25 Population dynamics (general)
60K35 Interacting random processes; statistical mechanics type models; percolation theory
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