Abstract
In the simple mean-field SIS and SIR epidemic models, infection is transmitted from infectious to susceptible members of a finite population by independent p-coin tosses. Spatial variants of these models are considered, in which finite populations of size N are situated at the sites of a lattice and infectious contacts are limited to individuals at neighboring sites. Scaling laws for these models are given when the infection parameter p is such that the epidemics are critical. It is shown that in all cases there is a critical threshold for the numbers initially infected: below the threshold, the epidemic evolves in essentially the same manner as its branching envelope, but at the threshold evolves like a branching process with a size-dependent drift. The corresponding scaling limits are super-Brownian motions and Dawson–Watanabe processes with killing, respectively.
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This work was supported by NSF grant DMS-0405102.
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Lalley, S.P. Spatial epidemics: critical behavior in one dimension. Probab. Theory Relat. Fields 144, 429–469 (2009). https://doi.org/10.1007/s00440-008-0151-0
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DOI: https://doi.org/10.1007/s00440-008-0151-0
Keywords
- Spatial epidemic
- Branching random walk
- Dawson–Watanabe process
- Critical scaling
Mathematics Subject Classification (2000)
- Primary: 60H30
- Secondary: 60K35