Asymptotic of the Critical Value of the Large-Dimensional SIR Epidemic on Clusters

Xiaofeng Xue ()
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Xiaofeng Xue: Beijing Jiaotong University

Journal of Theoretical Probability, 2018, vol. 31, issue 4, 2343-2365

Abstract: Abstract In this paper we are concerned with the susceptible-infective-removed (SIR) epidemic on open clusters of bond percolation on the square lattice. For the SIR model, a susceptible vertex is infected at rate proportional to the number of infective neighbors, while an infective vertex becomes removed at a constant rate. A removed vertex will never be infected again. We assume that at $$t=0$$ t = 0 the only infective vertex is the origin and define the critical value of the model as the supremum of the infection rates with which infective vertices die out with probability one; then, we show that the critical value under the annealed measure is $$\big (1+o(1)\big )/(2dp)$$ ( 1 + o ( 1 ) ) / ( 2 d p ) as the dimension d of the lattice grows to infinity, where p is the probability that a given edge is open. Furthermore, we show that the critical value under the quenched measure equals the annealed one when the origin belongs to an infinite open cluster of the percolation.

Keywords: SIR model; Critical value; Percolation; 60K35 (search for similar items in EconPapers)
Date: 2018
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DOI: 10.1007/s10959-017-0780-2

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