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

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  • Xiaofeng Xue

    (Beijing Jiaotong University)

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.

Suggested Citation

  • Xiaofeng Xue, 2018. "Asymptotic of the Critical Value of the Large-Dimensional SIR Epidemic on Clusters," Journal of Theoretical Probability, Springer, vol. 31(4), pages 2343-2365, December.
    • Handle: RePEc:spr:jotpro:v:31:y:2018:i:4:d:10.1007_s10959-017-0780-2
    DOI: 10.1007/s10959-017-0780-2
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    References listed on IDEAS

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    1. Cox, J. T. & Durrett, Richard, 1988. "Limit theorems for the spread of epidemics and forest fires," Stochastic Processes and their Applications, Elsevier, vol. 30(2), pages 171-191, December.
    2. Xue, Xiaofeng, 2017. "Law of large numbers for the SIR model with random vertex weights on Erdős–Rényi graph," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 486(C), pages 434-445.
    3. Xue, Xiaofeng, 2016. "Critical value for contact processes on clusters of oriented bond percolation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 448(C), pages 205-215.
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