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

Abstract

In this paper we are concerned with the SIR (Susceptible-Infective-Removed) epidemic on open clusters of bond percolation on the squared 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 there is only one infective vertex at t = 0 and define the critical value of the model as the maximum of the infection rates with which infective vertices die out with probability one, then we show that the critical value is 1 + o(1) /(2dp) as d → +∞, where d is the dimension of the lattice and p is the probability that a given edge is open. Our result is a counterpart of the main theorem in [4] for the contact process.

The critical infection rate of the high-dimensional two-stage contact process

“…In Section 4, we give the proof of lim sup d→+∞ 2dλ c (d, γ, δ) ≤ 1 + 1+δ γ . The proof is inspired by the approach introduced in [7]. We will introduce a two-stage SIR(susceptible-infected-recovered) model, the critical infection rate of which is an upper bound of λ c (d, γ, δ).…”

“…In this section we give the proof of lim sup d→+∞ 2dλ c (d, γ, δ) ≤ 1 + 1+δ γ . The proof is inspired by the approach introduced in [7].…”

“…According to Lemma 3.4 of [7], there exists M 1 , M 2 which do not depend on C 1 , C 2 and the dimension d of the lattice that…”

“…i B j ) P (B i )P (B j ).For the proof of Proposition 4.3, see Section 3 of[7]. At last we give the proof of Lemma 4.2.…”

Survival probabilities of high-dimensional stochastic SIS and SIR models with random edge weights

“…We find out that the SIR model is a useful auxiliary tool for us to accomplish our objective and similar conclusion holds for the SIR model simultaneously according to our proof. We are inspired a lot by the technique introduced in [14], which gives asymptotic behavior of the critical value of the high-dimensional SIR model on clusters of bond percolation.…”