In this paper we study threshold-one contact processes on lattices and regular trees. The asymptotic behavior of the critical infection rates as the degrees of the graphs growing to infinity are obtained. Defining λ c as the supremum of infection rates which causes extinction of the process at equilibrium, we prove that nλ T n c → 1 and 2dλ Z d c → 1 as n, d → +∞. Our result is a development of the conclusion that λ Z d c ≤ 2.18 d shown in [2]. To prove our main result, a crucial lemma about the probability of a simple random walk on a lattice returning to zero is obtained. In details, the lemma is that lim d→+∞ 2dP ∃n ≥ 1, S (d) n = 0 = 1, where S (d) n is a simple random walk on Z d with S (d) 0 = 0.

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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.

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