  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 \lambda_c as
the supremum of infection rates which causes extinction of the process at
equilibrium, we prove that n\lambda_c^{T^n}\rightarrow1 and
2d\lambda_c^{Z^d}\rightarrow1 as n,d\rightarrow+\infty. Our result is a
development of the conclusion that \lambda_c^{Z^d}\leq\frac{2.18}{d} shown in
\cite{Dur1991}. 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\rightarrow+\infty}2dP\big(\exists n\geq1,
S_n^{(d)}=0\big)=1, where S_n^{(d)} is a simple random walk on Z^d with
S_0^{(d)}=0.
