  This paper is concerned with contact process with random vertex weights on
regular trees, and study the asymptotic behavior of the critical infection rate
as the degree of the trees increasing to infinity. In this model, the infection
propagates through the edge connecting vertices $x$ and $y$ at rate
$\lambda\rho(x)\rho(y)$ for some $\lambda>0$, where $\{\rho(x),{x\in T^d}\}$
are $i.i.d.$ vertex weights. We show that when $d$ is large enough there is a
phase transition at $\lambda_c(d)\in(0,\infty)$ such that for
$\lambda<\lambda_c(d)$ the contact process dies out, and for
$\lambda>\lambda_c(d)$ the contact process survives with a positive
probability. Moreover, we also show that there is another phase transition at
$\lambda_e(d)$ such that for $\lambda<\lambda_e(d)$ the contact process dies
out at an exponential rate. Finally, we show that these two critical values
have the same asymptotic behavior as $d$ increases.
