  We study the threshold $\theta$ contact process on $\mathbb{Z}^d$ with
infection parameter $\lambda$. We show that the critical point
$\lambda_{\mathrm{c}}$, defined as the threshold for survival starting from
every site occupied, vanishes as $d\to\infty$. This implies that the threshold
$\theta$ voter model on $\mathbb{Z}^d$ has a nondegenerate extremal invariant
measure, when $d$ is large.
