Large deviations for the contact process in random environment

The asymptotic shape theorem for the contact process in random environment gives the existence of a norm $\mu$ on $\text{Rd}$ such that the hitting time $t(x)$ is asymptotically equivalent to $\mu (x)$ when the contact process survives. We provide here exponential upper bounds for the probability of the event $\{\frac{t(x)}{\mu (x)}\in ̸[1-ϵ,1+ϵ]\}$; these bounds are optimal for independent random environment. As a special case, this gives the large deviation inequality for the contact process in a deterministic environment, which, as far as we know, has not been established yet.