Branching bound for contact-process survival (${\lambda }_c(d)\ge (2d{)}^{-1}$)

With recovery rate one, the number of infected sites in a nearest-neighbour contact process has death rate equal to its size and birth rate at most $2d\lambda$ times its size. It is dominated by a linear birth-death branching process with these per-particle rates. In particular, from one infection its expected size is at most $e^{(2d\lambda -1)t}$, so its survival probability is zero when $2d\lambda <1$. This proves the displayed lower bound on the critical rate.