The truncation property and continuity for the long-range contact process on $\mathbb{Z}^d$

We consider a general class of contact processes on $\mathbb{Z}^d$ with potentially long-range interactions. By adapting well established renormalization arguments to the long-range setting we extend by now classical results for finite-range processes to this more general setting. Particularly, we provide general conditions on the decay of the interactions under which a supercritical process remains supercritical after truncation of the interaction parameter at a sufficiently large distance. Further, for the family of parameters satisfying this latter truncation property, we conclude that the probability of the process to never recover is continuous.