Extinction and Survival for a Generalized Contact Process with Deterministic Cures

We study a generalized contact process on ${\mathbb{Z}}^d$ parameterized by an infection rate $\lambda$ and a resetting probability $p\in [0,1]$, modeling deterministic cure times. Once a vertex is infected, its recovery is scheduled exactly one time unit later. Incoming attempts to an already-infected vertex successfully reset its recovery clock with probability $p$, and are ignored otherwise. This unifies spatial versions of classical Type I ($p=0$, non-paralyzable) and Type II ($p=1$, paralyzable) counters. Except in the fully resetting case, deterministic recovery deadlines destroy coordinatewise attractiveness and create a causal shielding effect. Using a first-moment bound on potential causal chains, we prove that the process dies out from finite configurations whenever $\lambda <1\mathrm{/}(2d)$, uniformly in $p\in [0,1]$. The same causal-chain bound yields local convergence to the empty configuration from arbitrary initial states. In dimensions $d\ge 2$, an oriented percolation exploration based on the first infective window establishes global survival for all $p\in [0,1]$ when $\lambda >-\log (1-p_c^{\mathrm{o}\mathrm{r}})$. Finally, in the purely non-resetting case $p=0$, we derive a delayed identity for the one-site density and prove that this density remains strictly between zero and one at every finite time.