On the Critical Parameters of Branching Random Walks

Given a discrete spatial structure X, we define continuous-time branching processes ${\left\{{\eta }_t\right\}}_{t\ge 0}$ that model a population breeding and dying on X. These processes are usually called branching random walks, and ${\eta }_t\left(x\right)$ denotes the number of individuals alive at site x at time t. They are characterised by breeding rates $k_{xy}$ (governing the rate at which individuals at x send offspring to y) and by a multiplicative speed parameter $\lambda$. These processes also serve as models for epidemic spreading, where $\lambda k_{xy}$ represents the infection rate from x to y. In this context, ${\eta }_t\left(x\right)$ represents the number of infected individuals at x at time t, and the removal of an individual is due to either death or recovery. Two critical parameters of interest are the global critical parameter ${\lambda }_w$, related to global survival, and the local critical parameter ${\lambda }_s$, related to survival within finite sets (with ${\lambda }_w\le {\lambda }_s$). In disease or pest control, the primary goal is to lower $\lambda$ so that the process dies out, at least locally. Nevertheless, a process that survives globally can still pose a threat, especially if sudden changes cause global survival to transition into local survival. In fact, local modifications to the rates can affect the values of both critical parameters, making it important to understand when and how they can be increased. Using results on the comparison of the extinction probabilities for a single branching random walk across different sets, we extend the analysis to the extinction probabilities and critical parameters of pairs of branching random walks whose rates coincide outside a fixed set $A\subseteq X$. We say that two branching random walks are equivalent if their rates coincide everywhere except on a finite subset of X. Given an equivalence class of branching random walks, we prove that if one process has ${\lambda }_w^{*}\ne {\lambda }_s^{*}$, then ${\lambda }_w^{*}$ is the maximal possible value of this parameter within the class. We describe the possible configurations for the critical parameters within these equivalence classes.