Epidemics with avoidance and isolation on $\mathbb{Z}^d$
The contact process with avoidance is a generalization of the classical contact process (SIS epidemic) that introduces a mechanism for healthy individuals to avoid their infected neighbors. Let $G$ be a directed graph. At each time $t$, each vertex is either healthy or infected and each edge is either active or inactive. Each infected vertex infects each healthy neighbor across each active edge at rate $\lambda$ and recovers at rate $1$, while each active edge pointing from an infected vertex to a healthy vertex becomes inactive at rate $\alpha$. An inactive edge becomes active when its tail vertex recovers. This model has been previously studied on $\mathbb{Z}$, the $n$-cycle $\mathbb{Z}_n$, and the $n$-star graph; here we extend the study of this model to lattices $\mathbb{Z}^d$, $d \geq 2$. We show that for every $d \geq 2$ and fixed $\alpha > 0$, there exist constants $\lambda(\alpha,d)^-$ and $\lambda(\alpha,d)^+$ such that for all $\lambda < \lambda(\alpha,d)^-$ the infection dies out almost surely while for all $\lambda > \lambda(\alpha,d)^+$ the infection persists indefinitely with positive probability. Furthermore, we show that both $\lambda(\alpha,d)^-$ and $\lambda(\alpha,d)^+$ scale like $1/d$ as $d \rightarrow \infty$ and that there exists a constant $C(\alpha)$ such that when $\lambda > C(\alpha)/d$ the process has a nontrivial invariant measure for $d$ sufficiently large. Our methods and most of our results also apply to the SIRS model and a related model in which infected vertices enter an isolated state at rate $\alpha$ and transition from both isolated and infected to healthy at rate $1$.