Contact process on interchange process

Abstract

We introduce a model of epidemics among moving particles on any locally finite graph. At any time, each vertex is empty, occupied by a healthy particle, or occupied by an infected particle. Infected particles recover at rate $1$ and transmit the infection to healthy particles at neighboring vertices at rate $\hat{I}»$. In addition, particles perform an interchange process with rate $\mathsf{v}$, that is, the states of adjacent vertices are swapped independently at rate $\mathsf{v}$, allowing the infection to spread also through the movement of infected particles. On ${\mathbb{Z}}^d$, we start with a single infected particle at the origin and with all the other vertices independently occupied by a healthy particle with probability $p$ or empty with probability $1-p$. We define $\hat{I}{»}_c(\mathsf{v},p)$ as the threshold value for $\hat{I}»$ above which the infection persists with positive probability and analyze its asymptotic behavior as $\mathsf{v}\to \infty$ for fixed $p$.

59 pages, 3 figures