Contact process on interchange process
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 $λ$. 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 $λ_c(\mathsf{v}, p)$ as the threshold value for $λ$ above which the infection persists with positive probability and analyze its asymptotic behavior as $\mathsf{v} \to \infty$ for fixed $p$.