First Passage Percolation with Recovery

Abstract

First passage percolation with recovery is a process aimed at modeling the spread of epidemics. On a graph $G$ place a red particle at a reference vertex $o$ and colorless particles (seeds) at all other vertices. The red particle starts spreading a \emph{red first passage percolation} of rate $1$, while all seeds are dormant. As soon as a seed is reached by the process, it turns red and starts spreading {red first passage percolation}. All vertices are equipped with independent exponential clocks ringing at rate ${\hat{I}}^3>0$, when a clock rings the corresponding \emph{red vertex turns black}. For $t\ge 0$, let $H_t$ and $M_t$ denote the size of the longest red path and of the largest red cluster present at time $t$. %, respectively. If $G$ is the semi-line, then for all ${\hat{I}}^3>0$ almost surely ${\mathrm{limsup}}_t\frac{H_t\log \log t}{\log t}=1$ and ${\mathrm{liminf}}_t{H}_t=0$. In contrast, if $G$ is an infinite Galton-Watson tree with offspring mean $\mathbf{m}>1$ then, for all ${\hat{I}}^3>0$, almost surely ${\mathrm{liminf}}_t\frac{H_t\log t}{t}\ge \mathbf{m}-1$ and ${\mathrm{liminf}}_t\frac{M_t\log \log t}{t}\ge \mathbf{m}-1$, while ${\mathrm{limsup}}_t\frac{M_t}{e^{ct}}\le 1$, for all $c>\mathbf{m}-1$. Also, almost surely as $t\to \infty$, for all ${\hat{I}}^3>0$ $H_t$ is of order at most $t$. Furthermore, if we restrict our attention to bounded-degree graphs, then for any $\epsilon >0$ there is a critical value ${\hat{I}}_c^3>0$ so that for all ${\hat{I}}^3>{\hat{I}}_c^3$, almost surely ${\mathrm{limsup}}_t\frac{M_t}{t}\le \epsilon$.

7 figures, 25 pages