First Passage Percolation with Recovery

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 $\gamma>0$, when a clock rings the corresponding \emph{red vertex turns black}. For $t\geq 0$, 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 $\gamma>0$ almost surely $\limsup_{t}\frac{H_t\log\log t}{\log t}=1$ and $\liminf_{t}H_t=0$. In contrast, if $G$ is an infinite Galton-Watson tree with offspring mean $\mathbf{m}>1$ then, for all $\gamma>0$, almost surely $\liminf_{t}\frac{H_t\log t}{t}\geq\mathbf{m}-1$ and $\liminf_{t}\frac{M_t\log\log t}{t}\geq \mathbf{m}-1$, while $\limsup_{t} \frac{M_t}{e^{c t}}\leq 1$, for all $c>\mathbf{m} -1$. Also, almost surely as $t\to \infty$, for all $\gamma>0$ $H_t$ is of order at most $t$. Furthermore, if we restrict our attention to bounded-degree graphs, then for any $\varepsilon>0$ there is a critical value $\gamma_c>0$ so that for all $\gamma>\gamma_c$, almost surely $\limsup_{t}\frac{M_t}{t}\leq \varepsilon$.