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 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 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 $t$. If $G$ is the semi-line, then for all $\gamma >0$ almost surely ${lim\; sup}_t\frac{H_{t}loglogt}{logt}=1$ and ${lim\; inf}_t{H}_t=0$. In contrast, if $G$ is an infinite Galton–Watson tree with offspring mean $m>1$ then, for all $\gamma >0$, almost surely ${lim\; inf}_t\frac{H_{t}logt}{t}\ge m-1$ and ${lim\; inf}_t\frac{M_{t}loglogt}{t}\ge m-1$, while ${lim\; sup}_t\frac{M_t}{e^{ct}}\le 1$, for all $c>m-1$. 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 ${lim\; sup}_t\frac{M_t}{t}\le \varepsilon$.