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 γ>0, when a clock rings the corresponding red vertex turns black. For t≥0, let Ht and Mt 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 γ>0 almost surely lim suptHtloglogtlogt=1 and lim inftHt=0. In contrast, if G is an infinite Galton–Watson tree with offspring mean m>1 then, for all γ>0, almost surely lim inftHtlogtt≥m−1 and lim inftMtloglogtt≥m−1, while lim suptMtect≤1, for all c>m−1. Also, almost surely as t→∞, for all γ>0Ht is of order at most t. Furthermore, if we restrict our attention to bounded-degree graphs, then for any ɛ>0 there is a critical value γc>0 so that for all γ>γc, almost surely lim suptMtt≤ɛ.