**Model**: Consider an infinite undirected connected graph $\mathcal{G}=(\mathcal{V},\mathcal{E})$. At time $t=0$, a given virus node $s\in\mathcal{V}$ starts infecting the network $\mathcal{G}$. Specifically, for each edge $(u,v)\in\mathcal{E}$, if either $u$ or $v$ becomes infected, then the other uninfected node on the edge $(u,v)$ will be infected after a random time following an exponential distribution with rate $\lambda$. Assume the spreading times for each edge are independent and identically distributed. 

**Question**: Now given a connected subgraph $G\subset\mathcal{G}$ ($s\in G$ ), what is the probability that all the nodes in $G$ are infected at time $t>0$? 

I tried to find a mathematical model to track such an infection process on different kinds of networks (trees, grids, regular networks, etc.). However, after checking the classical [epidemic models](http://en.wikipedia.org/wiki/Epidemic_model) (SIS, SIR, etc.), I found these models often use the mean field approximation to find the percentages of population in different states (Suspicious, Infected, etc.) without considering the network structures. Thus I think they might not be useful in modeling the infection process described above. 

Any insights or references to papers/books would be helpful and greatly appreciated.