What is the probability of a virus spreading through a network given a virus source node? 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 $t&gt;0$? 

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 (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.
there are many "spread" models, what you request does seem to have been studied. eg this paper builds a framework to analyze the difference &amp; distinguish between random virus spreading and network- (graph-) based spreading.


Network Forensics: Random Infection vs Spreading Epidemic Milling, Caramanis, Mannor, Shakkottai



  Computer (and human) networks have long had to contend
  with spreading viruses. Effectively controlling or curbing an
  outbreak requires understanding the dynamics of the spread.
  A virus that spreads by taking advantage of physical links
  or user-acquaintance links on a social network can grow explosively
  if it spreads beyond a critical radius. On the other
  hand, random infections (that do not take advantage of network
  structure) have very different propagation characteristics.
  If too many machines (or humans) are infected, network
  structure becomes essentially irrelevant, and the different
  spreading modes appear identical. When can we distinguish
  between mechanics of infection? Further, how can this be
  done efficiently? This paper studies these two questions.
  We provide sufficient conditions for different graph topologies,
  for when it is possible to distinguish between a random
  model of infection and a spreading epidemic model, with
  probability of misclassification going to zero. We further
  provide efficient algorithms that are guaranteed to work in
  different regimes.