I considered the problem of a form of CTMC evolving in a graph:


Consider a graph of $G(V,E)$ with $|V|=N$ nodes. Each node has a 1-0 CTMC associated with it:


* There is a vertex dependent rate $\mu\_i$ such that node $i$ moves from 1 to 0 at a Poisson rate $\mu\_i$.
* There is an edge-dependent Poisson rate $\lambda\_{ij}$ along the edge $e(i,j)\in E$ which takes $j$ from 0 to 1 whenever node $i$ is in state 1.
* As long as node $j$ is in state 1, its neighbours in state 0 could each move to state 1 at a Poisson rate of $\lambda\_{jk}$ where $k$ is one of the neighbours of $j$. A node in state 0 cannot influence the states of its neighbouring nodes.
* If node $j$ is in state 0, and nodes $i$ and $k$ are its neighbours in state 1, the probabilities of $i$ changing $j$'s state and of $k$ changing $j$'s state are mutually exclusive.


In such a setting, if $X(0)$ represents the set of nodes in state 1 at time 0 and $T\_j$ the random variable of the first time when $j$ moves into state 1, how do we determine $E(T\_j|X(0)=\{i\})$? 


Could someone provide me with relevant literature pertaining to such problems?



First hit time in a graph setting
This model (at least when all the $\lambda\_{ij}$ are the same and all the $\mu\_i$ are the same) is called the [Contact Process](https://en.wikipedia.org/wiki/Contact_process_(mathematics)).


