I was contemplating an epidemic model where infection and recovery rates are determined by links. Here node $i$ is infected first and recovers at a rate $\mu_i$. For all other nodes, the recovery is based on a Poisson process along the link where they are infected. Here $i$ could infect $j$ at rate $\lambda_{ij}$. $j$ recovers at rate $\mu_{ij}$ and while it is infected, it infects $l$ at rate $\lambda_{jl}$. All nodes recover when $i$ recovers - that is, when $l$ is infected through $j$, it recovers at rate $\mu_{jl}+\mu_{ij}+\mu_i$, as all rates are independent and Poisson.
I am interested in computing probabilities of paths of disruption - $\mathrm{P}(v_j=1\mid v_i=1, v_l=1)$. I think the probability that path $i\to j \to l$ is disrupted must be $\frac{\lambda_{ik}}{\lambda_{ik}+\mu_i}\frac{\lambda_{kj}}{\lambda_{kj}+\mu_i+\mu_{ij}}$, but how does one recover conditional probabilities of the form $\mathrm{P}(v_j=1\mid v_i=1, v_l=1)$?
