the context is epidemiology: there are 3 types of infected people in a human population (and the number of susceptibles is considered infinite).
At each time-step, an individual of type $i$ can become a type $j$ with probability $s_{ji}$ and can get out of the infected population (be "removed") with probability $1-s_{1i}-s_{2i}-s_{3i}$. Independently from this process, an individual of type $i$ recruits $\cal{F}_{ji}$ individuals from the susceptible population as type $j$ infected. At inception, there is one infected individual in type 1.
From what I understand (and that is not much), this is a typical multi-type branching process, from each we know the expected growth rate of the infected population, the extinction probability, etc...
But, is it possible from it, to derive the variance or the distribution of the number of $n^{th}$ generation infected individuals caused by the original infected individual after $t$ timesteps? For $n=2$ and $t=3$, for example, can one calculate $Pr_{2,3}(X=k)$, the probability that after 3 time-steps, there are, among the infected people in the population, $k$ individuals that were infected by someone themselves infected by the original infected individual ? or the variance of the number of such individuals?
For the expectation, I put the expectations of the two processes $s_{ji}$ and $\cal{F}_{ji}$ into matrix form and I look at the sum of all combinations of 3 matrices (3 time-steps), of which two are recruitment matrices: $\bf{F}\bf{F}\bf{S}+\bf{F}\bf{S}\bf{F}+\bf{S}\bf{F}\bf{F}$. The first column of that resulting matrix gives me the expected number of 2nd generation infected individuals in each type after 3 time-steps (from a type 1 initial individual).