Discrete Markov Epidemics

Hello everyone!

For a programming task I have to use a discrete time Markov Model. A population consists of n individuals of which m are initially infected.

The progression of the disease is illustrated in the graph below, where

s = susceptibles
i = new infectives
j remaining infectives of previous peiriod
d = diseased
r = removed

All probabilities are given: mu_t depends on i_t,j_t and d_t

Attachment 21224

I've already used a multivariate Markov chain with the tupels (s,i,j,d) as state space but the transition matrix becomes huge if
the population size increases.

I've also heard of total size approximation models that use the number of deaths as time scale that simplifies the model as only one infective is considered in one time step. The problem is that I need to know the distribution after a certain time.

I'm looking forward to receive any kind of idea to solve this. It has to be a markov chain - so no SDEs are possible.

Thanks in advance
C.Sager