If someone would be able to point me in a starting direction, I would be greatly appreciative.
Question
I am attempting to derive a closed form for the transition probabilities of the discrete time Markov Chain $(X_n, n\in\mathbb{N})$ in a infection/recovery type process.
Where,
- $Y_n$ : Number of newly infected during the $n^{th}$ period
- $X_n$ : Total number of infected at the start of $n^{th}$ period
- $c\in(0,1)$ : Infection rate
- $r\in(0,1)$ : Recovery rate
- $N\in\mathbb{N}$ : Total Population
And,
$${\left({Y}_{{n}}∣{X}_{{n}}\right)} \sim {Bin}{\left({N}-{X}_{{n}}, {c}\right)},∀{n}∈{\mathbb{N}}$$
$${\left({X}_{{{n}+{1}}}∣{X}_{{n}},{Y}_{{n}}\right)}\sim {Bin}{\left({X}_{{n}}+{Y}_{{n}},{\left({1}-{r}\right)}\right)}, \forall n\in\mathbb{N}$$
My attempt
My initial approach was simply to apply the Law of Total Probability
$$ \\\mathbb{P}{\left({X}_{{{n}+{1}}}={j}∣{X}_{{n}}={i}\right)}={∑_{{{k}={0}}}^{{{N}-{i}}}}\mathbb{P}{\left({X}_{{{n}+{1}}}={j}∣{X}_{{n}}={i},{Y}_{{n}}={k}\right)}\mathbb{P}{\left({Y}_{{n}}={k}∣{X}_{{n}}={i}\right)} \\=\sum _{k=0}^{N-i}\binom{i+k}{j}(1-r)^jr^{i+k-j}\binom{N-i}{k}c^k(1-c)^{N-i-k} \\=\left(\frac{1-r}{r}\right)^j r^i(1-c)^{N-i}\sum _{k=0}^{N-i}\binom{i+k}{j}\binom{N-i}{k}\left(r\times\frac{c}{1-c}\right)^k $$ I then attempted to expand the combinations in terms of their factorial definition, but that didn't yield a closed form solution as far as I could tell.
Not really sure where to go from here. Not sure if I've made a mistake in my approach or if there is some step that will help this summation cancel. I would really appreciate if someone could point out any mistakes I have made and/or give me some guidance in regards to an approach.