There is a complete graph, there are $n$ infected people out of a total of $N$. Every period, every infected person has some probability, $p$ of spreading to each of his neighbors(everyone who isn't infected since its a complete graph).
If there are $n$ infected people out of $N$, what is the probability that there are $x$ people are infected next period ($x>n$)?
I had in the past posted something similar which was completely wrong.
For $i=1,\dots, N-n$ let $B_i$ be iid random variables with Bernoulli distribution with parameter: $$r:=1-(1-p)^n$$
Here $N-n$ corresponds with the number of uninfected persons and $r$ is the probability that such a person will be infected during the period.
Then the number of persons that are infected after the period can be written as:$$X=n+\sum_{i=1}^{N-n}B_i$$
Here $\sum_{i=1}^{N-n}B_i$ has binomial distribution with parameters $N-n$ and $r$.
Based on that we can find: $$P(X=x)=P\left(\sum_{i=1}^{N-n}B_i=x-n\right)=\binom{N-n}{x-n}r^{x-n}(1-r)^{N-x}$$