In Tijms's book Understanding Probability, I found the following example.
Consider a population of $m$ individuals, some of whom are infected and the others are susceptible. Initially one individual is infected. The times between contacts between any two individuals of the population have an exponential distribution with rate $\mu$. The times between the various contacts are independent of each other. If a susceptible individual comes in contact with an infected individual, the susceptible individual becomes infected. Once infected, an individual stays infected. What stochastic process describes the spread of the disease?
Solution. The key to the solution is the fact that the minimum of $r$ independent random variables having exponential distributions with rates $\mu_1,\dots,\mu_r$ is again exponentially distributed with rate $\mu_1 + \dots + \mu_r$. Denote by the random variable $X(t)$ the number of infected individuals at time $t$. Then, by the memoryless property of the exponential distribution, the process $\{X(t)\}$ is a continuous-time Markov chain with state space $I = \{1, 2, \dots , m\}$. How do we find the transition rates? If there are $i$ infected individuals, the time until one of the $m -i$ susceptible individuals becomes infected is the smallest of $(m - i)i$ independent exponentials each having rate $\mu$ and hence this time has an exponential distribution with rate $(m − i)i\mu$. It now follows that the transition rates $q_{i,i+1}$ are given by $$ q_{i,i+1} = (m − i)i\mu \qquad \text{for }i = 1, 2, \dots, m - 1.$$ The other transition rates $q_{ij}$ are zero because of the assumption that an infected individual stays infected once it is infected. Note that state $m$ is an absorbing state with leaving rate $\nu_m = 0$.
I mostly agree with the solution, but I do not understand why $i$ infected individuals interact with $m-i$ susceptible individuals with probability 1.
I think that this solution does not consider the case that some of the $i$ infected individuals can interact between themselves (the wording of the problem does not exclude this possibility).
Shouldn't we keep in mind this thing when calculating the transition rates? If so, how to model this more complex disease propagation model?