Consider the following (simple) epidemic model: A population of size N consists of infected and susceptible individuals. During each time period, each of the N choose 2 possible pairs in the population will come in contact with probability p. If a pair is in contact and one person in the pair is infected and the other susceptible, then the disease will be transmitted to the infected person. Nobody is ever cured of the disease.
If there are k (k < N) infected individuals at time t in the population, what is
the probability that a specified susceptible person will become infected in the
period t->t + 1?
Don't see any not posted in problem description.
The Attempt at a Solution
Maybe I am making this too hard, but it seems like the answer should be just p. Say there are two people, one infected and one susceptible to the infection. The infected person will always stay infected with probability 1 and the person who is susceptible will become infected with probability p since the probability that they will come in contact with an infected person is just that, p.
A markov chain would look like this I presume, with I=infected, S=susceptible, but not infected
I | 1 0
S | p 1-p