Solving Markov Chain Problem: Probability of Infection

[daneault23]

Homework Statement

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?

Homework Equations

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 S
I | 1 0
S | p 1-p

 

[daneault23]

Nevermind guys. I think I got it. For every susceptible individual, there will be n-1 individuals, with the chance of coming in contact with any of them is p. If k of these individuals are infected, the probability of the susceptible individuals coming in contact with none of them is (1-p)^k. The probability that the susceptible individual will come in contact with at least 1 of the infected people from time t to time t+1 (and therefore become infected) is 1-(1-p)^k.

 

[Ray Vickson]

Homework Statement

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?

Homework Equations

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 S
I | 1 0
S | p 1-p

Start again: your transition probability matrix does not look anything like what the problem description dictates.

You need to worry about the following: if I = N-S are the # infected, how many of the C(N,2) pairs contain no infected individuals? How many contain exactly one infected? How many contain two infected? So, in the next time period, what is the probability distribution of the number of new infections?