Probability of Bob Being Infected with Spread of Disease

[mac11]

Assume there is a population of a given (even) size. One person is ‘infected’ in the beginning. During every ‘round’, everybody in the population pairs off and ‘interacts’ with her partner. If an infected person interacts with an uninfected person, the uninfected person is then infected. If two infected people interact, there is no change. Let’s say my favorite guy in this population is Bob. After n rounds, what is the probability Bob is infected?
Further, if an infected person interacts with an uninfected person, assume there is a known probability, p, that the uninfected person will get infected. What is the new probability Bob will be infected?
You can probably guess what this problem is attempting to model. I’m guessing the answer is recursive, so do your best.

 

[Office_Shredder]

Do they always interact with the same partner? If so, the odds of Bob being infected are quite low, because he's either going to get infected in the first round, or never

 

[jordi]

Assume there is a population of a given (even) size. One person is ‘infected’ in the beginning. During every ‘round’, everybody in the population pairs off and ‘interacts’ with her partner. If an infected person interacts with an uninfected person, the uninfected person is then infected. If two infected people interact, there is no change. Let’s say my favorite guy in this population is Bob. After n rounds, what is the probability Bob is infected?
Further, if an infected person interacts with an uninfected person, assume there is a known probability, p, that the uninfected person will get infected. What is the new probability Bob will be infected?
You can probably guess what this problem is attempting to model. I’m guessing the answer is recursive, so do your best.

The problem is not defined. At least, you need to add the topology.

 

[mac11]

Do they always interact with the same partner? If so, the odds of Bob being infected are quite low, because he's either going to get infected in the first round, or never

No, the partner is chosen randomely each round.

I think I explained the problem pretty well, not sure what you mean by 'adding a topology'.

But again, this happens in discrete 'rounds'. Obviously, the population will (theoretically) propagate at a rate of 2^n (doubling every round), but this is unlikely to happen because two infected people will meet and 'slow down' this rate. If the population is extremely large, however, you can expect the disease to spread exponentially at least initially.