Understanding the Contact Rate in SIS Epidemic Modelling

[rickywaldron]

For the probability of the number of infective's increasing in one time step, I found the answer is:
∆tβi(N-i)/N
where β is the contact rate, ∆t is the time step, i is number of infectives,N is total number of susceptible's and infective's

I can't quite see where this is coming from. β is the contact rate, so it makes sense to multiply by i since for more i, more chances of encounters. Then multiplying by N-i makes sense since more susceptible also means more chance of encounters.

But then why the division by N? I think my problem may be in understanding what the "contact rate" actually means

 

[Mandelbroth]

But then why the division by N? I think my problem may be in understanding what the "contact rate" actually means

I think it actually has to do with understanding what a probability is.

What is a probability? What does it tell you?

Answering those questions should tell you the answer to your question.