I'm trying to model how a population would grow wrt the following constraints:
1. Number of people within the population
2. How many people each person of the population informs
3. Probability of each person of the population informing a member outside the population.
4. There is an upper limit to the population growth, but this is not the most important constraint as of now.
It's almost like a virus growth model and I have some idea of where to begin, but it's been a while since I've solved these problems.
Because of the first constraint, this looks like an exponential growth model to begin with. Thus,
is the basic equation to be used.
The Attempt at a Solution
From the second constraint onwards, I'm having a problem forming the DE.
The growth is dependent on the probability of each person within the population telling others about it and the number of people each person will tell.
Thus, if [tex]\alpha (t)[/tex] is the probability of a person within the population telling others
and [tex]m(t)[/tex] are the number of people each person tells, then I think the model would look like this:
Is this right? Again, it's been a while since I've done this stuff, so I'm not sure, but I think I'm on the right track.
For the final constraint, if I have to put an upper limit on the model, I think I can do a (k-N) growth model where k is the upper limit.