A simple model of a disease in a population can be formulated using the premise that the time rate of change of the number of infected individuals is proportional to the product of the number of those infected with the number of those not infected. Let the total population be Nt and let the initial number of those infected be No. Let N(t) stand for the number of those infected at time t. Constant of proportionality is 1.
So I need to set up an IVP for N(t), t > or = 0
and solve it.
Heres what I have:
dN/dt = N(t)(Nt - No) note: (Nt - No) = #people not infected
N'/N = (Nt-No)
N(t) = e^((Nt-No)t + k)
This is where ive gotten to and I feel its wrong. This function says that the number of infected individuals goes to infinity but doesnt it converge to the total population Nt realistically?Any help is appreciated.