# Thread: Diff equation word problem

1. ## Diff equation word problem

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.

Chris

2. Originally Posted by chrsr345
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 Mr F says: This is poor notation. It's confusing and I doubt that what it represents is even correct.

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.

Chris
All the t's are obfusicating. The notation is poor. Your feelings are correct.

Let M be the total population. The DE is $\frac{dN}{dt} = N (M - N)$.

The solution is $N = \frac{A M e^{Mt}}{1 + A e^{Mt}} = \frac{A M}{e^{-Mt} + A}$ where A is an arbitrary constant.