logistics and integral calculas

Quote:

Originally Posted by softdrink_jr

Can somebody please help me with this homework question. I haven’t got much of an idea on what to do!

Infection rate of a disease has a constant of 0.004, this may be represented mathematically by -

dN/dt = 0.0004N(1000-N), N(0)=1

1. Using Integral calculus derive a model to determine the number of people with this disease and any time “t”.

2. Use this model to predict how long it will take for 75% of the population will be infected. Population = 1000

3. Discuss the limitations of the model for long term predictions.

$\frac{dN}{dt} = 0.0004N(1000-N)$
$\implies \frac{dN}{0.0004N(1000-N)} = dt$
taking the integral of both sides:
$\int \frac{dN}{0.0004N(1000-N)} = \int \left( {\frac{0.001}{0.0004N} + \frac{2.5}{1000-N}} \right)dN = t + c$
if my partial fraction is correct, then you can continue from here..