• Dec 14th 2009, 08:34 PM
Alice96
Under certain circumstances, the rate at which a disease spreads through a population of P individuals is proportional to the product of the number I of people infected and the number not infected

a) Write and equation for dI/dt

b) Invert to get dt/dI

c) Find t as a function of I, assuming time is 0 at the moment when half the population is infected

d) Find I as a function of t

Hint for part c: What happens when you combine 1/I+1/(P-1) into a single fraction.

Any help that you can offer is appreciated!
• Dec 14th 2009, 08:49 PM
dedust
Quote:

Originally Posted by Alice96
Under certain circumstances, the rate at which a disease spreads through a population of P individuals is proportional to the product of the number I of people infected and the number not infected

a) Write and equation for dI/dt

b) Invert to get dt/dI

c) Find t as a function of I, assuming time is 0 at the moment when half the population is infected

d) Find I as a function of t

Hint for part c: What happens when you combine 1/I+1/(P-1) into a single fraction.

Any help that you can offer is appreciated!

a) $\displaystyle \frac{dI}{dt}=I(P - I)$
b) $\displaystyle \frac{dt}{dI}=\frac{1}{I(P - I)}$
c) $\displaystyle \frac{dt}{dI}=\frac{1}{I(P - I)}=\frac{1}{PI} + \frac{1}{P(P-I)}$, so $\displaystyle dt = \frac{1}{PI} ~dI+ \frac{1}{P(P-I)}~dI$. solve this differential equation.
d) solve part (a)