Spread of a Virus Through a City

Date: 05/09/2002 at 04:49:24
From: Katy
Subject: Virus Calculation

A Flu-like virus is spreading through a city of population 
260,000 at a rate proportional to the product of the number of 
people already infected and the number of people still 
uninfected. If 600 people were infected initially and 30,000 
people were infected after 10 days, how many people will be 
infected after 25 days?

Date: 05/09/2002 at 05:46:21
From: Doctor Mitteldorf
Subject: Re: Viurs Calculation


Dear Katy,

First, translate the specification in the first sentence into a
differential equation.  Then use the second sentence to specify 
two boundary conditions for the solution.

Let p = 260,000, y = number of people infected, t = time.  Then 
the first sentence says 

   dy/dt = A y(p-y),   

where A is a constant to be determined.

This is called the "logistic equation", and it can be solved by
separating variables:  

   Integral [dy / (y(p-y))] = At + B, 

where B is another constant.

You can do the integral by "partial fractions", noting that

  1/(y(p-y)) 

can be written as (1/p) (1/y + 1/(p-y))

Finish solving the differential equation; then use the second
sentence to find A and B.  Note that when t=0, y=600; and when 
t=10, y=30,000. 

- Doctor Mitteldorf, The Math Forum
  http://mathforum.org/dr.math/