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### 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/
```
Associated Topics:
College Calculus
High School Calculus

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