Spread of a Virus Through a CityDate: 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/ |
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