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la6ki
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Homework Statement
The native Hawaiians lived for centuries in isolation from other peoples. When foreigners finally came to the islands they brought with them diseases such as measles, whooping cough, and smallpox, which decimated the population. Suppose such an island has a native population of 5000, and a sailor from a visiting ship introduces measles, which has an infection rate of 0.00005. Also suppose that the model for spread of an epidemic described in Example 3* applies.
a. Write an equation for the number of natives who remain uninfected. Let t represent time in days.
Homework Equations
*The model for spread of an epidemic described in Example 3: \frac{dy}{dt}=k(1-y/N)y
Where in modeling population growth y is the population, k is a growth constant, and N is the carrying capacity.
The Attempt at a Solution
Okay, so the general solution to this differential equation is:
y=\frac{N}{1+be^{-kt}}
where b=\frac{N-y_{0}}{y_{0}}
(y_{0} is the initial population size).
If we apply this model to the current problem, y could represent the number of people affected. Therefore, the number of people unaffected is x=N-y. Solving for x:
N-\frac{N}{1+be^{-kt}}
Setting y_{0}=1 (since there is one affected individual initially), the solution of the above equation is:
x=\frac{24955000}{4999+e^{0.00005t}}
Now, I thought I had solved the problem, but when I looked at the correct answer, it was:
x=\frac{24955000}{4999+e^{0.25t}}
I checked several times if I made a mistake anywhere in my calculations, but I couldn't find it (not that it's still not possible I keep missing something obvious; hopefully you guys will help me spot it if so).
I went back to the chapter where the logistic function was introduced for the first time and there it said (right after the general solution was given):
"This function was introduced in Section 4.4 on Derivatives of Exponential Functions in the form:
G(t)=\frac{m}{1+(m/G_{0})e^{-kmt}}
where m is the limiting value of the population, G_{0} is the initial number present, and k is a positive constant."
First of all, I can't see how this equation follows from N-\frac{N}{1+be^{-kt}} (or how the latter could be rewritten in this form. But second, I tried using it and still couldn't get the correct answer. I decided to go back to the chapter where this second form was first introduced and I found the following:
G(t)=\frac{m}{1+(m/G_{0}-1)e^{-kmt}}
Which is different from the one above it (there is an extra negative one in the denominator). This is all very confusing. First of all, is this a just a typo (that is, they just missed writing the negative in the first equation I wrote above) or am I missing something? Second, when I now plug in m and G_{0}, I do get the correct response. And the reason is, there is an extra m at the exponent of the e term. Where did it come from? I keep solving and resolving the differential equation \frac{dy}{dt}=k(1-y/N)y and simply can't get to this general solution.
I am very confused and have been stuck on this problem. Any input will be appreciated.