Need help setting up Logistics equation

[obperryo]

Please help me set up this logistics equation.

At t=0, half an alligator population (50,000) has contracted a disease that causes them to loose their teeth. Initial populations is 100,000. Disease spreads at rate of 1000 per day.

I know that dP/dt = kP(M-P), but I am confused in how what is given in the problem relates. I am confused by P, Po and M. A suggestion is to find the value of k by substituting P(0) and P'(0) in the logistic equation above.

Thanks,
Bob

 

[Defennder]

Are you sure this is supposed to be a logistic DE problem? Were you explicitly told it was or are you trying to see if it's logistic? There's a problem using the logistic model to mathematically simulate the above situation.

I'm sure you're aware that the logistic model is used to model birth and death rates of a population, but in this case it is not apparent that those infected with the disease would actually die or be excluded from the count of healthy aligators (corresponding to -kP^2 in the logistic DE) nor is there any indication of a "birth-rate" for aligators or some factor (such as migration etc.) which cause uninfected alligators to increase in numbers over time.

All these factors, as well as the above statement that "disease spreads at rate of 1000/day" , which is apparently a constant rate, appear to signify that the DE you should be using is a lot simpler than the logistic DE.

 

[obperryo]

Well, I am learning, and having a very hard time coming up with an equation for this story problem,so I am not sure on what I should do. This issue has come up before, where the questions don't make complete sense.

Here is the problem I am working exactly as it is written:
Suppose at t=0, half of an alligator population has contracted a disease that causes them to loose their teeth. The population is initially 100,000 and the disease spreads to other alligators at a rate of 1000 per day. How long will it take for the disease to spread to 80% of the population?

Here is a problem in the textbook that I believe the above question was modeled after:
Suppose that at time t=0, half of a "logistic" population of 100,000 persons have heard a certain rumor, and that the number of those who have heard it is then increased at the rate of 1000 persons per day. How long will it take for rumor to spread to 80% of the population?(Suggestion: Find the value of k by substituting P(0) and P'(0) in the logistic equation, Eq(3).
Eq (3) is dP/dt=kP(M-P)

If this don't make sense, how would you set it up for what it is. I will do that and move on. I have already spent an entire day trying to use what the book is teaching me on this problem.

Thanks for your help.

Oh, one last thought... maybe it could be looked at as just the birth rate of the disease (think disease instead of alligator). I think I only need to come up with how long it takes to grow from 50,000 to 80,000. Not sure, what do you think.

 

[HallsofIvy]

This doesn't even look to me like a "differential equation" problem! You are told that "the disease spreads at rate of 1000 per day" so it P(t) is the number of alligators with the disease on day t, then P(t+1)= P(t)+ 10000, with P(0)= 50000.

Hmmm, 80% of 100000= 80000. 80000- 50000= 30000 more alligators will have to get the disease to get up to 80%. Since "the disease spreads at rate of 1000 per day", that will take 30 days!

IF the problem said that a specific percentage of those that do NOT have the disease, or that do not have the disease and come into contact with those that do will contract the disease each day, then I could see modeling with a differential equation but not if there are a specific number who contract it each day.

 

[obperryo]

It think I figured out the approach..:

(in thousands)
P = 100
Po = 50
Rate = 1 = dP/dt
t = 0

What is t when P(t) = 80% of P?

dP/dt = kP(M-P)

1 = k*50(100-50), so k = .0004

Then, once you know k,

P(t) = M*P / Po + (M-Po)exp(-k*M*t)

So then with P(t) = 80, t = 34.6574

about 35 years

Does seem simple at this time... This keeps happening to me... why is it so difficult to see something that seems simple in the end.. any tips would be appreciated.

Thanks for your helps!

 

[HallsofIvy]

Okay, so you are saying that initially, when 50000 alligators already have the disease, 1000 more contract it the next day. That is very different from saying they contract it "at the rate of 1000 per day"!

One thing I always suggest is that people write out exactly what their variables mean and in what units. Is your time, "t", measured in years or days?

 

[obperryo]

Yes, I think the rate is meant to be the rate at t=0 and the only reason the initial population of 100,000 is given is so you can figure out that 80% is 80,000, thus you are looking for the time at a population fo 80,000.

I remember in one of my calculus classes this same thing happened.. That's why I think that the rate refers to t=0. And that is in days... yes, I should of.

Thanks again for the help.

 

[HallsofIvy]

So then with P(t) = 80, t = 34.6574

about 35 years

So if t is in days...

 

[obperryo]

So if t is in days...

ah, yes, 35 days, not years... mis-type copying from my paper.

Just goes to show, like you said, write it out... If I did at the top, I probably wouldn't of made the mistake.