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Logistic Growth of A Rumor Spreading


Date: 04/10/98 at 10:39:12
From: Hadhrami_Ab_Ghani
Subject: Re[2]: EXPONENTIAL PROBLEMS

In a country of 3,000,000 people, the prime minister suffers a heart 
attack, which the government does not officially publicize. Initially, 
50 governmental personnel know of the attack, but spread this 
information as a rumor. At the end of one week, 500 people know the 
rumor. Assuming logistic growth, find how many people know the rumor 
after two weeks.

What I want to know is the basic concept of exponent and some formulas 
related to this question.


Date: 04/11/98 at 09:43:08
From: Doctor Anthony
Subject: Re: Re[2]: EXPONENTIAL PROBLEMS

The logistic difference equation assumes that the rate of spread of a 
rumor is proportional to the number who know and the number who don't 
know. If we let x = proportion of the population who know (0 < x < 1), 
then:

     dx/dt = kx(1 - x)    where k is a constant.

        dx
     --------  = k*dt
     x(1 - x)

         1          A        B
     --------   =  ---  +  -----
     x(1 - x)       x      1 - x
 
     1 =  A(1 - x) + Bx

     x = 0   1 =  A
     x = 1   1 =  B
 
So we have:

     INT[1/x + 1/(1 - x)]dx = INT[k*dt]
 
     ln(x) - ln(1 - x) =  kt + constant
 
     ln(x/(1 - x)) = kt + constant

     x/(1 - x) = e^(kt + constant)

     x/(1 - x) = Ae^(kt)            

where A = constant.
 
Then at t = 0:

     x = 50/(3*10^6) and 1 - x = 1 (approx)

So:

     50/(3*10^6) = A

So we get:

     x/(1 - x) = 50/(3*10^6) e^(kt)
 
When t = 1:

     x = 500/(3*10^6) and 1 - x is still 1 (approx)

Then:
  
     500/(3*10^6) = 50/(3*10^6) e^k

     10 = e^k

And so:

     k = ln(10) = 2.3 (approx)

Thus, our equation becomes:

     x/(1 - x) = 50/(3*10^6) e^(2*3t)

We still let 1 - x = 1. Putting t = 2, we get:

     x =  50/(3*10^6) (100) 

     x =  5000/(3*10^6)

and so after 2 weeks, 5000 people will know the rumour.
 
-Doctor Anthony, The Math Forum
Check out out web site! http://mathforum.org/dr.math/   


Date: 04/11/98 at 01:16:29
From: Hadhrami_Ab_Ghani
Subject: Re[2]: EXPONENTIAL PROBLEMS

In my previous message, I had a simple mistake. According to the 
question, after one week, 500 people know the rumor. Actually the 
correct figure is 5000 people and not "500." I hope you may rewrite 
the methods to solve the question after the correction.

Secondly, can you please explain furthermore about:

         1       A       B
     -------- = --- + ------- 
     x(1 - x)    x     1 - x    

How do you derive this equation?

For the next question, what element or thing represented by x?

Lastly, the character: "^". I don't understand the meaning of it.  


Date: 04/11/98 at 11:53:08
From: Doctor Anthony
Subject: Re: Re[2]: EXPONENTIAL PROBLEMS

I have copied the calculation below and amended it to give 5000 after 
one week instead of 500. I have also put in more explanations. For 
example, x is the proportion of the population who know the rumor:

     x = Number who know/Total population

So after one week:
   
     x = 5000/(3*10^6)    

The symbol '^' means  'power':

     3^2 = 3 * 3 = 9
     3^3 = 3 * 3 * 3 = 27  and so on.

     3*10^6 = 3,000,000  = total population.

The logistic difference equation assumes that the rate of spread of a 
rumor is proportional to the number who know and the number who don't 
know. If we let x = proportion of the population who know (0 < x < 1), 
then:
 
     dx/dt = kx(1 - x)    where k is a constant.

        dx
     --------  = k*dt 
     x(1 - x)

Now split up the left hand side using partial fractions:

         1          A        B
     --------   =  ---  +  -----
     x(1 - x)       x      1 - x

Then:

         1          A(1 - x)        B(x)
     --------   =  ----------  +  --------
     x(1 - x)       x(1 - x)      x(1 - x)

So:

     1 =  A(1 - x) + Bx

     1 =  A + x(B - A) 

Thus:

      x = 0   1 =  A
      x = 1   1 =  B

                       1            1        1
so we can express  ---------  as   ---  + -------
                    x(1 - x)        x     (1 - x)

So we have:

     INT[1/x + 1/(1 - x)]dx = INT[k*dt]

     ln(x) - ln(1 - x) =  kt + constant
     
     ln(x/(1 - x)) = kt + constant
 
     x/(1 - x) = e^(kt+constant)

     x/(1 - x) = Ae^(kt)       where A = constant.

Then at t = 0:

     x = 50/(3*10^6)    and     1 - x = 1 (approx)

So: 50/(3*10^6) = A

So we get:

     x/(1 - x) = 50/(3*10^6) e^(kt)

When t = 1:
    
     x = 5000/(3*10^6)      and 1 - x is still 1 (approx)   
                            actual value is 0.99833

     5000/(3*10^6) = 50/(3*10^6) e^k

     100 = e^k

and so:
  
     k = ln(100) = 4.60517

Thus our equation becomes:

     x/(1 - x) = 50/(3*10^6) e^(4.605t)    

Putting t = 2, we get:

     x/(1 - x) =  50/(3*10^6) (10000)

     x/(1 - x) =  500000/(3*10^6)

               =  1/6

            6x = 1 - x 

            7x = 1

             x = 1/7

So 1/7 of the population knows the rumor, and 1/7 * 3*10^6 = 428,571 
people.

And so after 2 weeks, 428,571 people will know the rumor.

-Doctor Anthony, The Math Forum
Check out our web site! http://mathforum.org/dr.math/   
    
Associated Topics:
High School Calculus

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