Logistic Growth of A Rumor SpreadingDate: 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/ |
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