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jacobi1
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On a small island there are 25 inhabitants. One of these inhabitants, named Jack, starts a rumor which spreads around the isle. Any person who hears the rumor continues spreading it until he or she meets someone who has heard the story before. At that point, the person stops spreading it, since nobody likes to spread stale news.
In the first time increment, Jack randomly selects one of the other inhabitants, named Jill, to tell the rumor to. In the second time increment, both Jack and Jill each randomly select one of the remaining 23 inhabitants to tell the rumor to. (Note: they could conceivably pick each other again.) In the next time increment, there are 4 rumor spreaders, and so on. If a randomly selected person has already heard the rumor, that person stops spreading the rumor.
How many inhabitants out of 25 finally hear the rumor before it dies?
This is a simulation problem, and so I simulated it and got that about 80 percent of the inhabitants hear the rumor. However, I want a theoretical solution if possible, and that is what I am having trouble with.
I found the percentage of rumor spreaders as a function of time to be
\(\displaystyle f(t)=\left (1-\frac{25-2^{t}}{25^{t}}\right )^{25} - \frac{2^t}{25}\),
where the first term is the percentage of new spreaders, and the second term is the number of people who have already heard the rumor.
I tried to find the maximum number of people who were spreading the rumor, but I got an extremely complicated equation (here, n=25):
\(\displaystyle n^2 \left (n^t-n+2^{t} \right )^{n-1} \left (2^t \ln \frac{2}{n}+x+1 \right )=\ln 2 (2n^n)^t\), so...no progress there.
I also tried simulating it using a SIR model, but didn't get far.
How can I proceed?
Or is there another, simpler way to do it?
In the first time increment, Jack randomly selects one of the other inhabitants, named Jill, to tell the rumor to. In the second time increment, both Jack and Jill each randomly select one of the remaining 23 inhabitants to tell the rumor to. (Note: they could conceivably pick each other again.) In the next time increment, there are 4 rumor spreaders, and so on. If a randomly selected person has already heard the rumor, that person stops spreading the rumor.
How many inhabitants out of 25 finally hear the rumor before it dies?
This is a simulation problem, and so I simulated it and got that about 80 percent of the inhabitants hear the rumor. However, I want a theoretical solution if possible, and that is what I am having trouble with.
I found the percentage of rumor spreaders as a function of time to be
\(\displaystyle f(t)=\left (1-\frac{25-2^{t}}{25^{t}}\right )^{25} - \frac{2^t}{25}\),
where the first term is the percentage of new spreaders, and the second term is the number of people who have already heard the rumor.
I tried to find the maximum number of people who were spreading the rumor, but I got an extremely complicated equation (here, n=25):
\(\displaystyle n^2 \left (n^t-n+2^{t} \right )^{n-1} \left (2^t \ln \frac{2}{n}+x+1 \right )=\ln 2 (2n^n)^t\), so...no progress there.
I also tried simulating it using a SIR model, but didn't get far.
How can I proceed?
Or is there another, simpler way to do it?
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