Can Mathematical Models Predict the Winner in Ecolopolis' Mayoral Race?

[xipe]

Homework Statement

The election for mayor is coming soon in the small town of Ecolopolis (population 5000).The two candidates are Aaron Aardvark and Paula Platypus. Paula Platypus currently has a big lead, with 80% of the population ready to vote for her.
As usual, the politics turn ugly. Five days before the election, Aardvark’s campaigners (all
6 of them) decide that they will taint Platypus’ reputation by spreading the rumor that she has a
drinking problem. Naturally, everyone they tell believes this rumor and switches their allegiance over to Aardvark. What’s more, each person who hears the rumor then spreads the rumor to everyone they chat with.
The Platypus campaigners (all 6 of them) realize what is going on. Two days before the election, they go out and start telling the townspeople that the rumor is a lie. The people they talk with believe them and switch their allegiance back to Platypus (if they supported her originally),and then help refute the lie with others they chat with.
Who is likely to win the election? And how likely is it?

Homework Equations

None given

The Attempt at a Solution

I have tried multiple things to model this situation. However, I am thinking it required a higher understanding of second order differential equations. This is for an engineering class, but diff eq isn't required. Every project we have done this semester has been easily done with diff eq but our coursework hasn't gotten us to second order DE's yet. Any guidance AT ALL is extremely helpful. As of now I have two really, really weird equations that don't really do a good job at deciding who has a better chance of winning.

Also, we recently worked on a sensitivity analysis, and I think this deals with that too (For testing sensitivity of word of mouth variables), if that helps at all.

Thank you so much for any help you can offer.
xipe

 

[fresh_42]

I would try a SIR model as rumors behave very much like an epidemic. And the rumors (infection, immunity, convalescence) seems to be the only parameter you can say something about.

 

[xipe]

Thank you for the timely reply. I like what I have read from that link you posted, but I am unsure how to derive the solutions from a system of 3 DE's. I am just about half-way through my diff eq class and we haven't gotten so far as to attempt something like that. Any pointers would be greatly appreciated. The model is perfect though, three categories, all of which fit the bill.

 

[fresh_42]

You can look on the internet for examples, solutions and theory. I assume there can be found plenty as it is one of the more famous differential equation systems. (Just type in "SIR model" on your preferred search page.) Or what can always be done, is to make it discrete. Consider small time intervals (half a day or so) and calculate the changes in steps. This way you can use the equations without having to solve the system. (Just calculate $df(t)/dt$ as $\Delta f(t) / \Delta t\,$.) What's also possible, is to draw some vector fields: take a point in the phase space (actually many) and draw a vector in the direction of change. This might not be an exact method, but it illustrates the general behavior. In any case there will be a great dependence on the initial values and parameters you chose.