# This example simulates the spread of a rumor through a closed # population. The percentage of people who have not heard the # rumor, as a function of time, is plotted. # # You may run this example by doing: # # ode < rumor.ode | graph -T X -C # # or alternatively, to get a real-time plot, # # ode < rumor.ode | graph -T X -C -x 0 .25 -y 0 100 # The theoretical background for this model is as follows. # # Suppose a rumor spreads through a closed population of constant size # N+1. At time t the total population can be classified into three categories: # x persons who are ignorant of the rumor; # y persons who are actively spreading the rumor; # z persons who have heard the rumor but have stopped spreading it; # # Suppose that if two persons who are spreading the rumor meet then they stop # spreading it. # Suppose also that the contact rate between any two categories is constant, u. # # The equations # x' = -u * x * y, # y' = u * (x*y - y*(y - 1) - y*z) # give a deterministic model of the problem. # # When initially y = 1 and x = N, the number of people # who ultimately never hear the rumor is s, where s satisfies # 2N + 1 - 2s + N log(s/N) = 0. x' = -u * x * y y' = u * (x*y - y*(y-1) - y*( 100 + 1 - y - x)) x = 100 y = 1 u = 1 print t, x step 0, 0.25
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