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# 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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