Problem 4

An epidemic model due to Cooke [8] describes the fraction y(t) of a population which is infected at time t by the equation
y(t) = b  y(t-7) ( 1 - y(t) )  -  c  y(t)
Here b and c are positive constants. The equation is solved on [0,100] with history y(t) = a for t 0. The constant a satisfies 0 < a < 1 .

For all values of b and c, the solution y(t) = 0 is an equilibrium point. For b > c, the solution y(t) = 1 - c/b is a second equilibrium point. Solve this DDE for different values of b , c , and a. Verify that if b > c , the solution approaches the second equilibrium point, and otherwise it approaches the zero equilibrium point. The long-term behavior of the solution is independent of the delay; you might want to verify this computationally. The figure shows the approach of the solution to the second equilibrium point when b = 2 , c = 1, and a = 0.8 .



N. MacDonald, Time Lags in Biological Models, Springer-Verlag, Berlin, 1978.