Solving Delay Differential Equations with dde23

Problem 5

Another epidemic model due to Cooke et alia [3] that is more complicated than Problem 4 involves the size I(t) = y₁(t) of an infected population and the total size N(t) = y₂(t) of the population at time t. The equations are

y1¢(t) = l ( y2(t) - y1(t) )   y1(t) y2(t)  -   ( d + e+ g)  y1(t) y2¢(t) = b  e-a y2(t-T)  y2(t-T)   e-d1 T  -  d y2(t)  -  ey1(t)

They are solved on [0,25] with history y₁(t) = 2,y₂(t) = 3.5 for t £ 0 and parameter values a = 1, b = 80, d = 1, d₁ = 1, g = 0.5, e = 10, T = 0.2 .

As in Problem 2, it is convenient to pass the parameters to the function for evaluating the DDEs as global variables or to hard code them. In [3] the solution is investigated for a number of l, so pass it as a parameter through dde23. Values l = 12, 15, 20, 28 are of interest. You might find it interesting to compare your plots to those of Figure 4 in [3]. The following figure shows the case l = 12 .

References

[3]

K. Cooke, P. van den Driessche, and X. Zou, Interaction of maturation delay and nonlinear birth in population and epidemic models, J. Math. Biol., 39 (1999) 332-352.