# 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.