% Example 4.4 of H.J. Oberle and H.J. Pesch, Numerical treatment of 
% delay differential equations by Hermite interpolation, Numer. Math.,
% 37 (1981) 235-255.  This is a model for the spread of an infection
% due to Hoppensteadt and Waltman.  It is interesting because there are
% discontinuous changes in the equation at known times. Oberle and Pesch 
% solve the problem for several values of the parameter r, namely 0.2, 
% 0.3, 0.4, 0.5.  The example is also interesting in that values of the 
% derivative of the solution are required for another function of interest.

r = 0.5;
c = 1/sqrt(2);
opts = ddeset('Jumps',[(1-c), 1, (2-c)],...
              'RelTol',1e-5,'AbsTol',1e-8);
sol = dde23('exam5f',1,10,[0, 10],opts,r);

y10 = ddeval(sol,10);
fprintf('DDE23 computed     y(10) =%15.11f.\n',y10);
fprintf('Reference solution y(10) =%15.11f.\n',0.06302089869);

plot(sol.x,sol.y)
title(['Figure 5a. Hoppensteadt-Waltman model with r = ',...
       num2str(r),'.'])
xlabel('time t')
ylabel('y(t)')

Ioft = -(1/r)*(sol.yp ./ sol.y);
figure
plot(sol.x,Ioft)
title(['Figure 5b. Hoppensteadt-Waltman model with r = ',...
       num2str(r),'.'])
xlabel('time t')
ylabel('I(t)')