How to Extract Delayed State Terms in a Model with Distributed Delay?

I'm working on a model with distributed delays, and I'm using the ddesd function to solve the delay differential equations. My setup involves a distributed delay, defined by:

tau = 1;
gamma = 0.5;
number_of_delays = 11;   % should be odd
lags = linspace(tau-gamma,tau+gamma,number_of_delays);
tspan=[0 600];
sol=ddesd(@(t,y,Z)ddefunc(t,y,Z,lags),lags,[0.2; 0.08],tspan);
p=plot(sol.x,sol.y);
set(p,{'LineWidth'},{2;2})
title('y(t)')
xlabel('Time(days)'), ylabel('populations')
legend('x','y')

function yp = ddefunc(~,y,Z,lags)
a=0.1;
b=0.05;
c=0.08;
d=0.02;

yl1 = trapz(lags,Z(2,:));

yp = [a*y(1)-b*y(1)*yl1;
    c*y(1)*y(2)-d*y(2)];
end

In the case of discrete delays, we can easily extract the delayed state terms using the deval or interp1 commands. However, I'm unsure how to proceed with extracting or examining the delayed state terms for distributed delay case.

Answer 1

[Torsten]

So in the distributed case you want to extract

y2_delayed(t) = integral_{tau = t-1.5]^{tau = t-0.5} y2(tau) dtau 

thus the term yl1 in ddefunc ?

I plotted it as "Lag term" below.

tau = 1;
gamma = 0.5;
lags = [tau-gamma;tau+gamma];
tspan = [0 600];
sol = dde23(@ddefunc,lags,[0.2; 0.08; 0.08*2*gamma],tspan,ddeset('RelTol',1e-6,'AbsTol',1e-6));
figure(1)
p = plot(sol.x,sol.y(1:2,:));
set(p,{'LineWidth'},{2;2})
title('y(t)')
xlabel('Time(days)'), ylabel('populations')
legend('x','y')

figure(2)
p = plot(sol.x,sol.y(3,:));
set(p,{'LineWidth'},{2})
title('Lag(t)')
xlabel('Time(days)'), ylabel('Lag term')
legend('Lag')

function yp = ddefunc(~,y,Z)
  a=0.1;
  b=0.05;
  c=0.08;
  d=0.02;
  yp = [a*y(1)-b*y(1)*y(3);
        c*y(1)*y(2)-d*y(2);
        Z(2,1)-Z(2,2)];
end

[Muhammad]

Thank you

Answer 2

[Torsten]

What exactly do you want to extract ? y2, evaluated at (t-lags) ?

And you know that your equation can be solved using dde23 because you can treat

integral_{tau = t-1.5]^{tau = t-0.5} y2(tau) dtau 

in the same way as I suggested here:

https://uk.mathworks.com/matlabcentral/answers/2145819-how-to-modify-code-for-distributed-delay?s_tid=prof_contriblnk

?

In this case, you get an equation with discrete delays of length 1.5 and 0.5.

Hint:

d/dt (integral_{tau = t-1.5]^{tau = t-0.5} y2(tau) dtau) = y2(t-0.5) - y2(t-1.5)

[Muhammad]

Thank you for your response. My target is to extract delayed state like for discrete delay we extract y(t-tau).

t_eval = 600;  
y_delayed_deval = deval(sol, t_eval - lags);  % y(t-tau) 
or 
y_delayed_interp1 = interp1(sol.x, sol.y(2,:), t_eval - lags);

now for distributed delay I am not sure how we can extract this

[Muhammad]

yes I want to extract yl1 in ddefunc