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

Question

Muhammad 2024-8-22，9:42

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https://ww2.mathworks.cn/matlabcentral/answers/2147164-how-to-extract-delayed-state-terms-in-a-model-with-distributed-delay

编辑： Torsten 2024-8-22，17:21

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.

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Answer 1

Torsten 2024-8-22，14:00

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https://ww2.mathworks.cn/matlabcentral/answers/2147164-how-to-extract-delayed-state-terms-in-a-model-with-distributed-delay#answer_1503569

编辑：Torsten 2024-8-22，17:21

在 MATLAB Online 中打开

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

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Muhammad 2024-8-22，15:28

Thank you

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Answer 2

Torsten 2024-8-22，11:19

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https://ww2.mathworks.cn/matlabcentral/answers/2147164-how-to-extract-delayed-state-terms-in-a-model-with-distributed-delay#answer_1503489

移动：Torsten 2024-8-22，11:23

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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)

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Muhammad 2024-8-22，12:08

编辑：Muhammad 2024-8-22，12:09

在 MATLAB Online 中打开

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 2024-8-22，14:12

yes I want to extract yl1 in ddefunc

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