How to give the phase of the following delay differential equations by dde

where

and satisfies

It is the system switching between ODE and DDE, how can I get the phase picture?

Answer 1

[Arnav]

Hi @Tianyu Cheng,

You may model the system as the following function:

function dydt = ddesys(t, y, Z)
    S = y(1);
    I = y(2);
    S_lag = Z(1);
    I_lag = Z(2);
    l1 = 0.8 * heaviside(I - I_lag);
    l0 = 1 - l1;
    dSdt = 6 - S - I * exp(-(l0 * I + l1 * (I - I_lag))) * S;
    dIdt = I * exp(-(l0 * I + l1 * (I - I_lag))) * S - I;
    dydt = [dSdt; dIdt];
end

The system can be be solved using dde23 function as follows:

sol = dde23(@ddesys, [1 1], [S0 I0], [0 10]);

The phase plane shown above can be plotted by plotting the solutions obtained by varying the initial position over a grid of values.

You may refer to the MATLAB documentation page of dde23 function to learn more:

https://www.mathworks.com/help/matlab/ref/dde23.html