how to incorporate non instantaneous impulses

how can we incoporate non-instantaeous impulses in S where S(t)=g(t,S(kT)), and g(T+p,u) = (1-theta)*u in the interval t belongs to the interval[kT,kT+p), k belongs to z^+.

Answer 1

[Anushka]

Hello @jasmin,

To address the query on incorporating non-instantaneous impulses into the given system of equations in MATLAB, here is how you can approach the problem:

• Use MATLAB's 'ode45' function to solve the differential equations. You can refer to the below given documentation: https://www.mathworks.com/help/matlab/ref/ode45.html
• Write the equations 'S'(t)' and 'I'(t)' as a system. Here is a snippet of code you can refer to:

function dydt = diff_eqs(t, y, b, beta, alpha, gamma, omega, tau)
    S = y(1);
    I = y(2);
    dS = b - b*S - beta*S*I/(1 + alpha*S) + gamma*I*exp(-b*tau);
    dI = beta*exp(-b*omega)*S*I/(1 + alpha*S) - (b + gamma)*I;
    dydt = [dS; dI];
end

• After solving for one interval, update 'S' using 'g(t,S)'. You can refer to the below given code snippet:

function S_new = g(t, S_prev, theta, u)
    % Customize based on the problem
    S_new = (1 - theta) * u;
end

• Solve the problem iteratively for each interval [kT,(k+1)T], updating 'S' and 'I' at each step.

Hope this helps!