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The problem I have to solve is to choose optimally teh parameters of a system of differential equations. The problem is that one of the parameters depends on a function (which are the parameters I need to choose) and the value changes in every period. So, in order to choose the parameters I minimize the square error of the fitted model with some actual data. The problem is that the algorithm is really slow and I need to optimize it. In every step I need to redefine the ODE model in the following way:

syms S(t) I(t) R(t) D(t)

odeS = diff(S) == -beta*I*S/pop;

odeI = diff(I) == beta*I*S/pop - gamma*I-mu*I;

odeR = diff(R) == gamma*I;

odeD = diff(D) == mu*I;

% Transform the equations for the numerical solver

odes = [odeS; odeI; odeR; odeD];

odes2 = odeToVectorField(odes);

eq_mat = matlabFunction(odes2, 'Vars', {'t', 'Y'});

ic = [s0, i0, r0, d0];

tspan = [0, 100];

[t, y]= ode45(eq_mat, tspan, ic);

The problem is that in every period the value of beta changes and i need to run all the latter lines again and that takes time. I have tried other things but are even slower.

Stephan
on 22 Apr 2021

Edited: Stephan
on 22 Apr 2021

Are gamma and beta parameters that result from the gamma / beta functions? Or just scalar parameters? I suggest using other names if they are scalars, because Matlab inbuilt functions are called like that, which will produce errors.

Run this part only once - ideally in a seperate script.Save or copy the result into another script and optimize then without the symbolic calculation. Therefore use beta (and maybe the others) as an additional input.

syms S(t) I(t) R(t) D(t) Beta pop Gamma mu

odeS = diff(S) == -Beta*I*S/pop;

odeI = diff(I) == Beta*I*S/pop - Gamma*I-mu*I;

odeR = diff(R) == Gamma*I;

odeD = diff(D) == mu*I;

% Transform the equations for the numerical solver

odes = [odeS; odeI; odeR; odeD];

odes2 = odeToVectorField(odes);

eq_mat = matlabFunction(odes2, 'Vars', {'t', 'Y', 'Beta', 'pop', 'Gamma', 'mu'})

The numerical solution process is fast, but symbolic calculations are not. So this part should be the only one that runs repeated durng the optimization:

eq_mat = @(t,Y,Beta,pop,Gamma,mu)[-Gamma.*Y(1)-mu.*Y(1)+(Beta.*Y(1).*Y(2))./pop;-(Beta.*Y(1).*Y(2))./pop;Gamma.*Y(1);mu.*Y(1)]

s0 = 1;

i0 = -1;

r0 = 1;

d0 = -1;

ic = [s0, i0, r0, d0];

Gamma = 1;

pop = 2;

mu = -1;

Beta = 0.5;

tspan = [0, 100];

[t, y]= ode45(@(t,Y)eq_mat(t,Y,Beta,pop,Gamma,mu), tspan, ic);

plot(t,y)

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