# Reverse problem of finding time-varying parameters of an ODE with the help of solution data.

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Anshuman on 23 Nov 2022
Commented: Anshuman on 23 Nov 2022
Any way in which I can find the time-varying parameters of an ODE ; The solution to which is know in the form of time-series data.
Background:
Example: SIR model ode: as following.
% The SIR model differential equations.
def deriv(y, t, beta, gamma):
S, I, R = y
dSdt = -beta * S * I
dIdt = beta * S * I - gamma * I
dRdt = gamma * I
return dSdt, dIdt, dRdt
Then, I use odeint(deriv, y0, t, args=( beta, gamma)) (In python) to solve for S, I and R using minimisation of these parameters.
Question
But I want to ask, is there any way in which a reverse problem can be constructed such that; I have the data for S, I and R as time-sereis. and then we can calculate 'beta' and 'gamma' paramter from there?
Note beta and gamma should be time dependent.They can be assumed as sum of natural cubic splines; Where N_k(t), k=1 to K, are K natural cubic spline basis functions evaluated at K-2 equally spaced knots in addition to the boundary knots .
Can this problem be solved on matlab?
Thanks.
Anshuman on 23 Nov 2022
Actually I couldn't find a way to implement python packages for this time dependent problem; so I thought of switching to matlab.

Torsten on 23 Nov 2022
Use "lsqcurvefit" to fit the parameters together with an integrator (e.g. ODE45).
For an example, see Star Strider's code under
Here, four parameters from a Monod Kinetic are fitted against measurement data.
Anshuman on 23 Nov 2022
Sure, thanks a lot!