# Parameter Estimation for a System of Differential Equations

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Igor Moura on 1 Dec 2016
Commented: Star Strider on 24 Oct 2022
Hello. Ok, so I'm new to matlab and I've got a question regarding parameter estimation for a kinetic model. I have 4 different reactants and their concentrations are c1, c2, c3 and c4. I also have 4 differential equations, each one related to a concentration (c1, c2, c3 and c4, respectively -see below-) and experimental data for all these concentrations on 12 different times plus the initial condition. The k's are the rate coefficients. I want to solve this system of ODE's using ode45 and then use the output to compute the experimental data minus the observed data and use these results to estimate the values of k's using lsqnonlin, but apparently I can't solve these ODE's without numerical values for k -which is what I want to know-. Any help on how to set up the command to solve this?
function dcdt=batch(t,c,k)
dcdt=zeros(4,1);
dcdt(1)=-k(1)*c(1)-k(2)*c(1);
dcdt(2)= k(1)*c(1)+k(4)*c(3)-k(3)*c(2)-k(5)*c(2);
dcdt(3)= k(2)*c(1)+k(3)*c(2)-k(4)*c(3)+k(6)*c(4);
dcdt(4)= k(5)*c(2)-k(6)*c(4);
end
Data:
t c1 c2 c3 c4
0 1 0 0 0
0.1 0.902 0.06997 0.02463 0.00218
0.2 0.8072 0.1353 0.0482 0.008192
0.4 0.6757 0.2123 0.0864 0.0289
0.6 0.5569 0.2789 0.1063 0.06233
0.8 0.4297 0.3292 0.1476 0.09756
1 0.3774 0.3457 0.1485 0.1255
1.5 0.2149 0.3486 0.1821 0.2526
2 0.141 0.3254 0.194 0.3401
3 0.04921 0.2445 0.1742 0.5277
4 0.0178 0.1728 0.1732 0.6323
5 0.006431 0.1091 0.1137 0.7702
6 0.002595 0.08301 0.08224 0.835
Star Strider on 2 Jan 2021
Budda —
Make appropriate changes to the ‘kinetics’ function (specifically to the ‘C’ variable) to output only the variable you need. In that code, there are 4 columns, so address only the column you want returned in ‘C’.

Star Strider on 1 Dec 2016
See if the techniques in Monod kinetics and curve fitting will do what you want.
Star Strider on 24 Oct 2022
I looked at it just now, however it has already been answered.
Note that initial parameter estimates of zeros is never a good approach. Choose something nonzero and preferably not the same values for both parameters.