I have a system of ODEs that contains 3 unknown parameters that have to be optimised in order to fit the system to a given data set.
This is the system of ODEs that I have:
$\frac{dP}{dt} = -\alpha PS - \gamma P$
$\frac{dS}{dt} = \alpha PS - \beta S - \gamma S$
$\frac{dQ}{dt} = \beta S - \gamma Q$
($t$ = time and $\alpha$, $\beta$ and $\gamma$ are constants)
How can I come up with the values for $\alpha$, $\beta$ and $\gamma$ such that it'll allow the model to fit perfectly with the data that is given? Could anyone guide me through on the approach? It'll be good if there's a sample code that I can make reference to as I'm not very good with matlab. Thank you!
Maybe this works.
$\frac{d(P+S +Q)}{dt} = -\gamma(P+S + Q)$ this equation has solution with an exponential growth(you can solve it), it is quite easy to fit the curve and get $\gamma$.
Then
$\frac{1}{S}(\frac{dQ}{dt} + \gamma Q) = \beta$. Then get $\alpha$ in similar way.