My apologies if this is a silly question. I'm developing a model for an applied system and I'm stuck. This is a (simplified) version of my model.
$x'=m(y-x)-qx$
$y'=m(x-y)-qy+m(z-y)$
where $z=x{(t_0)=y(t_0)} $ and $m$ and $q$ are my parameters I want to estimate.
I was able to calculate the analytical solution of the form $\begin{bmatrix}x\\y\end{bmatrix}=c_1v_1e^{\lambda t}+c_2v_2e^{\lambda t}$.
Now I want to figure out how I'm actually going to input my data/likelihood in the model so that I can estimate m and q. My data are $qx$ and $qy$ at times $t=0, 1, ..., 5$, and the likelihood for them can be modeled via a Poisson distribution. I feel like I'm misunderstanding something, but I don't know how I would put in my expected values into this model. What am I missing?