Thanks Gilead, at this point I am not sure what is best way to go but I think I can accept you answer as it helped me a lot.

Thanks very much Gilead, I am going through your answer. Seems like problem can be solved but require some advanced tricks (my mathematics knowledge is in fancy stage). My purpose is dual estimation, where I can estimate states as well as parameters using a nonlinear variant of Kalman filter. I have discrete time series data for $y_5$.

@sleepless-in-beantown thanks. I agree initial description was not sufficient and it was bit fuzzy. I think with incremental inputs from you all I am moving in right direction. Coming back to the problem, often biological systems are partially observed with very few indirect measurements. So it is not surprising that I end-up with this kind of situation. Eventually I want to convert the set of equations in matrix form. One thing which is not clear to me how I should convert conditional (if-otherwise type) equations in matrix form?

Thanks everyone, I really meshed up the equations as I mixed the nonlinear version with linear one. I have updated my question for your consideration. Please let me know my question is now clear to you all.