This is my first question. Apologies if my descriptions or question is naive. I hope I'm providing the necessary detail.
I'm trying to formulate differential equations to model the process described in the figure (linked above). However, I'm having some trouble forming the correct differential equations to describe the process. The states are the subscripted capital letters, (e.g. $M_r, N_{n,n}$, etc). I think what is causing me problems is the mixture of two types of processes, one type governed by first-order rates,(e.g. $\lambda_b, \lambda_n, \lambda_s, \lambda_T$), the other type governed by constant time (e.g. $\tau_s, \tau_t)$.
For example, $N_{n,n}$ is formed by two first order rates, but it becomes $N_{c,n}$ through a time constant process where one $N_{n,n}$ takes $\tau_s$ time to become $N_{c,n}$.
Truly, I only need an equations for the sum two of the states, $N_{c,n} + N_{c,c}$, as a function of time.
The problem emerges for the differential equation for $N_{c,n}$. When formulated as shown in the figure and solved, the resulting function will become negative at $t=\tau_t$ when $\lambda_n$ is very small (compared to $\lambda_b$). That should be impossible in the process, so clearly I'm modeling this wrong. I suspect somehow I'm subtracting part of what is forming $N_{r,n}$ twice.
Any help with formulating the correct differential equation would be very helpful. Or if there is another approach I should take, please let me know. Thank you.