Compartmental models

Question

How can we prove the positivity and the boundlessness of compartments in a compartmental model.

Answer 1

Let's look if you can escape the positive orthant. At $S=0$ you have $S'=\mu T\ge0$, at $T=0$ also $T'=0$, at $I=0$ you get $I'=\alpha T\ge 0$, etc. Thus it is impossible to get from a strictly positive state to zero values in some components in forward time direction.

If you consider $N=S+T+I+M+R$, then $N'=-b_1S-b_2I-b_3M\le 0$, so that this can only decrease in forward time. Thus the single variables are all bounded by $N\le N_0$.