ODE solution positivity for systems biology models

Certainly, if that would help. The basic form is:
\frac{dT}{dt} = 1 - \theta_{12} T I \\ \frac{dI}{dt} = \theta_{12} T I - E I \\ \frac{dE}{dt} = \theta_3 - \theta_4 E \frac{Q^n}{1+Q^n} \\ \frac{dQ}{dt} = \theta_5 I \\ T(0) = \theta_6 e^{-\theta_{12}} \\ I(0) = \theta_6 \left(1-e^{-\theta_{12}} \right) \\ E(0) = 0 \\ Q(0) = 0.

The variable I is the problem variable that goes negative (or becomes very close to zero) and its log is the main observable.