State space model/description of blood sugar ratio & insulin ratio

Question

Currently trying to solve a state-space model for the blood sugar ratio and Insulin ratio of a person.

The following is presented in the problem:

$I(t)$ = ratio of Insulin in the body

$S(t)$ = ratio of Blood sugar in the body

The stable reference state is a fixed level with blood sugar level $= S_0$ and the insulin level being zero.

The following properties are presented for each state variable $(I(t), S(t))$:

Sugar:

1. Presence of insulin increases the sugar turnover (= reduced blood sugar content). The higher the sugar and insulin content, the higher the turnover. An assumption is made that blood level decreases proportionally to the product of the sugar and insulin content.
2. If the blood sugar level drops below the stable reference point($S_0$). Sugar will be released by the liver proportional to the sugar deficit.
3. An increase in blood sugar happens when consuming food. The effect of this is proportional to the time function $z(t)$

Insulin:

4. If the blood sugar level exceeds S0, insulin is released, proportional to the sugar surplus.
5. Insulin itself has a limited duration. Half of the free insulin becomes inactive within 10-25 min
6. If the patient is diabetic an external injection is to be made. Assumed that the amount of insulin is fed is proportional to a function of w(t), which describes the injection pattern.

The question is:

Provide a state-space description with the ratio of blood sugar content $(S(t))$ and the insulin content $(I(t))$ as state variables.

What i have tried/interpret: for Sugar:

1. $S'(t) = -k_1(S(t)-S_0)$
2. $k_3(S_0-S(t)) = S(t)$
3. $Z(t) = S(t)$

combined: $S'(t) = -k_1*k_2*I(t)*S0 + k_1*k_2*I(t)*S(t) - k_1*k_2*I(t)*z(t)$

For Insulin:

4. $k_3(S_0-S(t)) = I'(t)$
5. $k_4(I(t)) = I(t)$
6. $w(t) = I(t)$

Combined: $I'(t) = k_3(S_0-S(t)) + k_4(I(t) + w(t))$

Is the interpretation correct? If so, how do I continue to get the state-space variables $S(t)$ and $Y(t)$?