# Nonlinear Models of Biological Systems (3)

## Model description:

This example deals with a model described in **E. Walter, Identifiability of State Space Models. Berlin, Germany:Springer-Verlag, 1982.** The model is shown in Fig. 3. All fluxes are assumed linear except some leaving compartment 3. In particular, parameters $k_{13}$, $k_{23}$, and $k_{43}$ are assumed to be parametrically linearly controlled by the arrival compartment, i.e. $k_{13}(x_1) = k_{13}x_1$, $k_{23}(x_2) = k_{23}x_2$ and $k_{43}(x_4) = k_{43}x_4$.

The system-experiment model is

$$\begin{align*} \dot{x}_1(t) &= - (k_{01} + k_{21})x_1(t) + k_{12}x_2(t) + k_{13}x_1(t)x_3(t)\\ \dot{x}_2(t) &= k_{21}x_1(t) - (k_{02} + k_{12} + k_{32} +k_{42})x_2(t) - k_{23}x_3(t)x_2(t) + k_{24}x_4(t)\\ \dot{x}_3(t) &= k_{32}x_2(t) - [k_{03} + k_{13}x_1(t) + k_{23}x_2(t) - k_{43}x_4(t)]x_3(t) + k_{34}x_4(t) + u(t)\\ \dot{x}_4(t) &= k_{43}x_2(t) + k_{43}x_3(t)x_4(t) - (k_{24} + k_{34})x_4(t)\\ y_1(t) &= x_1(t) \\ y_2(t) &= x_2(t) \end{align*}$$

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## Publication details:

Title | Global identifiability of nonlinear models of biological systems |

Publication Type | Journal Article |

Authors | Audoly, S., Bellu G., D'Angio L., Saccomani M.P., and Cobelli C. |