`StructuralIdentifiability.jl`

is a Julia package for assessing structural parameter identifiability of parametric ODE models, both local and global.
This includes computation of identifiable functions of states and parameters. The package also offers functionality to assess local identifiability
in discrete-time models.
For an introduction to structural identifiability, we refer to [2].

The package can be installed from this repository by

```
using Pkg
Pkg.add("StructuralIdentifiability")
```

For information on using the package, see the stable documentation. Use the in-development documentation for the version of the documentation, which contains the unreleased features.

The package can be loaded by `using StructuralIdentifiability`

.

A parametric ODE system in the state-space from can be defined by the `@ODEmodel`

macro:

```
ode = @ODEmodel(
x1'(t) = -(a01 + a21) * x1(t) + a12 * x2(t) + u(t),
x2'(t) = a21 * x1(t) - a12 * x2(t) - x3(t) / b,
x3'(t) = x3(t),
y(t) = x2(t)
)
```

In this example:

`x1(t), x2(t), x3(t)`

are the**state variables**, they defined the state of the system and are assumed to be unknown;`u(t)`

is the**input/control variable**which is assumed to be known and generic (exciting) enough;`y(t)`

is the**output variable**which is assumed to be observed in the experiments and, thus, known;`a01, a21, a12, b`

are unknown scalar**parameters**.

Note that there may be multiple inputs and outputs.

The identifiability of the parameters in the model can be assessed by the `assess_identifiability`

function as follows

`assess_identifiability(ode)`

The returned value is a dictionary from the parameter of the model to one of the symbols

`:globally`

meaning that the parameter is globally identifiable`:locally`

meaning that the parameter is locally but not globally identifiable`:nonidentifiable`

meaning that the parameter is not identifiable even locally.

For example, for the `ode`

defined above, it will be

```
OrderedDict{Any, Symbol} with 7 entries:
x1(t) => :locally
x2(t) => :globally
x3(t) => :nonidentifiable
a01 => :locally
a12 => :locally
a21 => :globally
b => :nonidentifiable
```

If one is interested in the identifiability of particular functions of the parameter, one can pass a list of them as a second argument:

`assess_identifiability(ode, funcs_to_check = [a01 + a12, a01 * a12])`

This will return:

```
OrderedDict{Any, Symbol} with 2 entries:
a01 + a12 => :globally
a01*a12 => :globally
```

In the example above, we saw that, while some parameters may be not globally identifiable, appropriate functions of them (such as `a01 + a12`

and `a01 * a12`

)
can be still identifiable. However, it may be not so easy to guess these functions (even in this example). Good news is that this is not needed!
Function `find_identifiable_functions`

can find generators of all identifiable functions of a given model. For instance:

`find_identifiable_functions(ode)`

will return

```
3-element Vector{AbstractAlgebra.Generic.Frac{Nemo.QQMPolyRingElem}}:
a21
a01*a12
a01 + a12
```

which are exactly the identifiable functions we have found before. Furthermore, by specifying `with_states = true`

, one can compute the generating set for
all identifiable functions of parameters and states (in other words, all observable functions):

`find_identifiable_functions(ode, with_states = true)`

This will return

```
6-element Vector{AbstractAlgebra.Generic.Frac{Nemo.QQMPolyRingElem}}:
x2(t)
a21
a01*a12
a01 + a12
x3(t)//(a12*b + a21*b + b)
(-x1(t)*a21*b + x2(t)*a12*b + x2(t)*a21*b + x2(t)*b + x3(t))//(a21*b)
```

Local identifiability can be assessed efficiently even for the models for which global identifiability analysis is out of reach.
This can be done using the `assess_local_identifiability`

function, for example:

`assess_local_identifiability(ode)`

The returned value is a dictionary from parameters and state variables to `1`

(is locally identifiable/observable) and `0`

(not identifiable/observable) values. In our example:

```
OrderedDict{Any, Bool} with 7 entries:
x1(t) => 1
x2(t) => 1
x3(t) => 0
a01 => 1
a12 => 1
a21 => 1
b => 0
```

As for `assess_identifiability`

, one can assess local identifiability of arbitrary rational functions in the parameters (and also states) by providing a list of such functions as the second argument.

**Remark** The algorithms we used are randomized, the default probability of the correctness of the result is 99%, one can change it by changing the value of a keyword argument `p`

to any real number between 0 and 1, for example:

```
# pobability of correctness 99.9%
assess_identifiability(ode; p = 0.999)
```

Maintained by Gleb Pogudin (gleb.pogudin@polytechnique.edu)

[1]
Ruiwen Dong, Christian Goodbrake, Heather Harrington, and Gleb Pogudin,
*Differential elimination for dynamical models via projections with applications to structural identifiability*,
preprint, 2021.

[2]
Hongyu Miao, Xiaohua Xia, Alan S. Perelson, and Hulin Wu,
*On Identifiability of Nonlinear ODE Models and Applications in Viral Dynamics*,
SIAM Review, 2011.

[3]
Alexey Ovchinnikov, Anand Pillay, Gleb Pogudin, and Thomas Scanlon,
*Computing all identifiable functions for ODE models*,
preprint, 2020.

[4]
Alexandre Sedoglavic,
*A probabilistic algorithm to test local algebraic observability in polynomial time*,
Journal of Symbolic Computation, 2002.