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Calculating sensitivities lets you determine which species or parameter in a model is most sensitive to a specific condition (for example, a drug), defined by a species or parameter. Calculating sensitivities calculates the time-dependent sensitivities of all the species states with respect to species initial conditions and parameter values in the model.

Thus, if a model has a species `x`

, and two
parameters `y`

and `z`

, the time-dependent
sensitivities of `x`

with respect to each parameter
value are the time-dependent derivatives

$$\frac{\partial x}{\partial y},\frac{\partial x}{\partial z}$$

where, the numerator is the sensitivity output and the denominators are the sensitivity inputs to sensitivity analysis.

For more information on the calculations performed, see References.

Sensitivity analysis is supported only by the ordinary differential equation (ODE) solvers. The software calculates local sensitivities by combining the original ODE system for a model with the auxiliary differential equations for the sensitivities. The additional equations are derivatives of the original equations with respect to parameters. This method is sometimes called “forward sensitivity analysis” or “direct sensitivity analysis”. This larger system of ODEs is solved simultaneously by the solver.

SimBiology^{®} sensitivity analysis uses “complex-step
approximation” to calculate derivatives of reaction rates.
This technique yields accurate results for the vast majority of typical
reaction kinetics, which involve only simple mathematical operations
and functions. When a reaction rate involves a nonanalytic function,
this technique can lead to inaccurate results. In this case, either
sensitivity analysis is disabled, or sensitivity analysis warns you
that the computed sensitivities may be inaccurate. An example of such
a nonanalytic function is the MATLAB^{®} function `abs`

. If sensitivity analysis gives questionable
results on a model whose reaction rates contain unusual functions,
you may be running into limitations of the complex-step method. Contact MathWorks
Technical Support for additional information.

Models containing the following active components do not support sensitivity analysis:

Nonconstant compartments

Algebraic rules

Events

You can perform sensitivity analysis on a model containing repeated assignment rules, but only if the repeated assignment rules do not determine species or parameters used as inputs or outputs in sensitivity analysis.

You can calculate sensitivities using `sbiosimulate`

or
the `SimFunctionSensitivity object`

.

Set the following properties of the `SolverOptions`

property
of your `configset`

object, before running the `sbiosimulate`

function:

`SensitivityAnalysis`

— Set to`true`

to calculate the time-dependent sensitivities of all the species states defined by the`Outputs`

property with respect to the initial conditions of the species and the values of the parameters specified in`Inputs`

.`SensitivityAnalysisOptions`

— An object that holds the sensitivity analysis options in the configuration set object. Properties of`SensitivityAnalysisOptions`

are:`Outputs`

— Specify the species and parameters for which you want to compute the sensitivities. This is the numerator as described in About Calculating Sensitivities.`Inputs`

— Specify the species and parameters with respect to which you want to compute the sensitivities. Sensitivities are calculated with respect to the`InitialAmount`

property of the specified species. This is the denominator, described in About Calculating Sensitivities.`Normalization`

— Specify the normalization for the calculated sensitivities:`'None'`

— No normalization`'Half'`

— Normalization relative to the numerator (species output) only`'Full'`

— Full dedimensionalization

For more information about normalization, see

`Normalization`

.

After setting `SolverOptions`

properties, calculate
the sensitivities of a model by providing the `model object`

as
an input argument to the `sbiosimulate`

function.

The `sbiosimulate`

function returns a `SimData object`

containing the following
simulation data:

Time points, state data, state names, and sensitivity data

Metadata such as the types and names for the logged states, the configuration set used during simulation, and the date of the simulation

A `SimData object`

is a convenient way of keeping
time data, state data, sensitivity data, and associated metadata together.
A `SimData object`

has properties and methods associated
with it, which you can use to access and manipulate the data.

For illustrated examples, see:

Create a ```
SimFunctionSensitivity
object
```

using the `createSimFunction`

specifying
the `'SensitivityOutputs'`

and `'SensitivityInputs'`

name-value
pair arguments. Then execute the object. For an illustrated example,
see Calculate Sensitivities Using SimFunctionSensitivity Object.

Ingalls, B.P, and Sauro, H.M. (2003). Sensitivity analysis of
stoichiometric networks: an extension of metabolic control analysis
to non-steady state trajectories. J Theor Biol. *222(1)*,
23–36.

Martins, J.R.R.A., Sturdza, P., and Alanso, J.J. (Jan. 2001).
The connection between the complex-step derivative approximation and
algorithmic differentiation. AIAA Paper *2001–0921*.

Martins, J.R.R.A., Kroo, I.M., and Alanso, J.J. (Jan. 2000).
An automated method for sensitivity analysis using complex variables.
AIAA Paper *2000–0689*.

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