##  [Parameter Sensitivity](/parameter-sensitivity-0) 

 Definition

A quantitative description of how small variations in model parameters or inputs change model outputs or solutions, often expressed by derivatives (gradients, Jacobians) for local sensitivity or by variance-based measures for global sensitivity.

 

 

 

 

 

 





## Principle

Principle

Local linearization and the chain rule: for sufficiently small perturbations, output change ≈ Jacobian times parameter perturbation; global measures integrate or sample nonlinear responses over parameter ranges to capture broader effects.

 

 

 

 

 





## Demonstration

Demonstration

In a parametric model y(θ), the Jacobian ∂y/∂θ evaluated at an estimated θ predicts which parameters most affect predicted values; in epidemiological models, small changes in transmission rate may produce large changes in peak incidence if sensitivity is high.

 

 

 

 

## Misapplication

Misapplication

Interpreting a large local sensitivity as implying practical importance without considering parameter uncertainty ranges, nonlinear effects, or identifiability confuses sensitivity with real-world influence.

 

 

 

 

 





## Consequence

Consequence

Sensitivity analysis identifies influential parameters, informs experiment design, parameter estimation priorities, and model reduction by revealing parameters that can be fixed or constrained.

 

 

 

 

## Reversal

Reversal

Parameter insensitivity describes parameters whose variations produce negligible output change, indicating potential for simplification or that data cannot identify them robustly.

 

 

 

 

 





## Boundary

Boundary

Sensitivity concerns the mapping from parameters to outputs; it does not by itself quantify stochastic uncertainty or model misspecification and depends on the scale, metric, and perturbation regime used (local vs global).

 

 

 

 

 





## Semantic Tension

Semantic Tension

Closely related to condition numbers in linear settings but distinct: condition numbers quantify input-to-solution amplification for a problem, while parameter sensitivity targets how model outputs change with model parameters and may be local or global.

 

 

 

 

 





## Synthesis

Synthesis

Parameter sensitivity brings gradients, Jacobians, and global variance measures into a unified view: it quantifies which parameters control outputs under specified perturbation scales and thus guides identifiability, inference, and reduction.