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A Tutorial on Symbolic Structural Identifiability Analysis of ODE Models in Julia

This tutorial introduces a modern, reproducible computational framework using the Julia package StructuralIdentifiability.jl to perform symbolic structural identifiability analysis on ODE models, guiding researchers through theoretical concepts and practical workflows via seven diverse case studies to ensure reliable parameter estimation.

Original authors: Abdallah Alsammani

Published 2026-05-20

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Abdallah Alsammani

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a detective trying to solve a mystery. You have a crime scene (the data you collected) and a theory about what happened (a mathematical model of how the system works). Your goal is to figure out the exact details of the crime: the speed of the getaway car, the time of the robbery, the number of suspects.

This paper is a guide for detectives who use Ordinary Differential Equation (ODE) models. These are complex math formulas used to describe how things change over time, like how a virus spreads, how a drug moves through the body, or how a population grows.

The paper asks a very specific question before you even start looking for clues: "Is it actually possible to figure out the exact details of the crime, even if we had perfect, noise-free information?"

This concept is called Structural Identifiability.

Here is the breakdown of the paper's main ideas using simple analogies:

1. The Core Problem: The "Blindfolded" Detective

Sometimes, your math model is set up in a way that makes it impossible to know the true values of your variables, no matter how much data you collect.

The Analogy: Imagine you are watching a car drive away, but you can only see its headlights. You can tell how fast the car is moving (the output), but you can't tell if it's a fast car with a weak engine or a slow car with a super-charged engine. Two completely different setups produce the exact same result.
The Paper's Claim: If a model is "non-identifiable," it means different combinations of parameters (like speed and engine power) create the exact same outcome. If you try to estimate these numbers from real data, your computer will just guess randomly, and your results will be meaningless.

2. The Solution: A New Detective Tool (Julia & StructuralIdentifiability.jl)

The paper introduces a new, modern toolkit built in a programming language called Julia. Think of this toolkit as a "Truth Detector" for your math models.

How it works: Instead of guessing, this tool uses advanced math (symbolic computation) to analyze the equations before you collect any data. It checks: "If I change this number, does the output change in a unique way?"
The Workflow:
1. Write the Model: You tell the computer your equations (like writing down your theory of the crime).
2. Local Check: The tool does a quick scan to see if the numbers are even locally unique (like checking if the car's speed is unique in a small neighborhood).
3. Global Check: If the first check passes, it does a deeper scan to see if the numbers are unique everywhere (checking if the car's speed is unique across the whole city).
4. The Fix: If the numbers aren't unique, the tool tells you exactly which combinations of numbers are unique.

3. The "Magic" of Combinations

Often, you can't find the individual numbers, but you can find their product or sum.

The Analogy: Imagine you have a secret code where two numbers, $A$ and $B$ , are multiplied together to make a result. You can't tell if $A=2$ and $B=3$ , or if $A=6$ and $B=1$ . Both give you 6.
The Paper's Insight: The tool tells you: "Stop trying to find $A$ and $B$ separately. They are a mystery. But the product ( $A \times B$ ) is perfectly clear. Change your report to focus on the product."
Real-world Example from the Paper: In a model of how a virus spreads, you might not be able to tell how fast the virus infects cells ( $\beta$ ) versus how fast infected cells burst ( $p$ ). But the tool reveals that the product of these two ( $\beta \times p$ ) is the only thing that matters. You should report that product, not the individual numbers.

4. The Power of "More Eyes" (Adding Measurements)

The paper shows that sometimes the problem isn't the model; it's that you aren't looking at enough things.

The Analogy: If you only watch the car's headlights, you can't tell the engine type. But if you add a second camera that watches the car's exhaust pipe, you might finally be able to distinguish the engine.
The Paper's Examples:
- Viral Dynamics: If you only measure the amount of virus in the blood, you can't separate all the biological rates. But if you also measure the number of healthy cells, the mystery is solved, and every number becomes identifiable.
- Epidemics: If you only count new infections, you can't tell how much of the spread is from person-to-person versus from the environment (like dirty water). But if you also measure the virus in the water, the mystery is solved.

5. Why This Matters (The "Don't Waste Your Time" Rule)

The paper argues that scientists often skip this step. They build a model, run it on a computer, get a result, and publish it.

The Risk: If the model was structurally non-identifiable, the result they got was just a lucky guess or a mathematical artifact. It has no real-world meaning.
The Recommendation: Always run this "Truth Detector" before you try to estimate parameters. It saves time, prevents false conclusions, and tells you exactly what kind of data you need to collect to solve the mystery.

Summary of the Paper's Contribution

This paper is a tutorial. It teaches researchers how to use a specific software package (StructuralIdentifiability.jl) to:

Diagnose if their model is solvable.
Identify which parts of the model are solvable and which are not.
Reformulate the model (by grouping numbers together or adding new measurements) so that it becomes solvable.

It emphasizes that this isn't just theoretical math; it's a practical, step-by-step process that can be done on a laptop using open-source tools, ensuring that scientific models are built on solid ground before they are used to make predictions about diseases, drugs, or ecosystems.

Technical Summary: A Tutorial on Symbolic Structural Identifiability Analysis of ODE Models in Julia

1. Problem Statement

Mechanistic ordinary differential equation (ODE) models are fundamental tools in epidemiology, systems biology, pharmacokinetics, and biomedical engineering for describing dynamical processes. However, the reliability of parameter estimation and subsequent inference depends critically on structural identifiability: the theoretical capacity to uniquely recover unknown model parameters from ideal, noise-free, continuous observations.

The paper addresses the gap between the mathematical theory of structural identifiability and its practical application in modern scientific workflows. While theoretical frameworks exist (dating back to Bellman and Åström), their application is hindered by:

Software Barriers: Many existing tools rely on proprietary computer algebra systems (e.g., Maple, MATLAB), limiting reproducibility.
Expertise Requirements: Symbolic methods often demand specialized mathematical knowledge, creating a barrier for applied scientists.
Computational Complexity: Global identifiability analysis for large-scale nonlinear models has historically been computationally prohibitive.
Workflow Disconnection: Identifiability analysis is often treated as a separate diagnostic step rather than an integrated part of the modeling, simulation, and estimation pipeline.

Consequently, researchers frequently calibrate non-identifiable models, leading to non-unique parameter estimates, inflated uncertainty, and misleading biological interpretations.

2. Methodology

The paper presents a unified computational framework for symbolic structural identifiability analysis using the Julia programming language and the StructuralIdentifiability.jl package. The methodology integrates symbolic computation with the broader Julia SciML ecosystem (including ModelingToolkit.jl, DifferentialEquations.jl, and Catalyst.jl).

Mathematical Foundations

The analysis considers parametric ODE systems of the form:
$\dot{x}(t) = f(x(t), u(t); \theta), \quad x(0) = x_0(\theta), \quad y(t) = g(x(t), u(t); \theta)$
where $x$ are states, $u$ are inputs, $y$ are outputs, and $\theta$ are parameters. The core problem is determining the injectivity of the parameter-to-output map $\Phi$ .

The framework distinguishes between:

Local Identifiability: Parameters are uniquely determined within a neighborhood of the true value.
Global Identifiability: Parameters are uniquely determined over the entire parameter space.
Identifiable Functions: When individual parameters are non-identifiable, rational combinations of parameters (generators of the identifiable field $K_{id}$ ) may still be uniquely recoverable.

Algorithmic Approach

The workflow employs two primary algorithmic strategies:

Local Identifiability (Sedoglavic Test): A probabilistic, polynomial-time algorithm that evaluates symbolic sensitivities at a random point to verify the rank of a Jacobian matrix. This serves as a fast, initial screening.
Global Identifiability (Differential Algebra): A method based on differential elimination through projections (SIAN-style). It eliminates unobserved states to derive input-output relations, whose coefficients are identifiable functions of the parameters. This approach is computationally efficient for mid-sized nonlinear biological models compared to classical Gröbner basis methods.

Computational Workflow

The paper proposes a standardized pipeline implemented via four core functions:

@ODEmodel: A macro for declaring ODE systems, states, parameters, inputs, and outputs in near-mathematical syntax.
assess_local_identifiability: Executes the Sedoglavic test to flag locally non-identifiable parameters.
assess_identifiability: Performs the global test to distinguish between globally and locally identifiable parameters.
find_identifiable_functions: Computes the generators of the identifiable field $K_{id}$ , providing the necessary reparameterization (e.g., identifying that only the product $pq$ is recoverable, not $p$ and $q$ individually).

3. Key Contributions

The paper contributes a comprehensive tutorial and reproducible reference that bridges theoretical identifiability with practical implementation. Key contributions include:

A Unified Julia Framework: The integration of symbolic identifiability analysis into the open-source Julia SciML ecosystem, allowing seamless interoperability with numerical simulation, automatic differentiation, and Bayesian inference tools.
Standardized Workflow: A clear, step-by-step pipeline (Local Test $\to$ Global Test $\to$ Identifiable Function Computation) applicable to diverse mechanistic models.
Case Study Demonstrations: Seven detailed case studies from epidemiology, pharmacokinetics, and viral dynamics illustrating:
- Globally identifiable systems (e.g., SIR model with incidence data).
- Local-only identifiability due to exchange symmetries (e.g., two-compartment pharmacokinetic models).
- Structural non-identifiability and the recovery of identifiability via additional measurements (e.g., adding environmental reservoir data to SIWR models).
- Reparameterization strategies using identifiable combinations (e.g., estimating $\beta p$ instead of $\beta$ and $p$ separately in viral dynamics).
Practical Guidance: Concrete recommendations for experimental design, model reformulation, and the interpretation of identifiability results in applied settings.

4. Results

The application of the framework to the seven case studies yields specific structural insights:

SIR Model: Demonstrated that incidence data ( $\beta SI$ ) alone is sufficient for global identifiability of both transmission ( $\beta$ ) and recovery ( $\gamma$ ) rates, despite the unobserved recovered compartment.
Pharmacokinetics (2-Compartment): Confirmed the classic result that rate constants $a_{01}$ and $a_{12}$ are locally but not globally identifiable due to an exchange symmetry. The framework successfully identified the symmetric combinations ( $a_{01} + a_{12}$ and $a_{01}a_{12}$ ) as the globally identifiable quantities.
Viral Dynamics (Perelson Model): Showed that measuring only viral load ( $V$ ) renders the infection rate ( $\beta$ ) and production rate ( $p$ ) non-identifiable individually, with only the product $\beta p$ being recoverable. The analysis confirmed that adding a measurement of uninfected target cells ( $T$ ) restores global identifiability for all parameters.
SIWR and SEIR-H Models: Illustrated how aggregated incidence data confounds environmental transmission parameters. The framework demonstrated that adding specific measurements (e.g., environmental pathogen concentration or case incidence) resolves these non-identifiabilities.
Reparameterization: In all non-identifiable cases, the find_identifiable_functions tool successfully generated the minimal set of identifiable combinations, providing a constructive path for model reformulation.

5. Significance and Claims

The paper posits that structural identifiability analysis is a necessary prerequisite for reliable parameter estimation, uncertainty quantification, and mechanistic interpretation. Its significance is framed as follows:

Constructive Diagnostics: Identifiability analysis is not merely a diagnostic tool to reject models but a constructive one that guides experimental design (e.g., "which additional measurements are needed?") and model reformulation (e.g., "what combinations should be estimated?").
Reproducibility: By utilizing open-source Julia tools and providing archived, executable notebooks, the framework addresses the reproducibility crisis in computational modeling, ensuring that identifiability claims can be independently verified.
Accessibility: The tutorial lowers the barrier to entry for applied scientists, moving identifiability analysis from a niche theoretical exercise to a routine step in the modeling workflow.
Distinction from Practical Identifiability: The authors emphasize that structural identifiability is an intrinsic property of the model and measurement scheme, distinct from practical identifiability (which depends on data quality and noise). A model must be structurally identifiable before practical issues can be meaningfully addressed.

The paper concludes that modern symbolic methods, when integrated into open-source ecosystems, make structural identifiability analysis accessible and computationally practical, urging researchers to incorporate it as a standard preliminary step before parameter estimation.