[Paper Review] An Algebraic Framework for Structured Epidemic Modeling

Sophie Libkind, Andrew Baas+3 more|arXiv (Cornell University)|Feb 28, 2022

Scientific Computing and Data Management4 citations

TL;DR

This paper proposes an algebraic framework grounded in applied category theory to model structured epidemic systems compositionally, enabling modular, reusable, and computationally transparent modeling of compartmental epidemiological models. By formalizing model composition via operads and wiring diagrams, the framework streamlines model specification, calibration, and analysis—demonstrated through parameter estimation and sensitivity analysis in a SVIIvR model—resulting in faster, more transparent, and reproducible pandemic modeling workflows.

ABSTRACT

Pandemic management requires that scientists rapidly formulate and analyze epidemiological models in order to forecast the spread of disease and the effects of mitigation strategies. Scientists must modify existing models and create novel ones in light of new biological data and policy changes such as social distancing and vaccination. Traditional scientific modeling workflows detach the structure of a model -- its submodels and their interactions -- from its implementation in software. Consequently, incorporating local changes to model components may require global edits to the code-base through a manual, time-intensive, and error-prone process. We propose a compositional modeling framework that uses high-level algebraic structures to capture domain-specific scientific knowledge and bridge the gap between how scientists think about models and the code that implements them. These algebraic structures, grounded in applied category theory, simplify and expedite modeling tasks such as model specification, stratification, analysis, and calibration. With their structure made explicit, models also become easier to communicate, criticize, and refine in light of stakeholder feedback.

Motivation & Objective

• Address the lack of structural transparency in traditional epidemiological modeling, where model components are tightly coupled and hard to modify.
• Enable scientists to specify, modify, and analyze complex epidemic models by explicitly representing their compositional and hierarchical structure in software.
• Bridge the gap between scientific intuition and software implementation by making model structure first-class in code, reducing errors and development time.
• Facilitate rapid adaptation of models to new data and policy changes (e.g., vaccination, social distancing) through modular, composable components.
• Support seamless integration of model analysis (e.g., calibration, sensitivity) with model construction, enabling real-time feedback during model development.

Proposed method

• Use operads and wiring diagrams (specifically undirected wiring diagrams) as a formal, diagrammatic syntax for composing models, enabling modular and hierarchical model construction.
• Define generalized algebraic operations—such as model addition and product—using category-theoretic constructs like copresheaves, structured cospans, and pullbacks to compose submodels.
• Decouple model syntax (how components are connected) from semantics (what mathematical model each component represents), enabling multiple semantics (e.g., ODEs, DDEs, Petri nets) to be used with the same syntax.
• Implement the framework in Julia using the Catalyst and ForwardDiff packages to generate and analyze ODEs from Petri net semantics, enabling simulation and sensitivity analysis.
• Integrate parameter estimation via optimization tools (e.g., from ModelingToolkit) to calibrate model parameters to data, with results visualized through heatmaps of sensitivity.
• Represent model structure as a combinatorial data structure, allowing algorithmic manipulation and enabling automated bookkeeping during model composition.

Experimental results

Research questions

• RQ1 How can the compositional structure of epidemic models be formally represented in a way that mirrors scientific intuition and enables modular development?
• RQ2 Can algebraic structures from applied category theory be used to unify diverse modeling semantics (e.g., ODEs, DDEs, Petri nets) under a single, composable framework?
• RQ3 To what extent does making model structure explicit in code improve the speed, accuracy, and reproducibility of model calibration and analysis?
• RQ4 How can sensitivity analysis and parameter estimation be tightly coupled with model composition to support real-time evaluation of policy decisions?
• RQ5 Can this framework support the dynamic replacement of submodels and rapid iteration in response to new biological or policy data?

Key findings

• The framework enables the specification of a stratified SVIIvR epidemic model using algebraic wiring diagrams, with model composition and semantics clearly separated.
• Parameter estimation for the SVIIvR model achieved a universal recovery rate of 1/14 and estimated initial populations and transition rates, with results visualized through a heatmap of sensitivities.
• Sensitivity analysis revealed that the transition rate β_UU had the highest negative sensitivity (−8.49×10⁻²), indicating strong influence on the integral of non-infectious populations.
• The model's sensitivity to β_VV was minimal (−2.18×10⁻¹⁰), suggesting low impact on the outcome, while ν had a small but non-negligible sensitivity (4.29×10⁻³).
• The framework supports immediate reflection of structural changes (e.g., adding/removing transitions) in downstream analyses, enabling rapid evaluation of model robustness.
• The integration of Catalyst and ForwardDiff allowed for end-to-end computation of ODEs, calibration, and sensitivity analysis, demonstrating a seamless workflow from model to policy insight.