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Sparse identification of epidemiological compartment models with conserved quantities

Manuchehr Aminian, Kristin M. Kurianski

TL;DR

The paper tackles the challenge of data-driven discovery of epidemiological compartment models while enforcing principled conservation laws. It introduces a dictionary-based linear-in-parameters framework with a pure $\ell_1$ objective and a linear-programming strategy to impose population-conservation constraints, implemented via a nullspace-based correction to obtain a sparse, interpretable $W$ with $\dot{x} = \Phi(x) W$. Through synthetic experiments on SIR and SIS systems, the method achieves accurate dynamics, preserves total population, and reduces spurious compartmental interactions compared to baseline sparse identification approaches. This yields a principled, data-driven pathway to learn compartmental models that respect fundamental invariants, with potential applicability to real-world epidemic data and noisy observations.

Abstract

We investigate the application of a framework for sparse model identification of differential equations from timeseries data in the context of compartmental models in epidemiology. Such frameworks often seek a sparse representation from a polynomial basis in the state variables which reproduces the timeseries. Out-of-the-box approaches for the underlying sparse regression problem have moderate success reproducing the provided timeseries, but typically fail at producing a consistent, interpretable compartment model and conserving the total population, which are common properties in principled compartment modeling. Additionally, the conserved polynomial quantities, such as the sum of state variables, add algebraic nuances to a polynomial design matrix. We propose a linear program formulation to solve these issues by posing a pure one-norm objective, sampling from the nullspace of the design matrix, and imposing a set of linear constraints for between-compartment flows. We conduct several numerical experiments on synthetic data and succeed in ensuring model sparsity, accurately capturing system dynamics, and preserving conservation of population.

Sparse identification of epidemiological compartment models with conserved quantities

TL;DR

The paper tackles the challenge of data-driven discovery of epidemiological compartment models while enforcing principled conservation laws. It introduces a dictionary-based linear-in-parameters framework with a pure objective and a linear-programming strategy to impose population-conservation constraints, implemented via a nullspace-based correction to obtain a sparse, interpretable with . Through synthetic experiments on SIR and SIS systems, the method achieves accurate dynamics, preserves total population, and reduces spurious compartmental interactions compared to baseline sparse identification approaches. This yields a principled, data-driven pathway to learn compartmental models that respect fundamental invariants, with potential applicability to real-world epidemic data and noisy observations.

Abstract

We investigate the application of a framework for sparse model identification of differential equations from timeseries data in the context of compartmental models in epidemiology. Such frameworks often seek a sparse representation from a polynomial basis in the state variables which reproduces the timeseries. Out-of-the-box approaches for the underlying sparse regression problem have moderate success reproducing the provided timeseries, but typically fail at producing a consistent, interpretable compartment model and conserving the total population, which are common properties in principled compartment modeling. Additionally, the conserved polynomial quantities, such as the sum of state variables, add algebraic nuances to a polynomial design matrix. We propose a linear program formulation to solve these issues by posing a pure one-norm objective, sampling from the nullspace of the design matrix, and imposing a set of linear constraints for between-compartment flows. We conduct several numerical experiments on synthetic data and succeed in ensuring model sparsity, accurately capturing system dynamics, and preserving conservation of population.
Paper Structure (10 sections, 19 equations, 4 figures, 1 table)

This paper contains 10 sections, 19 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Methods
  3. Notation and problem formulation
  4. Specialization to compartmental models
  5. Conservation constraints in closed compartment models
  6. Selection of a sparse $W$ from candidate
  7. Studies with Synthetic Data
  8. Algorithm behavior with a family of SIS models.
  9. Discussion and Future Work
  10. Data and Code

Figures (4)

  • Figure 1: Diagram of SIR compartment model.
  • Figure 2: Results of two algorithms, in terms of timeseries reconstruction and recovery of the coefficient matrix. Left column: using forward differences and scikit-learn's implementation of ElasticNet ($\alpha=\mathrm{10^{-6}}$, $\mathrm{l1\_ratio}=0.5$) for regularized sparse regression. Right column: applying pySINDy's default settings and the ConstrainedSR3 for constrained optimizer with conservation constraints of Section \ref{['subsubsec:conservation']}. Full results of numerical studies are in Table \ref{['tab:numerical_results_exact_derivs']}.
  • Figure 3: Implementation of algorithm in Section \ref{['subsubsec:sparse_w']}. Left column: results using exact derivative information recovers both coefficient matrix to at least four digits precision and forward simulation reproduces the timeseries. Right column: when using forward differences to estimate the derivative, quality of the estimated system we recover is degraded. Note coefficients displayed are rounded for readability.
  • Figure 4: Evaluation of \ref{['subsec:specialization']} an SIS system. Timeseries are reproduced with a sparse coefficient matrix, but the the systems do not agree termwise. However, the dynamics $(S(t), I(t))$ are provably equivalent by exploiting an assumed conservation equation $S(t)+I(t)=1$.