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<p>We introduce a graph-theoretical approach to epidemiological modeling that automates the derivation of the differential equations associated with a model and the computation of its base and real-time reproduction numbers. Our framework defines a novel structure called "epidemiological hypergraphs", graphs extended with epidemiological characteristics in order to automatize their analysis. The main focus of this article is to index a few individuals of interest and explicitly track their secondary infections, emulating the granularity of agent-based models. This structure also removes the need for model-specific analysis, improving reproducibility and enhancing accessibility for epidemiologists. We validate consistency with the next-generation matrix approach for the base reproduction number while demonstrating superior analytical accuracy over the classical estimate <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathcal {R}_t = S / N \cdot \mathcal {R}_0\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="script">R</mi> <mi>t</mi> </msub> <mo>=</mo> <mi>S</mi> <mo stretchy="false">/</mo> <mi>N</mi> <mo>·</mo> <msub> <mi mathvariant="script">R</mi> <mn>0</mn> </msub> </mrow> </math></EquationSource> </InlineEquation> for the real-time reproduction number, especially in scenarios where parameters evolve in time or when variants are introduced. Our approach offers an adaptive mathematical framework for real-time epidemic tracking and intervention planning.</p>

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A Graph-Theoretical Framework for Automated Computation of Reproduction Numbers in Deterministic Epidemiological Models

  • 期刊论文

  • 发表日期：

    2026年7月23日

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  • Alexandre Simard,
  • Jacques Bélair

摘要

We introduce a graph-theoretical approach to epidemiological modeling that automates the derivation of the differential equations associated with a model and the computation of its base and real-time reproduction numbers. Our framework defines a novel structure called "epidemiological hypergraphs", graphs extended with epidemiological characteristics in order to automatize their analysis. The main focus of this article is to index a few individuals of interest and explicitly track their secondary infections, emulating the granularity of agent-based models. This structure also removes the need for model-specific analysis, improving reproducibility and enhancing accessibility for epidemiologists. We validate consistency with the next-generation matrix approach for the base reproduction number while demonstrating superior analytical accuracy over the classical estimate \(\mathcal {R}_t = S / N \cdot \mathcal {R}_0\) R t = S / N · R 0 for the real-time reproduction number, especially in scenarios where parameters evolve in time or when variants are introduced. Our approach offers an adaptive mathematical framework for real-time epidemic tracking and intervention planning.