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A Graph-Theoretical Framework for Automated Computation of Reproduction Numbers in Deterministic Epidemiological
Alexandre Simard1, Jacques Bélair2,3,4
1Département de mathématiques et de statistique, Université de Montréal, C.P. 6128 succursale Centre-ville, H3C 3J7, Montréal, QC, Canada. as.simard99@gmail.com.
Bulletin of Mathematical Biology
|July 23, 2026
Summary
This study introduces epidemiological hypergraphs, a novel graph-theory method for automated epidemic modeling. This approach enhances real-time tracking and intervention planning by improving accuracy in calculating reproduction numbers.
Area of Science:
- Epidemiology
- Graph Theory
- Mathematical Modeling
Background:
- Traditional epidemiological models often require complex, model-specific analyses.
- Calculating real-time reproduction numbers can be analytically challenging, especially with evolving parameters or variants.
Purpose of the Study:
- To develop an automated, graph-theoretical framework for epidemiological modeling.
- To introduce "epidemiological hypergraphs" for enhanced analysis of disease transmission.
- To improve the accuracy and accessibility of calculating base and real-time reproduction numbers.
Main Methods:
- Defined "epidemiological hypergraphs" as an extension of graph theory.
- Automated the derivation of differential equations for epidemiological models.
- Tracked secondary infections with agent-based model granularity.
Main Results:
- Validated consistency with the next-generation matrix approach for the base reproduction number.
- Demonstrated superior analytical accuracy over classical estimates for the real-time reproduction number.
- Showcased improved performance in scenarios with evolving parameters and variants.
Conclusions:
- The epidemiological hypergraph approach offers an adaptive framework for real-time epidemic tracking.
- This method enhances reproducibility and accessibility for epidemiologists.
- The framework supports more accurate intervention planning through precise reproduction number calculations.
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