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Basic concepts for the Kermack and McKendrick model with static heterogeneity.

Hisashi Inaba1

  • 1Department of Education, Tokyo Gakugei University, 4-1-1 Nukuikita-machi, Koganei-shi, Tokyo, 184-8501, Japan. inaba57@u-gakugei.ac.jp.

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Summary
This summary is machine-generated.

This study introduces a mathematical framework for infection-age-dependent epidemiological models, establishing foundational concepts and calculating key metrics like the basic and effective reproduction numbers for heterogeneous populations.

Keywords:
Basic reproduction numberEffective reproduction numberHerd immunity thresholdHeterogeneityKermack–McKendrick model

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Area of Science:

  • Epidemiology
  • Mathematical Biology
  • Public Health Modeling

Background:

  • The Kermack-McKendrick model is a cornerstone of epidemiological modeling.
  • Previous models often assumed homogeneous populations, limiting their applicability to real-world scenarios with diverse host characteristics.
  • Understanding infection-age dependency is crucial for accurate disease transmission dynamics.

Purpose of the Study:

  • To develop a rigorous mathematical framework for infection-age-dependent Kermack-McKendrick models with continuous state spaces.
  • To establish well-posedness for heterogeneous epidemiological models.
  • To define and compute fundamental epidemiological quantities such as the basic reproduction number, effective reproduction number, and herd immunity threshold.

Main Methods:

  • Development of a novel mathematical framework to formalize epidemiological concepts.
  • Analysis of model well-posedness under conditions with unbounded structural variables and domains.
  • Derivation of analytical results for pandemic thresholds.
  • Systematic procedures for computing effective reproduction number and herd immunity threshold, utilizing the separable mixing assumption.

Main Results:

  • Demonstrated mathematical well-posedness of the infection-age-dependent Kermack-McKendrick model.
  • Established pandemic threshold results based on the basic reproduction number.
  • Provided a computable method for the effective reproduction number and herd immunity threshold.
  • Illustrated model behavior with concrete examples under the separable mixing assumption.

Conclusions:

  • The developed framework provides a robust mathematical foundation for heterogeneous, age-structured epidemiological models.
  • The study offers practical tools for calculating critical epidemiological parameters essential for disease control and public health policy.
  • The findings are applicable to a wide range of infectious diseases where age structure and population heterogeneity play significant roles.