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

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.