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Dynamic properties of Kermack-McKendrick-like models
Hamidou A Diallo1, Khalil Ezzinbi1,2, Nisrine Outada1
1Department of Mathematics, Faculty of Sciences Semlalia, Cadi Ayyad University, Marrakesh, Morocco.
Plos One
|July 14, 2026
Summary
This study analyzes infectious disease dynamics using a detailed 13-compartment model, revealing how immunity duration, disease severity, and age impact epidemic spread and outcomes.
Area of Science:
- Epidemiology
- Mathematical Biology
- Infectious Disease Modeling
Background:
- Classical SIR models offer a foundation for infectious disease dynamics.
- Expanding models to include disease progression and immunity is crucial for accurate predictions.
Purpose of the Study:
- To rigorously analyze the dynamic properties of an extended Kermack-McKendrick compartmental model.
- To investigate the influence of immunity duration, clinical progression, and age structure on epidemic outcomes.
Main Methods:
- Developed a 13-compartment mathematical model for infectious diseases.
- Established mathematical results on model well-posedness and stability.
- Analyzed both permanent and temporary immunity scenarios.
- Incorporated an age-stratified multi-group extension.
Main Results:
- Derived the basic reproduction number (R0) and a first integral for final epidemic size.
- Proved global stability of the disease-free equilibrium for permanent immunity.
- Demonstrated the existence of an endemic equilibrium for R0 > 1 with temporary immunity.
- Showcased the impact of age structure on epidemic dynamics.
Conclusions:
- The study provides a robust mathematical framework for understanding complex infectious disease dynamics.
- Results highlight the critical roles of immunity, disease severity, and population structure in shaping epidemics.
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