Hamiltonian structure of compartmental epidemiological models

Area of Science:

• Mathematical Biology
• Epidemiology
• Dynamical Systems Theory

Background:

• Compartmental models are widely used to study infectious disease dynamics.
• Understanding the underlying mathematical structure of these models is crucial for accurate predictions.
• Previous studies have explored various mathematical frameworks for epidemiological modeling.

Purpose of the Study:

• To demonstrate that epidemiological compartmental models with constant populations are Hamiltonian dynamical systems.
• To introduce new interacting compartmental models with Hamiltonian structures.
• To utilize the Hamiltonian framework for finding exact analytical solutions for epidemiological models.

Main Methods:

• Representing epidemiological compartmental models as Hamiltonian dynamical systems.
• Identifying the Hamiltonian function (total population) and Poisson structures.
• Analyzing bi-Hamiltonian properties in specific model cases.
• Utilizing Casimir functions associated with Poisson structures for analytical solutions.

Main Results:

• All epidemiological compartmental models with constant populations are Hamiltonian systems.
• Certain models exhibit bi-Hamiltonian properties.
• New interacting compartmental models with inherent Hamiltonian structures were developed.
• Casimir functions derived from Poisson structures provide effective tools for finding exact analytical solutions.

Conclusions:

• The Hamiltonian framework offers a powerful approach to understanding and solving epidemiological models.
• This methodology can be applied to real-world scenarios, including the COVID-19 pandemic.
• The identified mathematical structures facilitate the development of more robust and predictable epidemiological models.