Hamiltonian structure of compartmental epidemiological models

TL;DR

This paper demonstrates that epidemiological compartmental models with constant population can be formulated as Hamiltonian systems, revealing new bi-Hamiltonian cases and providing tools for exact solutions, including COVID-19 dynamics.

Contribution

It introduces a Hamiltonian framework for epidemiological models, including new bi-Hamiltonian cases and explicit Poisson structures, facilitating analytical solutions.

Findings

Epidemiological models can be represented as Hamiltonian systems.

Some models exhibit bi-Hamiltonian structures.

Poisson structures enable exact analytical solutions, including for COVID-19.

Abstract

Any epidemiological compartmental model with constant population is shown to be a Hamiltonian dynamical system in which the total population plays the role of the Hamiltonian function. Moreover, some particular cases within this large class of models are shown to be bi-Hamiltonian. New interacting compartmental models among different populations, which are endowed with a Hamiltonian structure, are introduced. The Poisson structures underlying the Hamiltonian description of all these dynamical systems are explicitly presented, and their associated Casimir functions are shown to provide an efficient tool in order to find exact analytical solutions for epidemiological models, such as the ones describing the dynamics of the COVID-19 pandemic.