Analytical Mechanics Allows Novel Vistas on Mathematical Epidemic Dynamics Modelling

TL;DR

This paper explores the use of analytical mechanics to reframe and analyze the basic SIR epidemic model, revealing new perspectives and potential research directions by applying Hamiltonian and Lagrangian formalisms.

Contribution

It introduces a novel approach by recasting the SIR model within analytical mechanics, using re-parameterizations and Hamiltonian/Lagrangian formalisms to open new research avenues.

Findings

Re-parameterization of the SIR model using time scaling and coordinate transformation.

Application of Hamilton's equations and Hamilton's principle to epidemic dynamics.

Identification of new research directions through mechanics-epidemic analogy.

Abstract

This contribution aims to shed light on mathematical epidemic dynamics modelling from the viewpoint of analytical mechanics. To set the stage, it recasts the basic SIR model of mathematical epidemic dynamics in an analytical mechanics setting. Thereby, it considers two possible re-parameterizations of the basic SIR model. On the one hand, it is proposed to re-scale time, while on the other hand, to transform the coordinates, i.e.\ the independent variables. In both cases, Hamilton's equations in terms of a suited Hamiltonian as well as Hamilton's principle in terms of a suited Lagrangian are considered in minimal and extended phase and state space coordinates, respectively. The corresponding Legendre transformations relating the various options for the Hamiltonians and Lagrangians are detailed. Ultimately, this contribution expands on a multitude of novel vistas on mathematical epidemic dynamics modelling that emerge from the analytical mechanics viewpoint. As result, it is believed that interesting and relevant new research avenues open up when exploiting in depth the analogies between analytical mechanics and mathematical epidemic dynamics modelling.