Lagrange and Hamilton geometries applied to a dynamical sistem governing COVID-19 disease
Ana-Maria Boldeanu, Mircea Neagu
TL;DR
This work develops a geometric formulation of a six-compartment COVID-19 spreading model on $M = \mathbb{R}^6$ by applying a least-squares variational approach to obtain both Lagrange and Hamilton geometries. The authors define a least-squares Lagrangian $L(x,y) = \sum_{i=1}^6 (y^i - X^i(x))^2$ and, within the Lagrange framework, compute a canonical nonlinear connection $N$, Lagrangian $d$-torsions, and a Lagrangian Yang–Mills energy, using a Jacobi stability analysis via the deviation tensor in the KCC sense. Transitioning to Hamilton geometry on $T^*M$, they derive the Hamiltonian $H(x,p) = \tfrac{1}{4}\sum p_i^2 + \sum X^i(x) p_i$ and its nonlinear connection with $\mathbf{N} = J + J^T$, obtaining Hamiltonian $d$-torsions and zero-adapted Cartan components; they also introduce energy-hypersurfaces $\Sigma_\rho$ defined by a sum of squared nonlinear-connection entries. The paper presents a novel geometric viewpoint on epidemiological dynamics and identifies open questions about the epidemiological interpretation of the constructed Lagrange–Hamilton objects, suggesting visual exploration of the level surfaces for potential insights into disease spread.
Abstract
In this paper we develop, via the least squares variational method, the Lagrange-Hamilton geometry (in the sense of nonlinear connections, d-torsions and Lagrangian Yang-Mills electromagnetic-like energy) produced by a dynamical system governing the spreading of COVID-19 disease. The Jacobi stability of this dynamical system is also discussed.