A Geometric Formulation of Transmission Dynamics: Social Risk Metrics in Hilbert Spaces and Stability Analysis

Nonlinear dynamical systems often exhibit complex collective behavior arising from the interaction of multiple elementary modes. In this work we investigate the aggregation of a countable family of dynamical modes generated by Lotka-Volterra systems and study the resulting structure in a Hilbert space of observables. The classical Lotka-Volterra equations form a fundamental class of nonlinear models describing interacting populations through coupled differential equations, while Koopman operator theory provides a framework in which nonlinear dynamics can be represented as a linear evolution acting on observable functions.We show that a countable aggregation of such dynamical modes admits a well-defined limit in a Hilbert space when the coefficients belong to l2. The resulting aggregated observable evolves according to the associated Koopman semigroup, yielding a linear representation of the underlying nonlinear dynamics in the observable space. We further prove that the geometry induced by this aggregated dynamics admits a canonical class of equivalent metrics generated by coercive operators, ensuring that the stability topology of the system is independent of the particular metric chosen within this class.Finally, we illustrate the theoretical framework by introducing a social risk functional defined as a quadratic observable associated with the induced metric. This example demonstrates how application-specific quantities can naturally arise from the geometric structure generated by aggregated nonlinear dynamics.