Abstract

Polynomial nonlinearities arise naturally in many epidemiological models, where interactions between compartments are often represented through multivariate polynomial terms. As these models increase in complexity, they frequently involve high-dimensional variable spaces with redundant algebraic structure, complicating both analysis and computation. In this paper, we introduce an apolarity-based framework for reducing the algebraic complexity of polynomial epidemiological systems. By extending the notion of essential variables from homogeneous polynomials to systems of polynomial differential equations, we define the essential variable dimension of a model and provide a systematic method for identifying minimal variable representations using catalecticant matrices and differential operator duality. This approach is purely symbolic and preserves the exact structure of the governing equations without relying on numerical approximations. The methodology is applied to several epidemiological models of increasing complexity, including the classical SIR model, extended SIR systems, and models incorporating vaccination and vector-borne transmission. In each case, we demonstrate that the nonlinear interaction structure admits a reduced representation, revealing inherent algebraic redundancies. These results establish a novel connection between apolarity theory and mathematical epidemiology and provide a rigorous preprocessing tool for simplifying complex compartmental models.

Publication status

Published - 2026