Sufficient conditions for correctness of NGM heuristic and rank-one shortcut

Provide sufficient conditions under which the next-generation matrix (NGM) heuristic and the rank-one NGM shortcut method correctly determine the local stability domain of the disease-free equilibrium (DFE) for compartmental reaction-network models.

Background

The NGM approach replaces analysis of the Jacobian of infection dynamics by an auxiliary matrix constructed from "new infection" and transition terms. While widely used in ME, rigorous validity conditions for heuristic variants (including rank-one shortcuts) are not fully understood, especially for structured CRNs and polynomial kinetics.

This problem seeks general criteria guaranteeing these heuristic constructions match the true DFE stability domain, improving reliability and scope of NGM-based methods.