Counterexample to the "all new infection terms" choice in NGM

Construct a counterexample where defining the NGM "new infections" matrix F by collecting all positive-sign, nonlinear infection terms fails to satisfy the conditions of the NGM theorem of Diekmann–Heesterbeek–Metz and van den Driessche–Watmough, thereby demonstrating limits of the "all" choice.

Background

The authors adopt a unique, algorithmic choice for F by including all positive new infection terms, which empirically worked in their examples. However, classical NGM theory does not mandate a unique decomposition, and the "all" choice may not always meet its technical conditions.

A concrete counterexample would delineate the scope of the "all" choice and guide refinements of heuristic NGM constructions.