Persistence, Coexistence, and Boundary Transcritical Relays for Multi-Strain Epidemic Models

Persistence, coexistence, and boundary transcritical relays are usually studied through model-specific analyses in mathematical epidemiology, ecology, population dynamics, and chemical reaction network theory. Although these fields address closely related questions, they have developed largely independently. This separation is reflected, for example, in the limited mentions of the multi-strain epidemiologic models in ecology’s chemostats and gradostats literature, despite the fact that these are revealed to be very similar once the concept of siphons from chemical reaction network theory is integrated. Conversely, the next-generation matrices and invasion graphs from eco-epidemiology are not mentioned in chemical reaction network theory. Our contribution is firstly conceptual, terminological and definitional: we propose a common framework for the study of boundary phenomena in all positive ODE subfields. We introduce and formalize notions like reproduction and invasion functions attached to siphon faces, relay graphs, relay tables, and boundary transcritical relays. Some of these concepts are known in one of the above fields but largely absent from the others, while others appear to be new; taken together, they suggest a common language for the analysis of boundary phenomena in positive dynamical systems. The usefulness of the framework is illustrated on multi-strain epidemic models like the Feng–Gavish model, for which we derive explicit, testable conditions. For example, invasion of the less fit strain into the fitter strain’s equilibrium is sufficient for coexistence—unconditionally under permanent immunity and together with an explicit feasibility condition on a reduced coexistence polynomial otherwise (for this model, mutual invasibility also ensures persistence; whether invasion is also necessary for coexistence, and explicit further assumptions under which one or the other criterion works for a larger class of models, are still open). Our approach rests on four pillars: (i) Siphon (a CRN concept) geometry, namely, the fact that forward-invariant coordinate faces correspond to siphons, with the disease-free face being the intersection of minimal siphons. (ii) The recently established fact that a transversal Jacobian block on a siphon face is Metzler, which puts under spotlight the roles of its Perron eigenvectors. (iii) A bifurcation theorem linking eigenvalue crossing at a boundary transcritical invasion relay to the emergence of a positive branch on an adjacent face. (iv) Next-generation matrices (NGMs), an ME concept: on siphon faces, NGMs may be defined via regular splittings, and invasibility may be determined by comparing their spectral radii to >1.