The Similarity Between Epidemiologic Strains, Minimal Self-Replicable Siphons, and Autocatalytic Cores in (Chemical) Reaction Networks: Towards a Unifying Framework

Motivation: We aim to study the boundary stability and persistence of positive odes in mathematical epidemiology models by importing structural tools from chemical reaction networks. This is largely a review work, which attempts to congregate the fields of mathematical epidemiology (ME), and chemical reaction networks (CRNs), based on several observations. We started by observing that epidemiologic strains, defined as disjoint blocks in either the Jacobian on the infected variables, or as blocks in the next generating matrix (NGM), coincide in most of the examples we studied, with either the set of critical minimal siphons or with the set of minimal autocatalytic sets (cores) in an underlying CRN. We leveraged this to provide a definition of the disease-free equilibrium (DFE) face/infected set as the union of either all minimal siphons, or of all cores (they always coincide in our examples). Next, we provide a proposed definition of ME models, as models which have a unique boundary fixed point on the DFE face, and for which the Jacobian of the infected subnetwork admits a regular splitting, which allows defining the famous next generating matrix. We then define the interaction graph on minimal siphons (IGMS), whose vertices are minimal siphons, and whose edges indicate the existence of reactions producing species in one siphon from species in another. When this graph is acyclic, we say the model exhibits an Acyclic Minimal Siphon Decomposition (AMSD). For AMSD models whose minimal siphons partition the infection species, we show that the NGM is block triangular after permutation, which implies the classical max structure of the reproduction number $R_0$ for multi-strain models. In conclusion, using irreversible reaction networks, minimal siphons and acyclic siphon decompositions, we provide a natural bridge from CRN to ME. We implement algorithms to compute IGMS and detect AMSD in our Epid-CRN Mathematica package (which already contain modules to identify minimal siphons, criticality, drainability, self-replicability, etc.). Finally, we illustrate on several multi-strain ME examples how the block structure induced by AMSD, and the ME reproduction functions, allow expressing boundary stability and persistence conditions by comparing growth numbers to 1, as customary in ME. Note that while not addressing the general Persistence Conjecture mentioned in the title, our work provides a systematic method for deriving boundary instability conditions for a significant class of structured models.