Acyclic Minimal Siphon Decomposition (AMSD)

• AMSD is a structural concept that decomposes infected variables in chemical reaction networks by identifying disjoint minimal siphons via an acyclic interaction directed graph.
• It employs topological ordering to yield block diagonal or block lower triangular structures in the next-generation matrix, facilitating the analysis of strain-specific reproduction numbers.
• This approach provides actionable insights into epidemiological stability by linking minimal siphons with infection dynamics and multi-strain threshold behavior.

Searching arXiv for the specified paper and closely related terminology to ground the article in the cited source. Acyclic Minimal Siphon Decomposition (AMSD) is a structural notion proposed for positive ODE models that are representable as chemical reaction networks and interpreted epidemiologically through an infected–resident decomposition. In its basic form, AMSD is defined by constructing the Interaction Directed Graph on Minimal Siphons (IGMS) and requiring that this directed graph be acyclic. In the setting emphasized for mathematical epidemiology, AMSD becomes consequential when the minimal siphons also partition the infected variables: under that stronger hypothesis, the infection Jacobian admits a block-triangular next-generation structure, and the basic reproduction number satisfies the familiar multi-strain formula $R_0=\max_i \rho(K_{ii})$. The concept was introduced as part of a broader attempt to relate epidemiologic strains, critical minimal siphons, self-replicable siphons, and autocatalytic cores within a common reaction-network framework (Avram et al., 30 Oct 2025).

1. Formal setting and core definitions

The framework is developed for positive dynamical systems, meaning systems for which

$$\mathbb{R}^n_{\ge 0}:=\{x\in\mathbb{R}^n: x_i\ge 0,\ i=1,\dots,n\}$$

is forward invariant. The reaction-network representation is a chemical reaction network $(\mathcal S,\mathcal C,\mathcal R)$, where $\mathcal S$ is the set of species, $\mathcal C$ the set of complexes, and $\mathcal R$ the set of reactions, with ODE form

$X' = RHS(X)=\Gamma\, rts(X).$

The reaction-level representation is essential because siphons depend on reactant and product structure, not only on the stoichiometric matrix $\Gamma$.

The epidemiological decomposition separates variables into infected or invading variables $x$, which vanish at the disease-free equilibrium, and resident or non-infection variables $y$, which remain positive there. The system is written as

$$\mathbb{R}^n_{\ge 0}:=\{x\in\mathbb{R}^n: x_i\ge 0,\ i=1,\dots,n\}$$0

A proposed definition of a mathematical epidemiology model further requires a nonempty DFE index set, a unique boundary point on the corresponding face, the possibility that DFE stability changes with parameters, stability of $$\mathbb{R}^n_{\ge 0}:=\{x\in\mathbb{R}^n: x_i\ge 0,\ i=1,\dots,n\}$$1 at the DFE, and a regular splitting of the infection Jacobian

$$\mathbb{R}^n_{\ge 0}:=\{x\in\mathbb{R}^n: x_i\ge 0,\ i=1,\dots,n\}$$2

with next-generation matrix

$$\mathbb{R}^n_{\ge 0}:=\{x\in\mathbb{R}^n: x_i\ge 0,\ i=1,\dots,n\}$$3

A nonempty subset $$\mathbb{R}^n_{\ge 0}:=\{x\in\mathbb{R}^n: x_i\ge 0,\ i=1,\dots,n\}$$4 is a siphon or semilocking set if whenever a species in $$\mathbb{R}^n_{\ge 0}:=\{x\in\mathbb{R}^n: x_i\ge 0,\ i=1,\dots,n\}$$5 appears in a product complex, at least one species in $$\mathbb{R}^n_{\ge 0}:=\{x\in\mathbb{R}^n: x_i\ge 0,\ i=1,\dots,n\}$$6 appears in the corresponding reactant complex. A locking set is a nonempty subset $$\mathbb{R}^n_{\ge 0}:=\{x\in\mathbb{R}^n: x_i\ge 0,\ i=1,\dots,n\}$$7 such that every reaction has at least one species from $$\mathbb{R}^n_{\ge 0}:=\{x\in\mathbb{R}^n: x_i\ge 0,\ i=1,\dots,n\}$$8 in its reactant complex. A minimal siphon is a siphon containing no proper subset that is itself a siphon. The standard CRNT fact recalled in the paper is that a nonempty set is semilocking iff the corresponding boundary face is forward invariant.

Criticality is defined through conservation relations. A siphon $$\mathbb{R}^n_{\ge 0}:=\{x\in\mathbb{R}^n: x_i\ge 0,\ i=1,\dots,n\}$$9 is critical if there exists no nonzero vector $(\mathcal S,\mathcal C,\mathcal R)$0 with

$(\mathcal S,\mathcal C,\mathcal R)$1

and

$(\mathcal S,\mathcal C,\mathcal R)$2

The union of all minimal critical siphons is called the total siphon,

$(\mathcal S,\mathcal C,\mathcal R)$3

and the corresponding face

$(\mathcal S,\mathcal C,\mathcal R)$4

is the DFE face. If a boundary fixed point $(\mathcal S,\mathcal C,\mathcal R)$5 exists, it is the DFE. Within this framework, the infected set is structurally identified with the union of minimal critical siphons.

Two related CRNT notions play a central role. A siphon $(\mathcal S,\mathcal C,\mathcal R)$6 is self-replicating if there exists a nonzero flux $(\mathcal S,\mathcal C,\mathcal R)$7 such that

$(\mathcal S,\mathcal C,\mathcal R)$8

A restricted stoichiometric matrix $(\mathcal S,\mathcal C,\mathcal R)$9 is stoichiometrically autocatalytic if there exists $\mathcal S$0, $\mathcal S$1, such that $\mathcal S$2; a minimal square such submatrix is an autocatalytic core. The paper uses these notions to motivate a structural correspondence among strains, critical minimal siphons, self-replicable siphons, and minimal autocatalytic cores (Avram et al., 30 Oct 2025).

2. Interaction Directed Graph on Minimal Siphons

The Interaction Directed Graph on Minimal Siphons is the graph whose vertices are the minimal siphons $\mathcal S$3. There is a directed edge

$\mathcal S$4

whenever there exists a reaction net-producing at least one species in $\mathcal S$5 from reactants in $\mathcal S$6. The theorem-level formulation sharpens this criterion: for $\mathcal S$7, an edge $\mathcal S$8 is present iff there exists a reaction whose reactant set intersects $\mathcal S$9 and whose product set intersects $\mathcal C$0.

Operationally, for each reaction $\mathcal C$1, one inspects the supports of its reactants and products. If

$\mathcal C$2

then the edge $\mathcal C$3 is added. In this sense, IGMS records which minimal siphons can contribute to the production of species in other minimal siphons.

The graph-theoretic property used to define AMSD is acyclicity. An acyclic directed graph admits a topological ordering $\mathcal C$4, and the main theorem uses exactly such an ordering to permute infected variables blockwise. The graph therefore functions both as a structural summary of reaction-level interdependence and as the combinatorial device that induces triangular matrix structure when the required partition hypothesis is also satisfied.

The paper does not present a separate equivalence theorem for alternative characterizations of IGMS, but it makes explicit three perspectives: a reaction-based edge criterion, a block-dependence interpretation for linearized infection dynamics, and an algorithmic computation from the reaction list together with the previously computed minimal siphons. In the implementation, the function

$\mathcal C$5

returns edges, cycles, and a graph object (Avram et al., 30 Oct 2025).

3. Definition of AMSD and the block-triangular theorem

In the narrow definitional sense, a model exhibits an Acyclic Minimal Siphon Decomposition precisely when its IGMS is acyclic. The abstract formulation is therefore purely graph-theoretic: compute the minimal siphons, construct the IGMS, and check whether it contains a directed cycle.

The principal theorem requires more than acyclicity. Let $\mathcal C$6 be the infected set, and suppose that $\mathcal C$7 is a partition of $\mathcal C$8 into siphons,

$\mathcal C$9

Define the inter-block transition digraph on these $\mathcal R$0 by the same edge rule as above, and assume that this graph is acyclic. If $\mathcal R$1 is any topological ordering, reorder the infected variables by listing all species in $\mathcal R$2, then those in $\mathcal R$3, and so on. Under these hypotheses:

• $\mathcal R$4 is block diagonal,
• $\mathcal R$5 is block lower triangular,
• hence

$\mathcal R$6

is block lower triangular with $\mathcal R$7 diagonal blocks.

If the graph has no edges, then $\mathcal R$8 is block diagonal.

The corollary is immediate: $\mathcal R$9 This is the mechanism by which AMSD recovers the classical max-structure of the reproduction number in multi-strain models. If each diagonal block corresponds to a strain-specific subsystem, then writing

$X' = RHS(X)=\Gamma\, rts(X).$0

gives

$X' = RHS(X)=\Gamma\, rts(X).$1

The theorem is embedded in the regular-splitting framework for Metzler matrices. For a regular splitting $X' = RHS(X)=\Gamma\, rts(X).$2 with $X' = RHS(X)=\Gamma\, rts(X).$3 and $X' = RHS(X)=\Gamma\, rts(X).$4, the cited matrix-theoretic criterion is

$X' = RHS(X)=\Gamma\, rts(X).$5

This makes the block structure of $X' = RHS(X)=\Gamma\, rts(X).$6 directly relevant to invasion and DFE stability.

A technical nuance noted in the source is that the printed proof states that both $X' = RHS(X)=\Gamma\, rts(X).$7 and $X' = RHS(X)=\Gamma\, rts(X).$8 are block lower triangular, whereas the theorem statement asserts the stronger and epidemiologically standard conclusion that $X' = RHS(X)=\Gamma\, rts(X).$9 is block diagonal and $\Gamma$0 block lower triangular. The intended statement is clearly the latter, because the max-formula is derived from a block decomposition in which new-infection terms remain within blocks (Avram et al., 30 Oct 2025).

4. Structural interpretation in epidemiology and CRNT

The central epidemiological interpretation is that minimal critical siphons behave like strains or irreducible infected subsystems. This motivates the paper’s explicit definition of a $\Gamma$1-strain model as an ME model with $\Gamma$2 critical minimal siphons and, after permutation, a triangular block structure of the next-generation matrix with $\Gamma$3 blocks. In this formulation, “strain” is not introduced as a purely biological primitive; rather, it is structurally encoded by a critical minimal siphon.

The paper’s broader unifying claim is more cautious than a theorem of equivalence. It reports that, in all studied examples, epidemiologic strains correspond conceptually to both critical minimal siphons and minimal autocatalytic sets, and that the infected set at the DFE can be defined as the union of either all minimal siphons or all cores, which coincide in those examples. It also recalls CRNT results that every minimal critical siphon of a weakly reversible CRN is either drainable or self-replicating, that every minimal self-replicating siphon contains a minimal autocatalytic core, and that every autocatalytic core defines a self-replicating siphon equal to its species support. At the same time, the paper explicitly does not prove a universal theorem that minimal siphons always equal strains or cores.

Acyclicity of IGMS is interpreted as hierarchical dependence among infected subsystems. The source lists biological readings such as one strain producing another, mutation or conversion, catalytic creation of downstream infected classes, and one-way cross-feeding or hierarchical seeding. This suggests that AMSD formalizes a one-directional dependency architecture rather than reciprocal feedback among minimal siphons. The connection drawn to hierarchical seed-dependent autocatalytic systems follows the same logic.

AMSD is not presented as a persistence theorem. Its role is narrower and more structural: it provides a systematic way to derive boundary instability conditions and threshold decompositions for a significant class of models. The paper is explicit that it is not solving the general persistence conjecture for positive systems. The main payoff lies instead in relating DFE invasion analysis, boundary faces defined by siphons, and block-structured NGMs.

The notion of reproduction functions fits naturally into this picture. The paper defines $\Gamma$4 as any strictly positive eigenvalue of the NGM with resident variables left free, and at the DFE writes $\Gamma$5. Under a block decomposition, these positive eigenvalues arise from diagonal blocks, so AMSD provides structural support for strain-specific reproduction functions and for reading invasion thresholds blockwise (Avram et al., 30 Oct 2025).

5. Representative models and instructive counterexamples

The paper analyzes several models in which IGMS and AMSD are computed explicitly. Together, they show both the utility and the limits of the framework.

Model	Minimal siphons / IGMS	Structural consequence
Three-tier SDAS-type model	$\Gamma$6; chain $\Gamma$7	AMSD holds; NGM is triangular, indeed diagonal
“5cycles” example	$\Gamma$8, $\Gamma$9, $x$0; highly cyclic IGMS	AMSD fails, yet $x$1 is block diagonal
Combination/mutation example	$x$2, $x$3; overlapping siphons	Partition fails, yet $x$4 is triangular and $x$5
Rahman SI$x$6V model	$x$7, $x$8; no IGMS edges	Trivial AMSD; diagonal $x$9; $y$0
Kozlov/GKTW two-strain coinfection model	$y$1, $y$2; IGMS is a cycle	AMSD fails, yet $y$3 can be triangular
Gavish-type two-strain model	$y$4, $y$5; disjoint	Structurally AMSD-compatible; block-diagonal $y$6

The three-tier SDAS-type model is the cleanest positive instance. Its three disjoint minimal siphons induce an acyclic chain $y$7, and the source states that the NGM is triangular, indeed diagonal, in line with the theorem. The Rahman SI$y$8V model is similarly transparent: the minimal siphons $y$9 and $$\mathbb{R}^n_{\ge 0}:=\{x\in\mathbb{R}^n: x_i\ge 0,\ i=1,\dots,n\}$$00 partition the infected variables, IGMS has no edges, and

$$\mathbb{R}^n_{\ge 0}:=\{x\in\mathbb{R}^n: x_i\ge 0,\ i=1,\dots,n\}$$01

with reproduction functions

$$\mathbb{R}^n_{\ge 0}:=\{x\in\mathbb{R}^n: x_i\ge 0,\ i=1,\dots,n\}$$02

The Gavish-type two-strain model provides a larger block-diagonal example. The minimal siphons

$$\mathbb{R}^n_{\ge 0}:=\{x\in\mathbb{R}^n: x_i\ge 0,\ i=1,\dots,n\}$$03

are disjoint and partition the infection variables $$\mathbb{R}^n_{\ge 0}:=\{x\in\mathbb{R}^n: x_i\ge 0,\ i=1,\dots,n\}$$04. After permutation, the NGM has two diagonal blocks, and the paper identifies

$$\mathbb{R}^n_{\ge 0}:=\{x\in\mathbb{R}^n: x_i\ge 0,\ i=1,\dots,n\}$$05

so that

$$\mathbb{R}^n_{\ge 0}:=\{x\in\mathbb{R}^n: x_i\ge 0,\ i=1,\dots,n\}$$06

Equally important are the negative examples. In the “5cycles” network, the minimal siphons overlap and IGMS contains multiple cycles, so AMSD fails. Nevertheless,

$$\mathbb{R}^n_{\ge 0}:=\{x\in\mathbb{R}^n: x_i\ge 0,\ i=1,\dots,n\}$$07

is block diagonal. In the combination/mutation example, overlapping minimal siphons again preclude the theorem’s partition hypothesis, yet the NGM remains triangular. The Kozlov/GKTW two-strain coinfection model sharpens the point further: IGMS is cyclic and minimal siphons overlap in $$\mathbb{R}^n_{\ge 0}:=\{x\in\mathbb{R}^n: x_i\ge 0,\ i=1,\dots,n\}$$08, but the NGM can still be triangular, with

$$\mathbb{R}^n_{\ge 0}:=\{x\in\mathbb{R}^n: x_i\ge 0,\ i=1,\dots,n\}$$09

These examples establish a central nuance: AMSD is a sufficient structural condition for block triangularization and max-type threshold decomposition, not a necessary one. They also show why the partition assumption is essential in the theorem as stated. When minimal siphons overlap, the number of meaningful reproduction functions need not align cleanly with the number of minimal siphons, and biologically natural triangular orderings may exist without being directly captured by AMSD (Avram et al., 30 Oct 2025).

6. Computation, limitations, and open questions

The computational workflow is implemented in the Epid-CRN Mathematica package. The paper describes a pipeline that computes minimal siphons, identifies the DFE face from the union of minimal critical siphons, computes the Jacobians $$\mathbb{R}^n_{\ge 0}:=\{x\in\mathbb{R}^n: x_i\ge 0,\ i=1,\dots,n\}$$10 and $$\mathbb{R}^n_{\ge 0}:=\{x\in\mathbb{R}^n: x_i\ge 0,\ i=1,\dots,n\}$$11, obtains a regular splitting and NGM, and constructs IGMS to detect cycles. The routine bdAn[RN, rts] returns the ODE right-hand side, variables and parameters, minimal siphons mSi, $$\mathbb{R}^n_{\ge 0}:=\{x\in\mathbb{R}^n: x_i\ge 0,\ i=1,\dots,n\}$$12, $$\mathbb{R}^n_{\ge 0}:=\{x\in\mathbb{R}^n: x_i\ge 0,\ i=1,\dots,n\}$$13, the DFE E0, the NGM K, reproduction functions or nonzero eigenvalues R0A, full NGM data including $$\mathbb{R}^n_{\ge 0}:=\{x\in\mathbb{R}^n: x_i\ge 0,\ i=1,\dots,n\}$$14, and the infection variable order. The routine $$\mathbb{R}^n_{\ge 0}:=\{x\in\mathbb{R}^n: x_i\ge 0,\ i=1,\dots,n\}$$19 computes the interaction graph, its edge set, and its directed cycles.

An implicit procedure for AMSD detection is clear. Starting from a reaction network and rates, one computes the minimal siphons $$\mathbb{R}^n_{\ge 0}:=\{x\in\mathbb{R}^n: x_i\ge 0,\ i=1,\dots,n\}$$15, builds IGMS by comparing reactant and product supports against these siphons, and tests whether the resulting digraph is acyclic. If no cycle is present, the model has AMSD in the narrow definitional sense. If, in addition, the $$\mathbb{R}^n_{\ge 0}:=\{x\in\mathbb{R}^n: x_i\ge 0,\ i=1,\dots,n\}$$16 partition the infected variables, the block-triangular theorem applies; a topological sort then gives the variable ordering needed for blockwise analysis of

$$\mathbb{R}^n_{\ge 0}:=\{x\in\mathbb{R}^n: x_i\ge 0,\ i=1,\dots,n\}$$17

No formal complexity bounds are given. The source only states that algorithms to compute IGMS and detect AMSD have been implemented, alongside existing modules for minimal siphons, criticality, drainability, and self-replicability.

The main limitations are also explicit. AMSD does not characterize all triangularizable epidemic models, because cyclic IGMS and overlapping minimal siphons can coexist with triangular or block-diagonal NGMs. Overlap among minimal siphons creates ambiguity in block assignment and weakens the direct identification of strains with siphons. The framework therefore works best in models where minimal siphons are both strain-like and disjoint.

Several open questions are identified. One concerns whether reproduction functions are always increasing in resident variables at the DFE. Another asks which $$\mathbb{R}^n_{\ge 0}:=\{x\in\mathbb{R}^n: x_i\ge 0,\ i=1,\dots,n\}$$18-strain models satisfy a full competitive-exclusion or linear-complementarity partition of parameter space in terms of reproduction and invasion functions. A third concerns the exact role of IGMS cycles, since cycles do not necessarily destroy triangularity. A fourth concerns the absence of a general theorem establishing formal equivalence between siphons and autocatalytic cores in the epidemiological setting. These unresolved points delimit AMSD as a structurally informative framework rather than a complete theory of persistence, reducibility, or strain decomposition (Avram et al., 30 Oct 2025).