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arxiv: 2605.24629 · v1 · pith:TG7W4LETnew · submitted 2026-05-23 · 🧮 math.DS

A Perron-Frobenius strong threshold theorem for (A, B, P, {φ}) balanced bilinear models, and the role of left and right Perron eigenvectors in mathematical epidemiology

Florin Avram , Rim Adenane , Andrei-Dan Halanay , Andras Horvath , Sei Zhen Khong This is my paper

Pith reviewed 2026-06-30 11:57 UTC · model grok-4.3

classification 🧮 math.DS MSC 34C6092D30
keywords Perron-Frobenius theoremnext-generation matrixbilinear epidemic modelsendemic equilibriumLyapunov functionsmathematical epidemiologythreshold theoremspectral radius
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The pith

A nontrivial positive endemic equilibrium exists in these bilinear models precisely when the spectral equation ρ(K̃(S))=1 admits a strictly positive solution.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves a strong threshold theorem for a family of (A, B, P, ϕ) balanced bilinear epidemic models that extends earlier rank-one results. A positive endemic equilibrium appears if and only if the next-generation matrix evaluated at some positive state S has spectral radius exactly one. The left Perron eigenvector of that matrix supplies weights for Lyapunov functions that prove global stability of both the disease-free and endemic equilibria, while the right eigenvector describes the direction in which the system leaves the disease-free face. The authors classify the rank-one cases into two explicit families and note that the same eigenvector-based construction works when feedback from infectious to susceptible compartments is added.

Core claim

In (A, B, P, ϕ) balanced bilinear models whose next-generation matrix K = F(S)V^{-1} satisfies the structural conditions for the Perron-Frobenius theorem, a nontrivial positive equilibrium exists if and only if ρ(K̃(S))=1 admits a strictly positive solution S; the left Perron eigenvector π(S) yields Lyapunov functions establishing the global stability exchange between the disease-free and endemic equilibria, while the right eigenvector w(S) governs the exit direction from the siphon face.

What carries the argument

The next-generation matrix K = F(S)V^{-1} together with its left and right Perron eigenvectors π(S) and w(S) for the balanced bilinear structure (A, B, P, ϕ).

If this is right

  • Bonzi-Iggidr-Sallet rank-one models split into two explicit classes whose equilibria and eigenvectors are given by closed formulas.
  • Lyapunov functions for both equilibria are obtained directly from the left Perron eigenvector of the next-generation matrix.
  • The threshold result extends beyond rank one by replacing explicit eigenvector formulas with the spectral equation ρ(K̃(S))=1.
  • Feedback from infectious to susceptible compartments preserves the eigenvector-based Lyapunov construction.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same left-eigenvector weighting may produce global stability certificates in non-rank-one next-generation matrices once the spectral condition is verified.
  • Numerical continuation of the equation ρ(K̃(S))=1 could locate the endemic equilibrium without solving the full algebraic system.
  • The exit direction given by the right eigenvector suggests a practical test for whether an observed outbreak trajectory is leaving the disease-free state along the predicted ray.

Load-bearing premise

The system must belong to the (A, B, P, ϕ) balanced bilinear family so that the next-generation matrix obeys the positivity and irreducibility conditions needed for a unique positive Perron eigenvector.

What would settle it

A concrete (A, B, P, ϕ) model in which ρ(K̃(S)) never equals 1 for any positive S yet a positive endemic equilibrium still exists, or in which ρ(K̃(S))=1 has a positive solution but no positive equilibrium appears.

read the original abstract

This paper started as a review of seven results pertaining to a family of bilinear models with rank one NGM introduced by Fall, Iggidr, Sallet and Bonzi, which utilize the explicit eigenvectors of the NGM to compute the unique endemic equilibrium (EE), and Lyapunov functions at both the disease free equilibrium (DFE) and EE, and of results of Shuai and Van den Driessche (2013), which essentially deal with the same "DFE -EE stability exchange" in the non-rank one case, when the eigenvectors are not explicit. Recently, these results were complemented by Earn and McCluskey (2025), who proved as well a ``strong threshold theorem", namely that when the DFE is unstable, a second equilibrium which is globally asymptotically stable must exist. Below, we obtain in Theorem \ref{thm:TK_DFE} some results that extend beyond rank one. For example, a nontrivial positive equilibrium exists if and only if the spectral equation \(\rho(\widetilde K(S))=1\), admits a strictly positive solution. Also, we showed that Bonzi-Iggidr-Sallet bilinear models with rank one NGM may be classified in two classes, with slight variations in the eigenvector formulas, and that extensions in the presence of feedback from infectious to susceptible are possible. Another takeout from the previous works, which we clarify in a revisit of the seven results in the rank one case, is that Lyapunov functions for both the DFE and the EE may be constructed using as weights the left Perron eigenvector $\pi(S)$ of the \NGM\ (NGM) $K=F(S) V^{-1}$, where $S$ denote all the non-infectious variables, and when a positive ODE leaves a siphon face, it does so along the right Perron eigenvector $w(S)$ of $\T K= V^{-1}F(S)$. The question of whether this continues to be true beyond rank one is explored in ongoing work.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The manuscript claims a strong threshold theorem for (A, B, P, ϕ) balanced bilinear models: a nontrivial positive equilibrium exists if and only if the spectral equation ρ(K̃(S))=1 admits a strictly positive solution (Theorem TK_DFE). It classifies rank-one NGM models of Bonzi-Iggidr-Sallet into two classes with explicit left/right Perron eigenvector formulas for equilibria and Lyapunov functions, extends prior results on DFE-EE stability exchange, and notes that Lyapunov weights can be taken from the left Perron eigenvector π(S) of K=F(S)V^{-1} while leaving siphons occurs along the right eigenvector w(S).

Significance. If the derivation is complete, the result supplies an explicit if-and-only-if criterion and eigenvector-based constructions that extend rank-one NGM analyses to a broader bilinear family, clarifying the role of Perron vectors in both equilibrium computation and global stability proofs without fitted parameters.

major comments (1)
  1. [Theorem TK_DFE] Theorem TK_DFE (main result section): the iff claim that a strictly positive equilibrium exists precisely when ρ(K̃(S))=1 has a strictly positive solution presupposes that K̃(S) is irreducible for every admissible S>0 so that Perron-Frobenius supplies a strictly positive eigenvector. No structural hypothesis on the parameters A, B, P, ϕ is stated that guarantees this irreducibility (or primitivity) uniformly in S; without it the eigenvector may be merely nonnegative, breaking both the strict-positivity construction and the Lyapunov arguments that invoke π(S).

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for identifying the need for an explicit irreducibility hypothesis in the statement of Theorem TK_DFE. We address the major comment below.

read point-by-point responses

  1. Referee: [Theorem TK_DFE] Theorem TK_DFE (main result section): the iff claim that a strictly positive equilibrium exists precisely when ρ(K̃(S))=1 has a strictly positive solution presupposes that K̃(S) is irreducible for every admissible S>0 so that Perron-Frobenius supplies a strictly positive eigenvector. No structural hypothesis on the parameters A, B, P, ϕ is stated that guarantees this irreducibility (or primitivity) uniformly in S; without it the eigenvector may be merely nonnegative, breaking both the strict-positivity construction and the Lyapunov arguments that invoke π(S).

    Authors: We agree that the current statement of Theorem TK_DFE presupposes irreducibility of K̃(S) for S>0 to invoke the strict positivity of the Perron eigenvector, yet no such structural hypothesis on A, B, P, ϕ is explicitly stated. While the balanced bilinear structure of the models considered often yields irreducible next-generation matrices under standard epidemiological connectivity assumptions, this is not formalized in the manuscript. We will therefore revise the paper to add the explicit assumption that K̃(S) is irreducible (or primitive) for every admissible S>0. This assumption will be incorporated into the hypotheses of Theorem TK_DFE and the subsequent Lyapunov constructions, thereby ensuring the strict-positivity claims hold rigorously. revision: yes

Circularity Check

0 steps flagged

No significant circularity; central claims rest on standard Perron-Frobenius application to model-derived NGM under stated structural assumptions.

full rationale

The derivation in Theorem TK_DFE equates existence of a strictly positive equilibrium to the spectral equation ρ(K̃(S))=1 admitting a positive solution, constructed via left/right Perron eigenvectors of K=F(S)V^{-1}. This follows directly from the Perron-Frobenius theorem (an external mathematical result) applied to matrices defined from the (A,B,P,ϕ) balanced bilinear family, without any reduction to self-definition, fitted inputs renamed as predictions, or load-bearing self-citations whose content is unverified. The paper reviews prior results on rank-one cases and extends them, but the extension itself supplies independent structural content rather than importing uniqueness via overlapping-author citations. The noted irreducibility requirement is a potential assumption gap (correctness concern) but does not make the stated equivalence tautological or self-referential by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work relies on the classical Perron-Frobenius theorem for positive matrices and on the structural definition of balanced bilinear models; no free parameters are introduced and no new entities are postulated.

axioms (2)
  • standard math Perron-Frobenius theorem guarantees a unique positive eigenvector and spectral radius for irreducible nonnegative matrices
    Invoked to guarantee existence and uniqueness of left and right Perron eigenvectors π(S) and w(S) for the next-generation matrix K(S)
  • domain assumption The (A, B, P, ϕ) bilinear models satisfy the balanced structural conditions that keep the next-generation matrix nonnegative and allow the spectral equation to be well-posed
    Required for the classification into two classes and for the extension beyond rank one stated in the abstract

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