A Perron-Frobenius strong threshold theorem for $(A, B, P, \varphi)$ balanced bilinear model

Rim Adenane; Florin Avram; Andrei-Dan Halanay; Andras Horvath; Sei Zhen Khong; Rim Adenane; Florin Avram; Andrei-Dan Halanay; Andras Horvath; Sei Zhen Khong

doi:10.3934/mbe.2026098

Mathematical Biosciences and Engineering

2026, Volume 23, Issue 9: 2720-2770. doi: 10.3934/mbe.2026098

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A Perron-Frobenius strong threshold theorem for $(A, B, P, \varphi)$ balanced bilinear model

1.
Department of Mathematics, Laboratory of Analysis, Geometry and Applications, Faculty of Science, Ibn Tofail University, Kenitra, 14000, Morocco
2.
Department de Mathématiques, Université de Pau et des Pays de l'Adour, Pau, 64000, France
3.
Department of Mathematics and Computer Science, University of Bucharest, Bucharest, RO-010014, Romania
4.
Department of Computer Science, University of Turin, Turin, 10124, Italy
5.
Department of Electrical Engineering, National Sun Yat-sen University, Kaohsiung, 80424, Taiwan

Received: 29 April 2026 Revised: 17 July 2026 Accepted: 14 August 2026 Published: 09 September 2026

We establish a Perron–Frobenius threshold theorem for positive chemical ODEs, and apply it to a family of bilinear epidemic models with rank-one next-generation matrices (NGMs). The main results are: (ⅰ) the Jacobian on every siphon face of a chemical ODE is block-triangular, with a Metzler transversal block (Theorems 2–3); (ⅱ) a Perron–Frobenius strong threshold theorem (Theorem 6) characterizing existence, and under an irreducibility hypothesis uniqueness, of the endemic equilibrium via the spectral condition $ \rho(\widetilde K(S)) = 1 $, with no rank-one assumption; (ⅲ) an algebraic-independence result splitting the rank-one bilinear models of Fall, Iggidr, Sallet, and Bonzi into two distinct classes (Theorems 4–5), unifying and extending these authors' results together with those of Shuai and Van den Driessche; (ⅳ) explicit closed-form Lyapunov functions and equilibrium formulas for this rank-one class, built from the left and right Perron eigenvectors of the NGM (§5), with their modification under linear feedback from infectious to susceptible compartments (§6); and (ⅴ) a Kirchhoff matrix-tree formula expressing the same Perron eigenvectors as graph-theoretic diagonal cofactors, valid at arbitrary rank (§7).
- positive ODE,
- disease-free and endemic equilibrium,
- Metzler matrices,
- next-generation matrix,
- regular splitting,
- Perron–Frobenius eigenvectors
Citation: Rim Adenane, Florin Avram, Andrei-Dan Halanay, Andras Horvath, Sei Zhen Khong. A Perron-Frobenius strong threshold theorem for $(A, B, P, \varphi)$ balanced bilinear model[J]. Mathematical Biosciences and Engineering, 2026, 23(9): 2720-2770. doi: 10.3934/mbe.2026098

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Abstract

We establish a Perron–Frobenius threshold theorem for positive chemical ODEs, and apply it to a family of bilinear epidemic models with rank-one next-generation matrices (NGMs). The main results are: (ⅰ) the Jacobian on every siphon face of a chemical ODE is block-triangular, with a Metzler transversal block (Theorems 2–3); (ⅱ) a Perron–Frobenius strong threshold theorem (Theorem 6) characterizing existence, and under an irreducibility hypothesis uniqueness, of the endemic equilibrium via the spectral condition $ \rho(\widetilde K(S)) = 1 $, with no rank-one assumption; (ⅲ) an algebraic-independence result splitting the rank-one bilinear models of Fall, Iggidr, Sallet, and Bonzi into two distinct classes (Theorems 4–5), unifying and extending these authors' results together with those of Shuai and Van den Driessche; (ⅳ) explicit closed-form Lyapunov functions and equilibrium formulas for this rank-one class, built from the left and right Perron eigenvectors of the NGM (§5), with their modification under linear feedback from infectious to susceptible compartments (§6); and (ⅴ) a Kirchhoff matrix-tree formula expressing the same Perron eigenvectors as graph-theoretic diagonal cofactors, valid at arbitrary rank (§7).

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