A Perron-Frobenius strong threshold theorem for $(A, B, P, \varphi)$ balanced bilinear model
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Rim Adenane1, Florin Avram2, Andrei-Dan Halanay3
1Department of Mathematics, Laboratory of Analysis, Geometry and Applications, Faculty of Science, Ibn Tofail University, Kenitra, 14000, Morocco.
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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