A Perron-Volterra Lyapunov function for mathematical epidemiology reaction networks (MERN) with non-interacting rank-one strains

Abstract

This paper introduces a general framework for analyzing global asymptotic stability (GAS), competitive exclusion, coexistence, and persistence in mathematical epidemiology reaction networks. The authors construct Perron-Volterra Lyapunov functions to yield competitive-exclusion GAS partitions of the parameter space. This applies to bilinear m-strain models with non-interacting, irreducible rank-one infection blocks. The parameter space splits into at most m+1 generic partition sets where either the disease-free equilibrium (DFE) is GAS, or one dominant strain is GAS. Coexistence only occurs on non-generic tie surfaces with equal reproduction numbers. The method integrates chemical reaction networks and mathematical epidemiology through five pillars: * Siphon Sets: Forward-invariant boundary faces are defined by siphon sets, with the DFE face being the intersection of minimal siphons. * Metzler Jacobians: Transversal blocks on siphon faces are Metzler, emphasizing their Perron eigenvectors. * Next-Generation Matrices: Defined via regular splittings on siphon faces to determine invasibility via spectral radii. * Invasion Relays: Link eigenvalue crossings to emergent positive branches. * Lyapunov Construction: Combines Volterra entropy for resident variables with Perron-weighted linear functionals for invading blocks. A second GAS partition is provided for two-strain models with increasing concave incidence. The approach is algorithmically implemented in the Mathematica package EPIDCRN to recursively compute eigenvectors and build Lyapunov functions.