From the Volterra type Lyapunov functions of Rahman-Zou towards a competitive exclusion partition property for rank one models

TL;DR

This paper introduces a Perron-Volterra framework for stability analysis of multi-strain epidemic models with rank-one matrices, proving the competitive exclusion partition property and providing an algorithmic approach.

Contribution

It develops a systematic Lyapunov function construction method for rank-one models, extending to multiple strains and implementing it in a Mathematica package.

Findings

Proves the competitive exclusion partition property for two-strain models.

Provides an explicit Lyapunov function for each region in parameter space.

Extends the framework to models with multiple singleton strains and rank-one blocks.

Abstract

This paper presents a Perron-Volterra framework that unifies explicit Lyapunov constructions for multi-strain epidemic models with rank-one next-generation matrices. At each boundary equilibrium on a siphon face, the Lyapunov function consists of a Volterra entropy on resident variables plus a Perron-weighted linear functional on invaders, derived from the left Perron eigenvector of the transversal Jacobian. A balance identity cancels coupling terms, reducing global stability to recursive computation of invasion numbers on the siphon lattice. For two-strain models with concave, increasing incidence, we prove the competitive exclusion partition property (CEPP): the parameter space splits into four open regions, each possessing a unique globally asymptotically stable equilibrium (disease-free, single-strain, or coexistence) certified by an explicit Lyapunov function. The same mechanism extends to an arbitrary number of singleton strains and to models with one scalar strain and one irreducible rank-one block. We implement the algorithmic approach in the Mathematica package EpidCRN, which constructs candidate Lyapunov functions, verifies the balance identity, and partitions the parameter space recursively. For two rank-one matrix blocks, the standard ansatz fails; we characterize the obstruction and propose an augmented cross-equilibrium Lyapunov function. A local Lyapunov theorem for siphon faces is also provided. The framework offers a systematic stability analysis of rank-one models.