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On the global asymptotic stability of an infection-age structured competitive model

Simon Girel, Quentin Richard

Abstract

We investigate an infection-age structured competitive epidemiological model involving multiple strains. While classical results establish competitive exclusion when a unique maximal basic reproduction number exists, we provide here a complete characterization of the asymptotic behavior for an arbitrary number of populations without assuming uniqueness of the maximal reproduction number. By means of integrated semigroups theory, persistence results, and Lyapunov functionals, we establish global asymptotic stability of equilibria and extend previous results obtained for simpler (ODE) models. A key contribution lies in overcoming technical difficulties related to the definition and differentiation of Lyapunov functionals, as well as in refining arguments based on the LaSalle invariance principle.

On the global asymptotic stability of an infection-age structured competitive model

Abstract

We investigate an infection-age structured competitive epidemiological model involving multiple strains. While classical results establish competitive exclusion when a unique maximal basic reproduction number exists, we provide here a complete characterization of the asymptotic behavior for an arbitrary number of populations without assuming uniqueness of the maximal reproduction number. By means of integrated semigroups theory, persistence results, and Lyapunov functionals, we establish global asymptotic stability of equilibria and extend previous results obtained for simpler (ODE) models. A key contribution lies in overcoming technical difficulties related to the definition and differentiation of Lyapunov functionals, as well as in refining arguments based on the LaSalle invariance principle.

Paper Structure

This paper contains 8 sections, 8 theorems, 62 equations.

Table of Contents

  1. Introduction
  2. Assumptions and results
  3. Proof of the global stability
  4. Case $\mathcal{R}_{0,1}\leq 1$
  5. Case $\mathcal{R}_{0,1}>1$
  6. Existence of a global attractor
  7. Lyapunov functional
  8. Computation of the global attractor and conclusion

Key Result

Lemma 2.2

Suppose that Assumption Assum:1 holds. Then: For the second and third points, the constant $K_r$ can be taken as $K_r:=2r\sum_{k=1}^n \|\beta_k\|_{L^\infty}$.

Theorems & Definitions (15)

  • Lemma 2.2
  • Proposition 2.3
  • proof
  • Corollary 2.4
  • proof
  • Theorem 2.5
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 5 more