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.

• The paper establishes global asymptotic stability using advanced attractor analysis and Lyapunov functionals for infection-age structured PDE models.
• It applies integrated semigroup and persistence theories to manage multiple competing strains without requiring a unique maximal R0.
• The results offer clear criteria for competitive exclusion and strong persistence, providing insights for public health intervention strategies.

Global Asymptotic Stability in Multi-Strain Infection-Age Structured Competition

Introduction and Context

The paper investigates the large-time dynamics of a class of PDE epidemiological models for multiple competing infectious strains, structured by infection age. This extends classical ODE models of competitive exclusion to a larger and more general framework. The key advancement is a complete characterization of global asymptotic behavior without the restrictive assumption that the basic reproduction number ($R_0$) is uniquely maximized by a single strain. The authors synthesize several functional analytic and dynamical systems techniques—notably integrated semigroup theory, persistence frameworks, and Lyapunov functionals—to overcome the difficulties posed by non-uniqueness and the infinite-dimensional state space.

Model Formulation

The considered system couples an ODE (for susceptible density $S$) with a set of linear transport-type PDEs (for infected of age $a$, for each of $n$ strains), with infection-age-dependent transmission and removal rates. The model reads:

$$\begin{aligned} S'(t) &= \Lambda - \mu_S S(t) - S(t) \sum_{k=1}^n \int_0^\infty \beta_k(a) x_k(t,a) \, da \ \partial_t x_k(t,a) + \partial_a x_k(t,a) &= -\mu_k(a) x_k(t,a),\ x_k(t,0)&= S(t) \int_0^\infty \beta_k(a) x_k(t,a) da, \end{aligned}$$

where $S$ is the susceptible compartment, $x_k(t,a)$ is the density of individuals infected by strain $k$ and infected since $a$ units of time, $\beta_k(a)$ is the age-dependent transmissibility, and $S$0 is the removal/mortality rate.

Each strain $S$1 has a natural $S$2, expressed in terms of the time-dependent rates.

Main Results

Classification of Asymptotic Behavior

The dynamical landscape is comprehensively classified. For an arbitrary number $S$3 of competing strains:

1. Global asymptotic stability of disease-free equilibrium: If $S$4 for all $S$5, all solutions converge globally to the disease-free equilibrium $S$6 with extinction of all infected classes.
2. Competitive exclusion with possible coexistence: If the set $S$7 of strains with maximal $S$8 (possibly $S$9) is present, and $a$0, then the global attractor consists of all possible endemic equilibria where only strains in $a$1 persist. The limiting equilibrium is determined by the basin of attraction defined via initial conditions, extending classical ODE results on coexistence for the degenerate case $a$2.
3. Strong persistence: The system satisfies uniform (strong) persistence: whenever an initial infected density for a $a$3 strain is positive, it remains uniformly bounded away from zero asymptotically.

Technical Innovations

The novelty lies in the removal of the assumption that the maximal $a$4 is unique, the cornerstone of most earlier results (e.g., Martcheva & Li 2013, Magal et al. 2010). The authors provide a complete global stability and persistence analysis regardless of multiplicity and thus settle a significant open problem for infection-age PDE structured systems.

Key theoretical achievements:

• Induction-based global attractor analysis: Using invariance of subsets determined by absence/presence of strains, the argument proceeds via induction on the number of strains.
• Lyapunov functional construction: Generalized Lyapunov functionals are constructed that remain well-defined even under overlap of maximal $a$5. Technical challenges in their effective differentiation and definition are carefully handled.
• Extension of LaSalle invariance principles: The evolution on omega-limit and alpha-limit sets is rigorously connected to equilibrium selection, refining errors in prior PDE literature.

Strong uniform persistence is rigorously established, ruling out nontrivial attractors aside from equilibria. The omega-limit set for each admissible initial condition is a singleton equilibrium uniquely identified according to the support of initially active strains.

Analytical Framework

The proof leverages:

• Integrated semigroup theory for well-posedness and to define mild solutions in non-densely defined infinite-dimensional Banach state spaces.
• Spectral analysis for linearized (local) stability, extended here to the nonlinear (global) context via attractor theory.
• Dissipativity and asymptotic smoothness to establish existence of a compact global attractor.
• Persistence theory (Smith & Thieme, Hale & Waltman) to transition weak persistence to strong persistence due to attractor compactness.
• Sophisticated Lyapunov functionals that handle mass redistribution between strains and infection-age, using delicate density and regularity/approximation arguments.

The rigorous treatment of Lyapunov functionals, including careful construction to remain within the appropriate domain for all time, addresses errors found in some prior literature.

Implications and Significance

Theoretical

• This paper fully resolves the global stability and selection problem for infection-age structured competitive epidemic PDE systems with arbitrary numbers of strains and non-unique $a$6 maximizers, matching ODE-level understanding.
• The tools developed—for Lyapunov functionals and limiting behavior in non-cooperative, infinite-dimensional, non-irreducible PDE systems—are broadly applicable to ecological and epidemiological models with more complex structuring.

Practical

• Results justify the competitive exclusion principle at the PDE level even with tie in fitness ($a$7) between strains, so long as all strains with maximal $a$8 are initially present, coexistence equilibria can occur as in many ODE models.
• The explicit criteria and attractor analysis can inform public health intervention or pathogen strain control strategies where multiple variants or strains circulate with comparable fitness.
• The framework provides a template for analyzing evolutionary-epidemiological dynamics in more elaborate structured populations.

Future Directions

Several direct extensions become accessible:

• Within-host/between-host hybrid models: Combination of load- or immune-structure with infection-age.
• Feedback control or vaccination strategies: Incorporating time-dependent or adaptive interventions using attractor and persistence results.
• Stochastic analogs: Extending persistence and exclusion mechanisms to stochastic infection-age PDEs.
• Trait-structured or mutation-selection dynamics: Generalization to settings with a continuum of strains or mutational flows.

Conclusion

This work achieves a rigorous and comprehensive global dynamics classification for infection-age structured competitive epidemic models with any number of strains, without unique maximizer assumptions for $a$9. The analytic approach—merging advanced semigroup theory, persistence, and Lyapunov analysis—removes technical obstacles that hindered earlier treatments and provides a canonical result for non-ODE competitive exclusion in PDE epidemiological modeling. The work thus significantly strengthens the mathematical foundation for understanding multi-strain epidemic competition in structured populations.