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Long-time dynamics and threshold Phenomena for a free-boundary SIS Model with asymmetric kernels in advective periodic environments

Soufiane Bentout, Hoang-Hung Vo

Abstract

We study a nonlocal SIS epidemic model with free boundaries, advection, and spatial heterogeneity, where the dispersal kernels are not assumed to be symmetric. The model describes the evolution of susceptible and infected populations in a bounded infected habitat whose endpoints move according to nonlocal boundary fluxes. Our goal is to determine the sharp threshold between disease spreading and vanishing, and to characterize the long-time behavior of solutions. The analysis faces several essential difficulties. The linearization around the disease-free equilibrium gives rise to a genuinely coupled nonlocal system with drift, so the relevant spectral quantity cannot be reduced directly to a standard scalar eigenvalue problem. In addition, the presence of advection terms and possibly non-symmetric kernels destroys self-adjointness, so no useful variational characterization is available; in particular, classical Rayleigh quotient and minimax arguments cannot be applied. To overcome these difficulties, we employ the generalized principal eigenvalue theory for nonlocal operators developed by Coville and Hamel, together with the Harnack inequality for non-symmetric nonlocal operators established therein. This non-variational framework is particularly well suited to our setting. Combined with comparison principles, sub- and supersolution constructions, and uniform estimates on time-dependent spatial intervals, it allows us to derive the precise asymptotic behavior of the generalized principal eigenvalue with respect to the spatial domain and the diffusion rate, identify the sharp threshold and the critical habitat size, and determine the long-time dynamics of $S$ and $I$ via an $ω$-limit set approach. To the best of our knowledge, this is the first work on a free-boundary SIS epidemic model with non-symmetric nonlocal dispersal kernels, advection, and spatial periodicity.

Long-time dynamics and threshold Phenomena for a free-boundary SIS Model with asymmetric kernels in advective periodic environments

Abstract

We study a nonlocal SIS epidemic model with free boundaries, advection, and spatial heterogeneity, where the dispersal kernels are not assumed to be symmetric. The model describes the evolution of susceptible and infected populations in a bounded infected habitat whose endpoints move according to nonlocal boundary fluxes. Our goal is to determine the sharp threshold between disease spreading and vanishing, and to characterize the long-time behavior of solutions. The analysis faces several essential difficulties. The linearization around the disease-free equilibrium gives rise to a genuinely coupled nonlocal system with drift, so the relevant spectral quantity cannot be reduced directly to a standard scalar eigenvalue problem. In addition, the presence of advection terms and possibly non-symmetric kernels destroys self-adjointness, so no useful variational characterization is available; in particular, classical Rayleigh quotient and minimax arguments cannot be applied. To overcome these difficulties, we employ the generalized principal eigenvalue theory for nonlocal operators developed by Coville and Hamel, together with the Harnack inequality for non-symmetric nonlocal operators established therein. This non-variational framework is particularly well suited to our setting. Combined with comparison principles, sub- and supersolution constructions, and uniform estimates on time-dependent spatial intervals, it allows us to derive the precise asymptotic behavior of the generalized principal eigenvalue with respect to the spatial domain and the diffusion rate, identify the sharp threshold and the critical habitat size, and determine the long-time dynamics of and via an -limit set approach. To the best of our knowledge, this is the first work on a free-boundary SIS epidemic model with non-symmetric nonlocal dispersal kernels, advection, and spatial periodicity.

Paper Structure

This paper contains 13 sections, 11 theorems, 586 equations.

Table of Contents

  1. Introduction and main results
  2. Hypotheses
  3. Main results
  4. Well-posedness of the system
  5. Linearization and the eigenvalue
  6. Existence of a steady state for the disease--free operator
  7. Block-operator matrix form.
  8. Scalar effective operator at $\lambda=0$.
  9. Asymptotics of the principal eigenvalue for small diffusion and for vanishing interval
  10. Monotonicity and small-$d$ asymptotics of the principal eigenvalue
  11. Monotonicity: dependence on $M$, $B$ and domain length
  12. The basic reproduction number $\mathcal{R}_0$ and its relation to the principal eigenvalue
  13. Spreading-Vanishing phenomena

Key Result

Theorem 1

Assume that (J1)-(J2), (H2)--(H3) hold. Then, for any $T>0$, the free-boundary problem eq:SIS-free-boundary admits a unique classical solution on $[0,T]$ in the sense of the functional setting introduced above. Moreover, there exist positive constants $M_S$ and $M_I$, independent of $T$, such that

Theorems & Definitions (26)

  • Theorem 1: Well-posedness
  • Theorem 2: Spreading--vanishing theorem
  • Remark 1
  • Theorem 3
  • Lemma 4: Maximum principle
  • proof
  • Lemma 5: Comparison principle
  • proof : Proof of Theorem \ref{['Theorem1']}
  • Remark 2
  • Definition 6: Generalized principal eigenvalue
  • ...and 16 more