arXiv subjects

Soufiane Bentout

Publications and source records attributed to Soufiane Bentout.

2 recordsLinked to original sources

arXiv2026

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

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.

math.AP↗

arXiv2025

The eigentheory for nonlocal cooperative-advective system and its role in the study of free boundary system for directional epidemic models

In this paper, we propose and analyze a nonlocal cooperative reaction--diffusion system with free boundaries and drift terms, motivated by directional epidemic spread. Lacking a variational structure but requiring sharper regularity of solutions, the model poses substantial analytical challenges compared with previous works~\cite{Du,Berestycki2016a,Berestycki2016b,Cao2019,NguyenVo2022,Tang2024a,Tang2024b}. We first establish the well-posedness of the local problem and the global existence and uniqueness of classical solutions in $C^1$ space. We then study the associated nonlocal eigenvalue problem, proving the existence, simplicity, qualitative properties, and asymptotic behavior of the principal eigenvalue. The analysis employs Fredholm theory, the Crandall--Rabinowitz bifurcation theorem, and Hadamard-type derivative formulas to describe its parameter dependence and connection with the basic reproduction number~$R_0$. Building on this spectral characterization, we show that the system admits a \emph{sharp vanishing--spreading dichotomy} in its long-term dynamics. When $R_0\le1$, all solutions vanish; for $R_0>1$, the outcome depends on the initial domain size~$h_0$ and the free-boundary expansion rate~$μ$. There exists a critical habitat length~$\mathcal L^\ast$ such that if $h_0<\mathcal L^\ast$, a threshold $\widehatμ>0$ separates vanishing ($μ\in(0,\widehatμ]$) from spreading ($μ>\widehatμ$). In the spreading regime, solutions converge to the unique positive steady state, while in the vanishing regime they decay uniformly to zero. These results provide a rigorous framework for the threshold dynamics of cooperative--advective nonlocal systems and offer mathematical insight for further studies in epidemic modeling, ecological invasion, and population dynamics.

math.AP↗