非局部流行病模型的动力学研究：带有新自由边界条件的扩散-反应系统，第一部分：扩散-消失二元性
Dynamics of a nonlocal epidemic model with a new free boundary condition, part 1: Spreading-vanishing dichotomy
AI总结：
本文提出了一种新的自由边界条件，研究非局部流行病模型的长期动力学行为，揭示了扩散与消失的二元性及其精确阈值。
摘要：
本文研究了带有自由边界条件的非局部流行病模型的长期动力学行为，其中病原体密度$u(t,x)$和感染人类密度$v(t,x)$根据一个带有非局部扩散的反应扩散系统在区间$[g(t), h(t)]$上演变，该区间代表通过其边界$x=g(t)$和$x=h(t)$扩展的流行病区域。此类基于Cao等人\cite{fb27}提出的自由边界条件的模型曾被若干研究考虑过。受Feng等人\cite{fb20}和Long等人\cite{fb5}最近工作的启发，我们提出了一种新的自由边界条件，其中流行病区域的扩展率由$h'(t)$和$g'(t)$决定，与病原体$u$通过范围边界处的向外通量（如\cite{fb27}中所述）和区域内感染个体的加权总人口数$v$（如\cite{fb5}中所述）的线性组合成正比。我们证明了在该新自由边界条件下，系统是良好的，其长期动力学行为由扩散-消失二元性所特征化。此外，我们得到了该二元性的精确标准，包括初始数据$(u_0,v_0)$的精确阈值；并通过研究一个相关的特征值问题，我们还发现了与扩散率相关的精确阈值，这补充了Nguyen和Vo\cite{fb7}的相关结果。这是两部分系列中的第一部分。在第二部分中，我们将确定当发生扩散时模型的扩散速度，并对一些典型的核函数类，我们将获得精确的加速扩散速率。
英文摘要：
This paper investigates the long-time dynamics of a nonlocal epidemic model with free boundaries, where a pathogen with density $u(t,x)$ and the infected humans with density $v(t,x)$ evolve according to a reaction-diffusion system with nonlocal diffusion over a one dimensional interval $[g(t), h(t)]$, which represents the epidemic region expanding through its boundaries $x=g(t)$ and $x=h(t)$, known as free boundaries. Such a model with free boundary conditions based on those of Cao et al. \cite{fb27} was considered by several works. Inspired by recent works of Feng et al. \cite{fb20} and Long et al. \cite{fb5}, we propose a new free boundary condition, where the expansion rate of the epidemic region, determined by $h'(t)$ and $g'(t)$, is proportional to a linear combination of the outward flux of the pathogen \(u\) through the range boundary (as in \cite{fb27}) and the weighted total population of infected individuals \(v\) within the region (as in \cite{fb5}). We prove that the system under this new free boundary condition is well-posed, and its long-time dynamical behavior is characterized by a spreading-vanishing dichotomy. Moreover, we obtain sharp criteria for this dichotomy, including a sharp threshold in terms of the initial data $(u_0,v_0)$; and by studying a related eigenvalue problem, we also find a sharp threshold in terms of the diffusion rate, which complements related results in Nguyen and Vo \cite{fb7}. This is Part $1$ of a two part series. In Part $2$, we will determine the spreading speed of the model when spreading occurs, and for some typical classes of kernel functions, we will obtain the precise rates of accelerated spreading.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
