一种具有非局部空间感染的流行病模型
An epidemical model with nonlocal spatial infections
AI总结：
本研究将基于最近邻作用的SIR模型扩展为非局部非线性偏微分方程模型，开发了稳态分析、演化计算、矩诊断及SINDy数据驱动方法，为流行病时空量化研究提供工具。
摘要：
SIR模型是流行病学中最具代表性的仓室模型之一。将这种常微分方程（ODE）框架推广为空间分布的偏微分方程（PDE）模型是一项相当大的挑战。在本研究中，我们将其中一位作者在文献~\cite{vaziry2022modelling}中提出的、基于最近邻空间相互作用的模型，扩展为SIR原型的非局部非线性偏微分方程变体。随后我们致力于开发一套能为该偏微分方程框架提供洞见的工具。稳态及其稳定性分析为研究感染的早期空间增长提供了视角。演化计算动力学可实现感染与恢复的时空进展可视化，让人们能够了解非局部核的不同参数（例如其宽度参数）的影响。我们在一维和二维场景中都探索了这些特性。在模型降维层面，我们开发了一系列可解释的基于矩的诊断指标，用以观察这些指标如何反映感染总人数、流行病的震中及其传播情况。最后，我们提出了一种基于稀疏非线性动力学识别（SINDy）的数据驱动方法，来识别这类量的近似闭式动力学方程。这些方法可为进一步的时空研究铺平道路，实现对流行病的量化分析。
英文摘要：
The SIR model is one of the most prototypical compartmental models in epidemiology. Generalizing this ordinary differential equation (ODE) framework into a spatially distributed partial differential equation (PDE) model is a considerable challenge. In the present work, we extend a recently proposed model based on nearest-neighbor spatial interactions by one of the authors in~\cite{vaziry2022modelling} towards a nonlocal, nonlinear PDE variant of the SIR prototype. We then seek to develop a set of tools that provide insights for this PDE framework. Stationary states and their stability analysis offer a perspective on the early spatial growth of the infection. Evolutionary computational dynamics enable visualization of the spatio-temporal progression of infection and recovery, allowing for an appreciation of the effect of varying parameters of the nonlocal kernel, such as, e.g., its width parameter. These features are explored in both one- and two-dimensional settings. At a model-reduction level, we develop a sequence of interpretable moment-based diagnostics to observe how these reflect the total number of infections, the epidemic's epicenter, and its spread. Finally, we propose a data-driven methodology based on the sparse identification of nonlinear dynamics (SINDy) to identify approximate closed-form dynamical equations for such quantities. These approaches may pave the way for further spatio-temporal studies, enabling the quantification of epidemics.
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