具有时空周期性和奇异性SIS传染病模型扩散特征研究

Epidemic problems has been causing increasing attention. Mathematically, it is a hard and important work for a long time to prevent and control the spread of infectious diseases by establishing the dynamic model and carrying on the theoretical analysis and numerical simulation. The project will proceed from the kinetic modeling point of view, using SIS epidemic models as the main research object, and using the theory of the following subjects involving nonlinear analysis, partial differential equations and numerical simulation as basic tools. We plan to focus on the spreading feature of infectious area. Specifically, we consider respectively epidemic models involving heterogeneous time-periodic environment with moving boundary, and the nonlinear recovery rate incorporating allocation of limited hospital resources in heterogeneous environment. The qualitative properties and dynamical behaviors of the solutions are investigated, and the formula of the basic reproduction number which related to such factors as the temporal and regional characteristics is introduced. In addition, the sufficient conditions of spreading and vanishing are explored. On the basis of above theoretical analysis, the numerical simulations are given, and the key factors influencing the transmission of infectious diseases are analyzed. The new findings will provide new ideas and methods for the research of epidemic models, and also provide certain theoretical reference for the prevention and control of various infectious diseases. We also hope that it can enrich and perfect more valuable and innovative research results.

传染病问题日益受到重视。在数学上，通过建立传染病动力学模型，描述传染病病毒的蔓延和消退，并对其进行理论分析和数值模拟，以期预防和控制疾病的扩散和传播。本项目拟从传染病动力学角度出发，以SIS传染病模型为主要研究对象，综合应用非线性分析、偏微分方程和数值分析等理论工具，重点围绕疾病传播区域的扩张特征进行研究，分别研究时空周期性和奇异性环境中具自由边界的传染病模型，以及受有限医疗资源配置和利用影响的具非线性恢复率的传染病模型。首先探讨模型解的定性性质和动力学行为，并建立与时间、空间区域特征等因素有关的基本再生数的计算公式，给出传染病病毒蔓延和消退的充分条件。其次，在理论分析的基础上给出数值模拟结果,并分析传染病病毒蔓延和消退的一些关键影响因素。本项目试图改进和发展现有的一些研究思路和方法，得到更多有价值和新意的研究成果，为多种传染病的预防和治疗提供理论保障。

本项目重点围绕病毒扩散的特征进行研究，侧重研究了周期异质环境下蚊媒疾病、新型冠状病毒的空间扩散特征。根据病毒的扩散特征，我们建立了相应的数学模型，这些模型具有实际的传染病学背景和特征，能够比较客观的反应病毒的传播规律。我们研究了非均质区域中的具交错扩散项影响的传染病模型，通过相关的谱半径和特征值问题，给出了具非常系数的传染病模型的基本再生数及其有关性质，并利用上下解迭代序列等方法得到交错扩散的程度对病毒蔓延与消退的影响。随着新冠病毒在全球大流行，结合中国防疫新冠的干预措施，我们建立了四级相应机制下新冠病毒的阶段性防控模型，分阶段评估了封城效果和感染风险。利用梯度算子刻画了人口流动对病毒传播的影响。从理论上说明封城和限制不必要的旅行在武汉疫情防控中的积极效应。数值模拟表明，封城、限制旅行、保持社交距离等干预措施分阶段实施既能及时有效控制疫情的大规模扩散，又能及时复苏经济，保持社会稳定。