周期反应扩散传染病模型的空间动力学

By using reaction-diffusion model to describe and explore the mechanism of spatial spread of infectious diseases is an important way of research on the disease prevention and control. The compartmental epidemic model systems usually do not have monotonicity, and hence, the comparison principles are no longer valid, which leads to some essential difficulties to study the transmission dynamics of infectious diseases. In addition, in the investigation of the growth of population and the transmission of infectious diseases, the periodic factors, such as alternation of day with night and seasonality, etc, can not be ignored, which gives rise to non-autonomous systems to characterize the spread of diseases. Therefore, it is of important significance to study the spatial dynamics of some time-periodic reaction-diffusion epidemic models. In this project, we study the dynamics of time-periodic reaction-diffusion epidemic models by means of dynamical system, nonlinear functional analysis, functional differential equations, linear and semilinear parabolic equations. More precisely, firstly, we will establish the existence and non-existence of time-periodic reaction-diffusion SIR epidemic models with demographic structure and the spatial discrete case of them. Secondly, by developing new methods, we study the existence and linear determinability of asymptotic speeds of spread for the time-periodic reaction-diffusion SIR epidemic models. Finally, we will introduce the basic reproduction number and establish the threshold dynamics for time-periodic and nonlocal delayed reaction-diffusion multigroup epidemic models.

通过反应扩散模型来描述和探究传染性疾病的空间传播机理是疾病预防和控制研究的一个重要方法。仓室传染病模型通常不具有单调性，致使比较原理不成立，这对系统的动力学研究带来了本质困难。此外，昼夜更替、季节变迁等周期变化因素在种群增长与疾病传播中不容忽视，这就需要用非自治系统来描述疾病的传播。因此，研究时间周期反应扩散传染病模型具有重要的意义。本项目将借助动力系统、非线性分析、泛函微分方程、线性与半线性抛物方程等理论，研究周期扩散传染病模型的传播动力学，具体研究内容包括：首先，建立带有种群结构的周期扩散SIR模型及其空间离散情形的周期行波解的存在与非存在性；其次，通过发展新方法研究其渐近传播速度的存在性与线性可确定性；最后，给出周期非局部时滞反应扩散多群体SIR模型的基本再生数定义，以此建立有界区域上系统的阈值动力学。

本项目围绕时间周期反应扩散传染病模型的传播动力学、受季节更替影响的确定性传染病模型的阈值动力学、时间周期的非局部时滞反应扩散单种群模型的空间动力学、生态流行病模型的持久性展开了研究，项目进展顺利，完成了预期目标。主要结果：（1）建立了时间周期反应扩散传染病模型的传播信息：周期行波解的存在性与不存在性，以及临界周期行波解的存在性，阐释了季节因素、疾病潜伏期等对最小波速的影响。定义了时空非齐次的SIS型反应扩散方程对流传染病模型的基本再生数，分析其阈值动力学，并讨论时、空因素，对流项和扩散能力对基本再生数的影响，进而揭示其对疾病传播的影响。建立了时空非齐次的非局部时滞反应扩散登革热模型，通过理论分析和数值模拟探究了登革热传播的动力学行为。（2）在有界区域上建立了具有年龄结构的单种群模型的动力学性态，给出物种持续生存和灭绝的充分条件，探讨各参数对种群演化的影响；发展新方法，在全空间上建立了具有年龄结构的非单调单种群模型的渐近传播速度和周期行波解。（3）建立了食饵种群具有传染性疾病的非局部时滞反应扩散捕食-食饵模型，通过理论方法结合数值模拟分别分析了子系统和全系统的动力学行为，揭示了传染性疾病对生态系统的调节作用。