基于随机空间传染病系统动力学研究

The current project applies a lot of mathematical theories and methods, including the differential equations, stochastic analysis, epidemiology, symbolic and numerical calculation, etc., to study the dynamical behavior of the stochastic reaction-diffusion epidemic systems...Starting with the epidemiological transmission of the specific epidemic (e.g. H7N9, HFMD), we consider random perturbations, and the way of diffusion or migration of the susceptibles, infectives and recovers. Then, we globally analyze the interactions and relations among them to establish the corresponding stochastic reaction-diffusion epidemic systems...We give the theorems to the existence and stability of the positive solutions of the systems, which involve the effects of time delay and Allee. By using nonlinear analysis method, we establish the amplitude equations close to the onset of the bifurcations, and the complex dynamics caused by bifurcations can be analyzed. As a result, we expose the mechanism of the pattern formation and selection, and further clarify the distribution structure of the species, persistence or extinction of the diseases after spatial diffusion. Then, we study the epidemic trend, find out the key spreading factor, and evaluate the effectiveness of the interventions. The results will provide the theoretical basis for the prevention and control of the diseases.

本项目综合运用微分方程、随机分析、流行病学及符号与数值计算等理论和方法，研究基于随机干扰（包括Brownian运动、Markov链）和空间扩散（包括利用Laplace算子刻画局部扩散、利用卷积算子刻画非局部扩散）作用下的传染病系统的动力学行为。.从具体传染病（如H7N9，手足口病）的流行学传播途径入手，考虑随机因素，以及易感者、感染者和康复者等种群的扩散与迁移方式，全面分析种群之间相互作用及关系，建立相应的随机反应扩散传染病系统。.给出随机反应扩散传染病系统（包括受时滞、Allee效应等因素影响）正解的存在唯一性和稳定性判定定理；利用非线性分析方法建立分支处的振幅方程，详细分析分支带来的复杂动力学行为，揭示系统斑图形成和选择的机理。进一步阐明种群受各因素影响下在空间扩散后分布结构和疾病持续、灭绝等过程。找出传播关键因素，评估各种干预措施有效性，为有关疾控部门提供理论依据。

本项目以传染病和种群系统为研究背景，针对传染病模型的传播机制和种群系统的扩散分布，分别利用随机微分方程和偏微分方程构建相应的动力学模型，并展开深入研究。根据考虑传染病的防控、随机扰动环境和时滞等因素，(1)建立了考虑接种疫苗和治疗控制的具有一般形式的疾病发病率的SIRS传染病模型；(2)研究了含时滞的具有Monod-Haldane型功能反应函数的空间扩散种群系统的稳定性、Hopf分支条件；(3)研究了在随机扰动下含时滞的具有Harrison型功能反应函数的种群系统的渐近稳定性、有界性、持久性等；(4)建立了考虑种群密度依赖死亡率的含Allee效应的具有Holling-II型功能反应函数的空间扩散种群系统。本项目的研究既丰富随机生物数学理论，又为有关疾控部门提供理论依据。