一类传染病模型的若干动力学行为分析

Based on a kind of epidemic problem arising from mathematical biology,we shall consider a kind of SIS patch model. By using classical qualitaive methods in dynamical systems, we are going to consider the impact of spatial heterogenity, habitat connectivity and disease transmission risk level on the spread of the disease. We mainly want to consider the follownig problems: 1. Consider the aymptotic behavior of the endemic equlibriums of the system as the diffusion coeffient of the infected individuals tends to zero and infinity, when the habitat has both high risk patches and low risk patches; 2. Consider the aymptotic behavior of the endemic equlibriums of the system as the diffusion coeffients of both susceptible and infected individuals tend to zero and infinity, when the habitat has both high risk patches and moderate risk patches;2. Consider the aymptotic behavior of the equlibriums of the system as the diffusion coeffients of both susceptible and infected individuals tend to zero and infinity, when the habitat has high risk patches, moderate risk patches, and low risk patches; The results to be obtained will clear for the clearer understanding of the dynamics of this particular disease model.

本项目针对生物数学中出现的传染病问题，以一类SIS斑块模型为研究对象，拟利用动力系统中经典的定性分析的办法，研究该类系统的空间异质性、时空连通性及疾病传播风险水平对系统动力学行为的影响。本项目主要研究如下三个问题：1、生存空间中只存在高风险斑块及低风险斑块，且已受感染个体的扩散系数趋于零时，系统的EE平衡解的渐进行为；2、生存空间中只存在高风险斑块及中等风险斑块，且种群个体易受感染及已受感染个体的扩散系数趋于无穷大及零时,系统的EE平衡解的渐进行为；3、生存空间中含有高风险斑块、中等风险斑块及低风险斑块时，且种群个体易受感染及已受感染个体的扩散系数的扩散系数趋于无穷大或零时,系统的平衡解的渐进行为。本项目预期得到的理论结果，将有助于人们清楚的认识这类传染病问题的内在的动力学行为。

研究内容包括三部分：1、通过相应的反应扩散模型来了解和研究物种在非均匀空间环境中的扩散以及相应的种群动力学问题。利用半线性偏微分方程组标准的Hopf分支定理，考虑空间齐次和非齐次周期轨道对唯一正平衡点的分支的存在性。结果表明扩散Gierer-Meinhardt模型有丰富的振动模式。研究了带有扩散项的Holling III类功能反应的捕食系统模型，具扩散的两种群相互作用模型，分别详细的分析了Turing，Hopf分支等动力学性质。2. 研究了几类具体的传染病模型。一类SEIRS流行病学模型，模型带有隔离和非线性发生率，二是在溶解性和非溶解性机制下的一类具有免疫时滞的HBV感染动力学模型。3. 研究了捕食者-食饵-共生者模型，具体到瓢虫、蚜虫和蚂蚁之间的关系。在棉田生态系统的基础上建立并研究了以能量为单位的种群能流数学模型，发现了丰富的动力学性质。