具有异质性扩散率的传染病模型分析

摘要

In order to describe the influence of individual movement in the large scale space on the spread of the disease,the propagation of infectious diseases on the heterogeneous meta population network is studied.Due to the difference of urban scale,the discrepancy of traffic conditions and so on,there is some heterogeneity in the diffusion rate.The epidemic dynamics model with heterogeneity reaction diffusion equation is established by taking the city as the node,which more accurately describes the impact of the internet structure and people's travel on the spread of disease.By analyzing the qualitative and stabil-ity of the system,it is found that there is a disease-free equilibrium,and the epidemic threshold for the instability of the disease-free equilibrium is obtained.When the epidemic threshold is reached,the system has a unique endemic equilibrium, which is always locally asymptotically stable.Finally,numerical simulations verify the correctness of the theory.Studies indicate that increasing the diffusion rates can speed up the spread of the disease.So,controlling the rate of population travel in large cities can control the outbreak of the disease to a certain extent.%为了刻画大尺度空间范围内的个体移动对疾病传播的影响,主要研究了异质集合种群网络的传染病传播过程.由于城市之间存在规模以及交通发达程度等方面的异质性,扩散率也存在一定的异质性.以城市作为节点,建立了具有异质性扩散率的传染病动力学模型,更加准确地刻画了网络结构以及人们的出行对疾病传播的影响.通过对系统进行定性及稳定性分析,发现系统存在一个无病平衡点,并得到了无病平衡点不稳定的阈值条件.当达到疾病爆发的阈值条件时,系统存在唯一的地方病平衡点,并且地方病平衡点是局部渐近稳定的.通过数值模拟验证了理论的正确性.研究表明,增大扩散率能加快疾病的流行速度,因此,通过有效控制大城市的人口出行率可以在一定程度上控制疾病的爆发.