【作者】 何敏华；

【导师】 张端明；

【作者基本信息】 华中科技大学 ， 凝聚态物理， 2006， 硕士

【摘要】 复杂网络广泛地存在于自然界和人类社会中,从Internet、WWW到化学反应、生物食物链、再到人类社会的人际关系等都呈现出复杂网络的拓扑特性,如:小世界特性和无标度特性等。基于复杂网络的流行病传播动力学模型同样吸引着许多科研工作者的目光,研究结果表明相对于规则网络而言,传染病在复杂网络的传播具有较小的传播阈值,甚至在无标度网络上阈值消失。本文综述了复杂网络的基本概念、典型的网络机制模型,尤其是阐述了目前在国际上流行的主要几种基于复杂网络上流行病模型的基本内容及其传播行为的规律。其中重点介绍了基于小世界和无标度网络上的流行病模型。在此基础上,我们在二维规则网络上构建了移动人群中流行病传播模型。利用了数值模拟的方法研究了流行病在到达稳态后的若干演化规律:感染人群比值随时间的演化规律;感染人数到达最大值所用时间与免疫时间和人群密度的相应关系;平均感染人数与人群密度δ和人群移动概率p的相应关系。有趣的是,我们还确定了若干临界参数的数值,如临界移动概率p c,临界人群密度δc等。其次,建立了近似等价于网络模型的时间演化平均场方程理论。与通常平均场理论不同的是,病人与健康人在人群中接触的概率不再是一个常数,而是人群密度的函数。并利用了平均场近似方程和我们的网络模型的等价性,得到了接触概率的具体函数形式。最后,我们利用了数值模拟的方法研究了疾病传播过程中的暂态行为。当人群密度超过临界密度时,新增感染态人数随时间呈指数爆发性的增长。研究表明,其指数系数与人群移动的概率呈现幂次关系,而具体的幂指数又取决于人群密度,具有复杂的非线性递减的关系。与稳态情况不同,我们发现平均场理论不能描述疾病传播的暂态行为。更多还原

【Abstract】 Complex networks describe a wide range of systems in nature and society. Frequently cited examples include internet, WWW, a network of chemicals linked by chemical reactions, social relationship networks, et al. Studies show that these complex networks exhibit some characters, such as Small-World effect and scale free property and so on. These characters can not be explained by regular or random networks theory. The research of epidemical model in complex networks has also attracted many researchers’attention. It is shown that there is smaller threshold of disease spreading in other complex networks compared with regular lattice, even more the absence of thresholds of percolation and epidemic.We summarize the basic concept and typical mechanism model of complex networks, and we especially introduce some main epidemical models and the spreading behavior of epidemical model on complex networks, and we pay more attention to the newest investigation of disease spreading in small world and scale free networks.On this base, we establish the epidemical model of mobile individuals in regular lattice. The behaviors of steady states of epidemic propagation are studied using large scale simulations. The influence of these parameters (the automata density, the jumping probability etc.) on epidemic spreading is studied, and the critical jumping probability and the critical population density are observed.Second, we establish the Mean-Field equations which are good agreement with our network model. Here the efficient contact rate is not a constant but a function of the population density, the point is different with conventional Mean-field equations. Through an approximate equivalence relation between network model and mean-field equations, the concrete form of the efficient contact rate is obtained.At last, the critical behaviors of epidemic propagation are studied using large scale simulations. The ration of new infected population exponentially increases with time above the critical population density. The exponential coefficient shows power-law relations with the jumping probability, and its exponent decreases non-linearly with the population density. And we also found that the mean-field equation can’t describe the critical behavior of epidemic spreading, which is different with steady state.更多还原