Epidemic spreading

We present an analysis of six deterministic models for epidemic spreading. The evolution of the number of individuals of each class is given by ordinary differential equations of the first order in time, which are set up by using the laws of mass action providing the rates of the several processes that define each model. The epidemic spreading is characterized by the frequency of new cases, which is the number of individuals that are becoming infected per unit time. It is also characterized by the basic reproduction number, which we show to be related to the largest eigenvalue of the stability matrix associated with the disease-free solution of the evolution equations. We also emphasize the analogy between the outbreak of an epidemic with a critical phase transition. When the density of the population reaches a critical value the spreading sets in, a result that was advanced by Kermack and McKendrick in their study of a model in which the recovered individuals acquire permanent immunization, which is one of the models analyzed here.

本研究针对六种传染病传播的确定性模型展开分析。各类别个体的数量演化由关于时间的一阶常微分方程描述，此类方程通过质量作用定律（mass action law）构建，用以确定定义各模型的多项过程的速率。传染病传播特征可通过新增病例频率表征，即单位时间内新增感染的个体数量。该传播特征亦可用基本再生数（basic reproduction number）表征，本研究证明其与演化方程无病解对应的稳定性矩阵的最大特征值相关。本研究同时强调，传染病暴发与临界相变存在类比关系。当种群密度达到临界值时，传染病传播便会启动，这一结论由Kermack与McKendrick在其针对康复个体可获得永久免疫的模型的研究中提出，而该模型正是本研究分析的六种模型之一。