传染病传播

摘要：

我们分析了六个传染病传播的确定性模型。每个类别的个体数量的演化由时间上的一阶常微分方程给出，该方程是通过使用质量作用定律建立的，该定律提供了定义每个模型的几个过程的速率。传染病传播的特征是新病例的频率，即单位时间内感染的人数。它的特征还在于基本再生数，我们表明，与无病解的发展方程的稳定性矩阵的最大特征值。我们还强调流行病的爆发与关键的相变之间的类比。当种群密度达到临界值时，扩散开始，这是Kermack和McKendrick在研究恢复个体获得永久免疫的模型时提出的结果，这是本文分析的模型之一。

英文摘要：

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.

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