We consider the SIR model and study the first time the number of infected individuals begins to decrease and the first time this population is below a given threshold. We interpret these times as functions of the initial susceptible and infected populations and characterize them as solutions of a certain partial differential equation. This allows us to obtain integral representations of these times and in turn to estimate them precisely for large populations.

中文翻译：

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SIR模型的两个关键时刻

我们考虑 SIR 模型并研究受感染个体的数量第一次开始减少以及该群体第一次低于给定阈值。我们将这些时间解释为初始易感人群和感染人群的函数，并将它们表征为某个偏微分方程的解。这使我们能够获得这些时间的积分表示，进而为大量人口精确估计它们。