To mitigate the harmful effects of the COVID-19 pandemic, world countries have resorted – though with different timing and intensities – to a range of interventions. These interventions and their relaxation have shaped the epidemic into a multi-phase form, namely an early invasion phase often followed by a lockdown phase, whose unlocking triggered a second epidemic wave, and so on. In this article, we provide a kinematic description of an epidemic whose time course is subdivided by mitigation interventions into a sequence of phases, on the assumption that interventions are effective enough to prevent the susceptible proportion to largely depart from 100% (or from any other relevant level). By applying this hypothesis to a general SIR epidemic model with age-since-infection and piece-wise constant contact and recovery rates, we supply a unified treatment of this multi-phase epidemic showing how the different phases unfold over time. Subsequently, by exploiting a wide class of infectiousness and recovery kernels allowing reducibility (either to ordinary or delayed differential equations), we investigate in depth a low-dimensional case allowing a non-trivial full analytical treatment also of the transient dynamics connecting the different phases of the epidemic. Finally, we illustrate our theoretical results by a fit to the overall Italian COVID-19 epidemic since March 2020 till February 2021 i.e., before the mass vaccination campaign. This show the abilities of the proposed model in effectively describing the entire course of an observed multi-phasic epidemic with a minimal set of data and parameters, and in providing useful insight on a number of aspects including e.g., the inertial phenomena surrounding the switch between different phases.

为了减轻 COVID-19 大流行的有害影响，世界各国采取了一系列干预措施（尽管时间和强度不同）。这些干预措施及其放松使疫情形成了多阶段的形式，即早期入侵阶段，通常随后是封锁阶段，封锁阶段的解锁引发了第二波疫情，等等。在本文中，我们提供了一种流行病的运动学描述，其时间过程通过缓解干预措施细分为一系列阶段，假设干预措施足以有效地防止易感比例在很大程度上偏离 100%（或任何其他相关级别）。通过将此假设应用于具有感染后年龄以及分段恒定接触率和康复率的一般 SIR 流行病模型，我们为这种多阶段流行病提供了统一的处理方法，显示了不同阶段如何随时间展开。随后，通过利用广泛的传染性和恢复内核，允许简化（无论是常微分方程还是延迟微分方程），我们深入研究低维情况，允许对连接不同相的瞬态动力学进行非平凡的全面分析处理的流行病。最后，我们通过拟合 2020 年 3 月至 2021 年 2 月（即大规模疫苗接种运动之前）意大利整体 COVID-19 疫情来说明我们的理论结果。这表明所提出的模型能够用最少的数据和参数集有效地描述观察到的多相流行病的整个过程，并提供对许多方面的有用见解，包括例如围绕疾病之间切换的惯性现象。不同的阶段。