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时间连续和时间离散的SIR模型再探讨：理论与应用

Time-continuous and time-discrete SIR models revisited: theory and applications.

作者信息

Wacker Benjamin, Schlüter Jan

机构信息

Next Generation Mobility Group, Department of Dynamics of Complex Fluids, Max-Planck-Institute for Dynamics and Self-Organization, Am Fassberg 17, D-37077 Göttingen, Germany.

Institute for Dynamics of Complex Fluids, Faculty of Physics, Georg-August-University of Göttingen, Friedrich-Hund-Platz 1, D-37077 Göttingen, Germany.

出版信息

Adv Differ Equ. 2020;2020(1):556. doi: 10.1186/s13662-020-02995-1. Epub 2020 Oct 7.

DOI:10.1186/s13662-020-02995-1
PMID:33042201
原文链接:https://pmc.ncbi.nlm.nih.gov/articles/PMC7538854/
Abstract

Since Kermack and McKendrick have introduced their famous epidemiological SIR model in 1927, mathematical epidemiology has grown as an interdisciplinary research discipline including knowledge from biology, computer science, or mathematics. Due to current threatening epidemics such as COVID-19, this interest is continuously rising. As our main goal, we establish an implicit time-discrete SIR (susceptible people-infectious people-recovered people) model. For this purpose, we first introduce its continuous variant with time-varying transmission and recovery rates and, as our first contribution, discuss thoroughly its properties. With respect to these results, we develop different possible time-discrete SIR models, we derive our implicit time-discrete SIR model in contrast to many other works which mainly investigate explicit time-discrete schemes and, as our main contribution, show unique solvability and further desirable properties compared to its continuous version. We thoroughly show that many of the desired properties of the time-continuous case are still valid in the time-discrete implicit case. Especially, we prove an upper error bound for our time-discrete implicit numerical scheme. Finally, we apply our proposed time-discrete SIR model to currently available data regarding the spread of COVID-19 in Germany and Iran.

摘要

自1927年克马克（Kermack）和麦肯德里克（McKendrick）引入其著名的流行病学SIR模型以来，数学流行病学已发展成为一门跨学科研究学科，涵盖了生物学、计算机科学或数学等领域的知识。由于当前诸如COVID-19等具有威胁性的流行病，这种兴趣在持续上升。作为我们的主要目标，我们建立了一个隐式时间离散的SIR（易感人群-感染人群-康复人群）模型。为此，我们首先引入其具有时变传播率和恢复率的连续变体，并作为我们的第一项贡献，深入讨论其性质。基于这些结果，我们开发了不同的可能的时间离散SIR模型，与许多主要研究显式时间离散格式的其他工作不同，我们推导了隐式时间离散SIR模型，并且作为我们的主要贡献，展示了与连续版本相比独特的可解性和其他理想性质。我们充分表明，时间连续情况下的许多理想性质在时间离散隐式情况下仍然有效。特别是，我们证明了我们的时间离散隐式数值格式的一个上误差界。最后，我们将我们提出的时间离散SIR模型应用于德国和伊朗目前关于COVID-19传播的可用数据。

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