{"title":"Exact solutions to SIR epidemic models via integrable discretization","authors":"Atsushi Nobe","doi":"10.1063/5.0152442","DOIUrl":null,"url":null,"abstract":"An integrable discretization of the SIR model with vaccination is proposed. Through the discretization, the conserved quantities of the continuous model are inherited to the discrete model, since the discretization is based on the intersection structure of the non-algebraic invariant curve defined by the conserved quantities. Uniqueness of the forward/backward evolution of the discrete model is demonstrated in terms of the single-valuedness of the Lambert W function on the positive real axis. Furthermore, the exact solution to the continuous SIR model with vaccination is constructed via the integrable discretization. When applied to the original SIR model, the discretization procedure leads to two kinds of integrable discretization, and the exact solution to the continuous SIR model is also deduced. It is furthermore shown that the discrete SIR model geometrically linearizes the time evolution by using the non-autonomous parallel translation of the line intersecting the invariant curve.","PeriodicalId":16174,"journal":{"name":"Journal of Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":1.2000,"publicationDate":"2024-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Mathematical Physics","FirstCategoryId":"101","ListUrlMain":"https://doi.org/10.1063/5.0152442","RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"PHYSICS, MATHEMATICAL","Score":null,"Total":0}
引用次数: 0
Abstract
An integrable discretization of the SIR model with vaccination is proposed. Through the discretization, the conserved quantities of the continuous model are inherited to the discrete model, since the discretization is based on the intersection structure of the non-algebraic invariant curve defined by the conserved quantities. Uniqueness of the forward/backward evolution of the discrete model is demonstrated in terms of the single-valuedness of the Lambert W function on the positive real axis. Furthermore, the exact solution to the continuous SIR model with vaccination is constructed via the integrable discretization. When applied to the original SIR model, the discretization procedure leads to two kinds of integrable discretization, and the exact solution to the continuous SIR model is also deduced. It is furthermore shown that the discrete SIR model geometrically linearizes the time evolution by using the non-autonomous parallel translation of the line intersecting the invariant curve.
本文提出了带疫苗接种的 SIR 模型的可积分离散化方法。通过离散化,连续模型的守恒量被继承到离散模型中,因为离散化是基于守恒量定义的非代数不变曲线的交集结构。离散模型的前向/后向演化的唯一性是通过正实轴上兰伯特 W 函数的单值性来证明的。此外,还通过可积分离散化构建了带有疫苗接种的连续 SIR 模型的精确解。当应用于原始 SIR 模型时,离散化过程会导致两种可积分离散化,并推导出连续 SIR 模型的精确解。此外,离散 SIR 模型还利用与不变曲线相交的直线的非自主平行平移将时间演化几何线性化。
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