Initial value estimation of uncertain differential equations and zero-day of COVID-19 spread in China
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Initial value estimation of uncertain differential equations and zero-day of COVID-19 spread in China Waichon Lio1 · Baoding Liu1 Accepted: 7 September 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Assume an uncertain process follows an uncertain differential equation, and some realizations of this process are observed. Parameter estimation for the uncertain differential equation that fits the observed data as much as possible is a core problem in practice. This paper first presents a problem of initial value estimation for uncertain differential equations and proposes an estimation method. In addition, the method of moments is recast for estimating the time-varying parameters in uncertain differential equations. Using those techniques, a COVID-19 spread model based on uncertain differential equation is derived, and the zero-day of COVID-19 spread in China is inferred. Keywords Uncertainty theory · Uncertain statistics · Uncertain differential equation · COVID-19
1 Introduction Based on uncertainty theory (Liu 2007), Liu (2008) initialized uncertain differential equation as a type of differential equations involving uncertain processes. Under linear growth and Lipschitz condition, Chen and Liu (2010) proved an existence and uniqueness theorem of solution of uncertain differential equation. Following that, Gao (2012) proved the theorem again under local linear growth and Lipschitz condition. Furthermore, an analytic solution to linear uncertain differential equations was derived by Chen and Liu (2010), and some analytic methods to nonlinear uncertain differential equations were presented by Liu (2012) and Yao (2013b). Yao and Chen (2013) made an important contribution for verifying that the solution of an uncertain differential
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Baoding Liu [email protected] Waichon Lio [email protected]
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Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China
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W. Lio, B. Liu
equation can be represented by a family of solutions of ordinary differential equations (this important work was named as Yao–Chen Formula later), and then the methods for calculating extreme value, first hitting time and time integral of the solution of uncertain differential equation were provided by Yao (2013a). To estimate the unknown parameters in uncertain differential equation that fits the observed data as much as possible, several methods were proposed, for example, the method of moments (Yao and Liu 2020), least squares estimation (Sheng et al. 2019), generalized moment estimation (Liu 2020b), uncertain maximum likelihood (Liu and Liu 2020), and minimum cover estimation (Yang et al. 2020). Recently, many scholars applied uncertain statistics to modelling COVID-19 pandemic. For instance, Liu (2020a) used uncertain regression analysis to forecast the cumulative numbers of COVID-19 infections in China, while Ye and Yang (2020) used uncertain time series. Following that, Chen et al. (2020) presented an uncertain SIR model, and Jia and Chen (2020) proposed an uncerta
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