Deterministic and stochastic models for the epidemic dynamics of COVID-19 in Wuhan, China

Damilola Olabode; Jordan Culp; Allison Fisher; Angela Tower; Dylan Hull-Nye; Xueying Wang; Damilola Olabode; Jordan Culp; Allison Fisher; Angela Tower; Dylan Hull-Nye; Xueying Wang

doi:10.3934/mbe.2021050

Mathematical Biosciences and Engineering

2021, Volume 18, Issue 1: 950-967. doi: 10.3934/mbe.2021050

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Deterministic and stochastic models for the epidemic dynamics of COVID-19 in Wuhan, China

Department of Mathematics and Statistics, Washington State University, Pullman, WA 99164, USA
^† First authors with equal contribution

Received: 31 August 2020 Accepted: 02 December 2020 Published: 04 January 2021

In this paper, deterministic and stochastic models are proposed to study the transmission dynamics of the Coronavirus Disease 2019 (COVID-19) in Wuhan, China. The deterministic model is formulated by a system of ordinary differential equations (ODEs) that is built upon the classical SEIR framework. The stochastic model is formulated by a continuous-time Markov chain (CTMC) that is derived based on the ODE model with constant parameters. The nonlinear CTMC model is approximated by a multitype branching process to obtain an analytical estimate for the probability of a disease outbreak. The local and global dynamics of the disease are analyzed by using the deterministic model with constant parameters, and the result indicates that the basic reproduction number $ \mathcal{R}_0 $ serves as a sharp disease threshold: the disease dies out if $ \mathcal{R}_0\le 1 $ and persists if $ \mathcal{R}_0 > 1 $. In contrast to the deterministic dynamics, the stochastic dynamics indicate that the disease may not persist when $ \mathcal{R}_0 > 1 $. Parameter estimation and validation are performed to fit our ODE model to the public reported data. Our result indicates that both the exposed and infected classes play an important role in shaping the epidemic dynamics of COVID-19 in Wuhan, China. In addition, numerical simulations indicate that a second wave of the ongoing pandemic is likely to occur if the prevention and control strategies are not implemented properly.
- COVID-19,
- SEIR model,
- ODE model,
- CTMC model,
- disease extinction, disease outbreak,
- second wave
Citation: Damilola Olabode, Jordan Culp, Allison Fisher, Angela Tower, Dylan Hull-Nye, Xueying Wang. Deterministic and stochastic models for the epidemic dynamics of COVID-19 in Wuhan, China[J]. Mathematical Biosciences and Engineering, 2021, 18(1): 950-967. doi: 10.3934/mbe.2021050

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Abstract

In this paper, deterministic and stochastic models are proposed to study the transmission dynamics of the Coronavirus Disease 2019 (COVID-19) in Wuhan, China. The deterministic model is formulated by a system of ordinary differential equations (ODEs) that is built upon the classical SEIR framework. The stochastic model is formulated by a continuous-time Markov chain (CTMC) that is derived based on the ODE model with constant parameters. The nonlinear CTMC model is approximated by a multitype branching process to obtain an analytical estimate for the probability of a disease outbreak. The local and global dynamics of the disease are analyzed by using the deterministic model with constant parameters, and the result indicates that the basic reproduction number $ \mathcal{R}_0 $ serves as a sharp disease threshold: the disease dies out if $ \mathcal{R}_0\le 1 $ and persists if $ \mathcal{R}_0 > 1 $. In contrast to the deterministic dynamics, the stochastic dynamics indicate that the disease may not persist when $ \mathcal{R}_0 > 1 $. Parameter estimation and validation are performed to fit our ODE model to the public reported data. Our result indicates that both the exposed and infected classes play an important role in shaping the epidemic dynamics of COVID-19 in Wuhan, China. In addition, numerical simulations indicate that a second wave of the ongoing pandemic is likely to occur if the prevention and control strategies are not implemented properly.

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