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Simple Summary: This paper proposes a modified SEIR model to study COVID-19 in Wuhan. The modified model is calibrated by the public number of COVID-19 hospitalization cases in Wuhan. The paper further uses this model to estimate the earliest date of COVID-19 infection in Wuhan, which is in agreement with some existing results. Based on SEIR (susceptible–exposed–infectious–removed) epidemic model, we propose a modified epidemic mathematical model to describe the spread of the coronavirus disease 2019 (COVID-19) epidemic in Wuhan, China. Using public data, the uncertainty parameters of the proposed model for COVID-19 in Wuhan were calibrated. The uncertainty of the control basic reproduction number was studied with the posterior probability density function of the uncertainty model parameters. The mathematical model was used to inverse deduce the earliest start date of COVID-19 infection in Wuhan with consideration of the lack of information for the initial conditions of the model. The result of the uncertainty analysis of the model is in line with the observed data for COVID-19 in Wuhan, China. The numerical results show that the modified mathematical model could model the spread of COVID-19 epidemics. [ABSTRACT FROM AUTHOR]