Jiawen Hou1, Jie Hong2, Boyun Ji3, Bowen Dong3, Yue Chen4, Michael P Ward5, Wei Tu6, Zhen Jin7, Jian Hu2, Qing Su2, Wenge Wang2, Zheng Zhao2, Shuang Xiao2, Jiaqi Huang2, Wei Lin8, Zhijie Zhang9. 1. Institute of Science and Technology for Brain-Inspired Intelligence, Fudan University, Shanghai, 200433, China; Centre for Computational Systems Biology and Research Institute of Intelligent Complex Systems, Fudan University, Shanghai, 200433, China. 2. Department of Epidemiology and Health Statistics, School of Public Health, Fudan University, Shanghai, 200032, China; Key Laboratory of Public Health Safety, Ministry of Education, Shanghai, 200032, China. 3. School of Mathematical Sciences, SCMS, and SCAM, Fudan University, Shanghai, 200433, China. 4. Department of Epidemiology and Community Medicine, Faculty of Medicine, University of Ottawa, 451 Smyth Rd, Ottawa, Ontario, Canada. 5. Faculty of Veterinary Science, The University of Sydney NSW, Sydney, Australia. 6. Department of Geology and Geography, Georgia Southern University, Statesboro, GA, 30460, USA. 7. Complex Systems Research Center, Shanxi University, Taiyuan, Shan'xi, 030006, China. 8. Institute of Science and Technology for Brain-Inspired Intelligence, Fudan University, Shanghai, 200433, China; Centre for Computational Systems Biology and Research Institute of Intelligent Complex Systems, Fudan University, Shanghai, 200433, China; School of Mathematical Sciences, SCMS, and SCAM, Fudan University, Shanghai, 200433, China; Shanghai Key Laboratory for Contemporary Applied Mathematics, LNMS (Fudan University), And LCNBI (Fudan University), Shanghai, 200433, China; State Key Laboratory of Medical Neurobiology, And MOE Frontiers Center for Brain Science, Institutes of Brain Science, Fudan University, Shanghai, 200032, China. Electronic address: wlin@fudan.edu.cn. 9. Department of Epidemiology and Health Statistics, School of Public Health, Fudan University, Shanghai, 200032, China; Key Laboratory of Public Health Safety, Ministry of Education, Shanghai, 200032, China. Electronic address: zhj_zhang@fudan.edu.cn.
Dear Editor,COVID-19 has widely transmitted both nationwide and internationally since its outbreak which was firstly reported in December 2019 in Wuhan city, China. As of July 2, 2020, about 85,263 confirmed cases have been reported in China, of which 68,135 cases are from Hubei province [1]. The past six months have witnessed a considerable number of studies regarding the transmission dynamics of COVID-19 in China and in other countries as well. In the first quarter of 2020, most of the studies endeavored to establish compartmental or/and statistical models, compute the basic reproduction number (), and predict the future epidemic of COVID-19 [2,3], while only a few studies focused on the early stage of the epidemic dynamics in Hubei province. The accumulated evidences in the past several months definitely showed a strong heterogeneity of the epidemics among different regions and at different stages, indicating that the transmission dynamics of COVID-19 were coined by exceptionally spatiotemporal characteristics [4]. However, the specific mechanism of transmission and disease control remains largely unclear.Here, we introduce a model, which actually was established in early February, describing spatiotemporal transmission dynamics of COVID-19 at the early stage in China. Within the model, we divided the transmission dynamics into three epidemic substages defined by population movement during Lunar New Year, implementation of control measures and locations. The continuous-time but piecewise model was based on the Susceptible-Exposed-Infectious-Removed (SEIR) meta-population model with three substages [5]. Substage 1 commenced on December 1, 2019 when the first COVID-19 case was detected, and ended on January 9, 2020, prior to the mass Lunar New Year population movement. Substage 2 of the epidemic from January 10 to 22, 2020 covered the period of mass migration of people returning to their hometown for the Lunar New Year. Finally, substage 3 commenced from January 23, 2020 when a lockdown was in effect in Wuhan. During substages 1 and 2, we focused on the epidemic dynamics in Wuhan while in substage 3 the transmission process was investigated nationwide. In substage 3, considering that the tightened control policies reduced the contact frequencies of residents within each province, an attenuation effect was added to the infection risk from exposed individuals in their incubation period and an enhancement term was added to the case recovery rate due to the fact that the medical resources in Wuhan were exhausted first but improved later on with aid from other regions of the country. To further describe the infectivity of COVID-19, we applied the controlled reproduction number () instead of in substage 3. This multi-stage SEIR model was sketchily illustrated in Fig. 1
(a) and the details of the model were provided in Ref. 5.
Fig. 1
A sketch for the computational model and its simulation results. (a) A schematic description of the transmission processes with normal social contact (upper panel) and via public transportations (lower panel). (b) Estimated controlled reproduction numbers () in different regions of mainland China. Here, the cities of Tibet and Xinjiang are excluded due to a lack of available data. (c) Estimated exposed numbers in Wuhan on January 10, 2020 change with different pairs of values for variables and .
A sketch for the computational model and its simulation results. (a) A schematic description of the transmission processes with normal social contact (upper panel) and via public transportations (lower panel). (b) Estimated controlled reproduction numbers () in different regions of mainland China. Here, the cities of Tibet and Xinjiang are excluded due to a lack of available data. (c) Estimated exposed numbers in Wuhan on January 10, 2020 change with different pairs of values for variables and .The key parameters of the established SEIR model was estimated using the epidemiological data collected from the National, Hubei Provincial and Wuhan Municipal Health and Family Planning Commissions and the population movement data estimated by Baidu Huiyan (http://huiyan.baidu.com), an APP based on big-data technology which covers more than 1.1 billion mobile devices and more than 150 million points of information (POIs). For Wuhan, the estimated for substages 1 and 2 and for substage 3 were 4.90, 3.00, and 4.75, respectively. The estimated for the other cities in Hubei province except Wuhan was 2.07. The key parameters for the other regions in mainland China were estimated in Ref. [5] and depicted in Fig. 1(b).To address the concern of under-reported cases at the early stage of the epidemic, we designed a source tracing strategy implemented through transportation in substage 2. In the source tracing algorithm, we considered a simplified SE model to describe the transmission during the population flow via public transportations. Here, we used parameters estimated in substage 2 for Wuhan and the numbers of exposed individuals in the other provinces at the beginning of substage 3 to infer the number of exposed individuals on January 10 in Wuhan, i.e., retrospective-estimation in time. Considering the compact space in the vehicles, a correction factor, , was introduced to describe the infectivity when the contacts were much closer in the vehicles than in the general society. With the estimated parameters, the estimated number of exposed individuals on January 10, 2020 in Wuhan was regarded as a function of and , the number of infected individuals on January 10, 2020. As is shown in Fig. 1(c), the number of initial exposed individuals decreases significantly as increases, and decreases in a linear mode as increases. Additionally, the estimated number of exposed individuals was much greater than the number of the reported infectious individuals on January 17, 2020 (7 days or an incubation period after January 10, 2020) in Wuhan.Our numerical results implied that, at the early stage of the outbreak, a number of under-reported individuals were likely to be existent in an out-of-hospital manner, which was somewhat in agreement with the results in Ref. [6]. These hardly-undetected, exposed individuals probably became a major source of intra- or/and inter-provincial infection. Under-reporting these exposed individuals naturally resulted in an underestimation of infectious individuals in substages 1 and 2, and also led to the abnormally rapid increase of the infectious number in substage 3 in Wuhan. Finally, it is concluded that of practical significance is using piecewise models for retrospective estimations, accurate predictions, precise evaluations of this COVID-19 pandemic and its control measures.
Funding
This work was supported by the from ZJZ (81673239, 81973102), WL (11925103 & 61773125), the National Major Infectious Disease Project of the Ministry of Science and Technology of China (2017ZX10305501002), the (Grant no. 2018YFC0116600), and the (Grant no. 18DZ1201000).
Disclaimer
The funder of the study had no role in study design, data collection, data interpretation, or writing of the report. The corresponding authors had final responsibility for the decision to submit for publication.
Fig. 1