doi: 10.3934/cpaa.2021155
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Transmission dynamics and high infectiousness of Coronavirus Disease 2019

1. 

Institute of NBC Defense of PLA, Beijing 102205, China

2. 

Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing 100029, China

3. 

School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China

4. 

Ministry of Education Key Laboratory of Environment and Health, State Key Laboratory of Environmental Health (Incubating), School of Public Health, Tongji Medical College, Huazhong University of Science and Technology, Wuhan 430063, China

5. 

Institute of Biotechnology, Academy of Military Medical Sciences, Beijing 100071, China

6. 

XiangYa School of Public Health, Central South University, Changsha, Hunan 410078, China

* Corresponding author

Contributed equally to this article

Received  February 2021 Revised  July 2021 Early access September 2021

Coronavirus disease 2019 (COVID-19) has rapidly spread around the world since the early 2020. Recently, a second wave of COVID-19 has resurged in many countries. The transmission dynamics and infectiousness of the COVID-19 pandemic remain unclear, and developing strategies to mitigate the severity of the pandemic is a top priority for global public health. According to the infection mechanism of COVID-19, a novel susceptible-asymptomatic-symptomatic-recovered (SASR) model with control variables in a patchy environment was proposed not only to consider the key characteristics of asymptomatic infection and the effects of seasonal variation but also to incorporate different control measures for multiple transmission routes. The basic reproduction number $ R_{0} $ was established to describe the spreading behavior in the natural state over a long time horizon, and the natural reproduction number $ R_{n} $, which describes the development trend of the disease during a short time in the future, was defined according to the actual propagation characteristics. In addition, the effective reproduction number $ R_{e} $ considering the control strategies was proposed to evaluate the impact of non-pharmaceutical interventions. The results of numerical simulations for COVID-19 cases in Wuhan, China, based on the SASR model indicate that $ R_{0} $ was 3.58, $ R_{n} $ ranged from 2.37 to 4.91, and $ R_{e} $ decreased gradually from 4.83 on December 8, 2019 to 0.31 on March 8, 2020, reaching 1.40 on January 23, 2020, when the lockdown was lifted in Wuhan. We further concluded that the total number of infections, including asymptomatic infections, was approximately 301, 804 as of March 8, 2020, in Wuhan, China. In particular, this article proposes a dynamic method to distinguish the impact of natural factors and human interventions on the development of the pandemic, and provides a theoretical basis for fighting the global COVID-19 pandemic.

Citation: Shunxiang Huang, Lin Wu, Jing Li, Ming-Zhen Xin, Yingying Wang, Xingjie Hao, Zhongyi Wang, Qihong Deng, Bin-Guo Wang. Transmission dynamics and high infectiousness of Coronavirus Disease 2019. Communications on Pure and Applied Analysis, doi: 10.3934/cpaa.2021155
References:
[1]

N. Bacaër and S. Guernaoui, The epidemic threshold of vector-borne diseases with seasonality, J. Math. Biol., 53 (2006), 421-436.  doi: 10.1007/s00285-006-0015-0.

[2]

Y. BaiL. Yao and T. Wei, Presumed asymptomatic carrier transmission of COVID-19, JAMA, 323 (2020), 790-808.  doi: 10.1001/jama.2020.2565.

[3]

J. F. ChanS. Yuan and K. H. Kok, A familial cluster of pneumonia associated with the 2019 novel coronavirus indicating person-to-person transmission: a study of a family cluster, The Lancet, 395 (2020), 514-523.  doi: 10.1016/S0140-6736(20)30154-9.

[4]

C. Corduneanu, Almost Periodic Functions, New York, 1994.

[5]

COVID-19 prevention and control expert group of Chinese preventive medicine association. The latest understanding of epidemiological characteristics in COVID-19, Chinese J. Viral Diseases, 10 (2003), 86–92(Chinese).

[6]

O. DiekmannJ. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio $R_{0}$ in the models for infectious disease in heterogeneous populations, J. Math. Biol., 28 (1990), 365-382.  doi: 10.1007/BF00178324.

[7]

A. Fink, Almost Periodic Differential Equations, Springer, Berlin, 1974.

[8]

A. B. GumelS. Ruan and T. Day, Modelling strategies for controlling SARS outbreaks, Proceedings of the Royal Society of London Series B Biological Sciences, 271 (2020), 2223-2232.  doi: 10.1098/rspb.2004.2800.

[9]

Z. D. GuoZ. Y. Wang and S. F. Zhang, Aerosol and surface distribution of severe acute respiratory syndrome coronavirus 2 in Hospital Wards, Wuhan, China, Emerg. Infect. Dis., 26 (2020), 1586-1591.  doi: 10.3201/eid2607.200885.

[10]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, Providence, 1988.

[11]

J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer, New York, 1993.

[12]

X. HaoS. Cheng and D. Wu, Reconstruction of the full transmission dynamics of COVID-19 in Wuhan, Nature, 584 (2020), 420-424.  doi: 10.1038/s41586-020-2554-8.

[13]

X. HeE. H. Y. Lau and P. Wu, Temporal dynamics in viral shedding and transmissibility of COVID-19, Nat Med, 26 (2020), 672-675.  doi: 10.1038/s41591-020-0869-5.

[14]

J. HellewellS. Abbott and A. Gimma, Feasibility of controlling COVID-19 outbreaks by isolation of cases and contacts, The Lancet Global Health, 8 (2020), 488-496.  doi: 10.1016/S2214-109X(20)30074-7.

[15]

C. HuangY. Wang and X. Li, Clinical features of patients infected with 2019 novel coronavirus in Wuhan, China, The Lancet, 395 (2020), 497-506.  doi: 10.1016/S0140-6736(20)30183-5.

[16]

G. M. HwangP. J. Mahoney and J. H. James, A model-based tool to predict the propagation of infectious disease via airports, Travel Medicine Infectious Disease, 10 (2020), 32-42. 

[17]

S. M. KisslerC. Tedijanto and E. Goldstein, Projecting the transmission dynamics of SARS-CoV-2 through the post-pandemic period, Science, 368 (2020), 860-868.  doi: 10.1126/science.abb5793.

[18]

J. S. LavineM. Poss and B. T. Grenfell, Directly transmitted viral diseases: modeling the dynamics of transmission, Trends Microbiol, 16 (2020), 165-172.  doi: 10.1016/j.tim.2008.01.007.

[19]

J. LeeG. Chowell and E. Jung, A dynamic compartmental model for the Middle East respiratory syndrome outbreak in the Republic of Korea: A retrospective analysis on control interventions and superspreading events, J. Theor. Biol., 408 (2016), 118-126.  doi: 10.1016/j.jtbi.2016.08.009.

[20]

Y. LiuZ. Ning and Y. Chen, Aerodynamic analysis of SARS-CoV-2 in two Wuhan hospitals, Nature, 582 (2020), 557-560.  doi: 10.1038/s41586-020-2271-3.

[21]

L. MarcC. Ted and C. Ben, Transmission dynamics and control of severe acute respiratory syndrome, Science, 300 (2003), 1966-1970. 

[22]

V. J. MunsterM. Koopmans and N. van Doremalen, A Novel coronavirus emerging in China-key questions for impact assessment, N. Engl. J. Med., 382 (2020), 692-694. 

[23]

R. A. Neher, R. Dyrdak and V. Druelle et al., Potential impact of seasonal forcing on a SARS-CoV-2 pandemic, Schwzerische medizinische Wochenschrift, 150 (2020).

[24]

S. NovoR. Obaya and A. M. Sanz, Topological dynamics for monotone skew-product semiflows with applications, J. Dyn. Differ. Equ., 25 (2013), 1201-1231. 

[25]

A. Pan, L. Liu and C. Wang et al., Association of Public Health Interventions With the Epidemiology of the COVID-19 Outbreak in Wuhan, China, JAMA, (2020), 9 pp.

[26]

P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48. 

[27]

L. QiangB. G. Wang and X. Q. Zhao, Basic reproduction ratios for almost periodic compartmental models with time delay, J. Differ. Equ., 269 (2020), 4440-4476. 

[28]

J. Read, J. Bridgen and D. Cummings et al., Novel coronavirus 2019-nCoV: early estimation of epidemiological parameters and epidemic predictions, Phil. Trans. R. Soc. B, 376 (2021), 9 pp. doi: 10.1098/rstb.2020.0265.

[29]

H. L. Smith, Monotone Dynamical System: An Introduction to the Theory of Competitive and Cooperative Systems, American Mathematical society, Providence, 1995.

[30]

B. Tang, X. Wang and Q. Li et al., Estimation of the Transmission Risk of 2019-nCov and Its Implication for Public Health Interventions. Social ence Electronic Publishing, 2020.

[31]

Z. TongA. Tang and K. Li, Potential presymptomatic transmission of SARS-CoV-2, Zhejiang Province, China, 2020, Emerg. Infect. Dis., 26 (2020), 1052-1054.  doi: 10.3201/eid2605.200198.

[32]

B. G. Wang and X. Q. Zhao, Basic reproduction ratios for almost periodic compartmental epidemic models, J. Dyn. Differ. Equ., 25 (2013), 535-562.  doi: 10.1007/s10884-013-9304-7.

[33]

S. WangY. Liu and M. Liu, Research progress of basic regeneration number in COVID-19, Chinese Science Bulletin, 65 (2020), 2334-2341.  doi: 10.1360/TB-2020-0413.

[34]

WHO, Middle East respiratory syndrome coronavirus (MERS-CoV).

[35]

WHO, Statement on the second meeting of the International Health Regulations (2005) Emergency Committee regarding the outbreak of novel coronavirus (2019-nCoV), 30 January 2020 Statement Geneva, Switzerland.

[36]

W. Wang and X. Q. Zhao, Threshold dynamics for compartmental epidemic models in periodic environments, J. Dyn. Differ. Equ., 20 (2008), 699-717.  doi: 10.1007/s10884-008-9111-8.

[37]

J. T. WuK. Leung and G. M. Leung, Nowcasting and forecasting the potential domestic and international spread of the 2019-nCoV outbreak originating in Wuhan, China: a modelling study, The Lancet, 395 (2020), 689-697.  doi: 10.1016/S0140-6736(20)30260-9.

[38]

Z. YangZ. Zeng and K. Wang, Modified SEIR and AI prediction of the epidemics trend of COVID-19 in China under public health interventions, J. Thorac. Dis., 12 (2020), 165-174.  doi: 10.21037/jtd.2020.02.64.

[39]

X. Q. Zhao, Dynamical Systems in Population Biology, Springer-Verlag, New York, 2017.

[40]

N. ZhuD. Zhang and W. Wang, For the China Novel Coronavairus Investigating and Reaseach Team. A Novel coronavirus from patients with pneumonia in China, 2019., N. Engl. J. Med., 382 (2020), 727-733. 

show all references

References:
[1]

N. Bacaër and S. Guernaoui, The epidemic threshold of vector-borne diseases with seasonality, J. Math. Biol., 53 (2006), 421-436.  doi: 10.1007/s00285-006-0015-0.

[2]

Y. BaiL. Yao and T. Wei, Presumed asymptomatic carrier transmission of COVID-19, JAMA, 323 (2020), 790-808.  doi: 10.1001/jama.2020.2565.

[3]

J. F. ChanS. Yuan and K. H. Kok, A familial cluster of pneumonia associated with the 2019 novel coronavirus indicating person-to-person transmission: a study of a family cluster, The Lancet, 395 (2020), 514-523.  doi: 10.1016/S0140-6736(20)30154-9.

[4]

C. Corduneanu, Almost Periodic Functions, New York, 1994.

[5]

COVID-19 prevention and control expert group of Chinese preventive medicine association. The latest understanding of epidemiological characteristics in COVID-19, Chinese J. Viral Diseases, 10 (2003), 86–92(Chinese).

[6]

O. DiekmannJ. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio $R_{0}$ in the models for infectious disease in heterogeneous populations, J. Math. Biol., 28 (1990), 365-382.  doi: 10.1007/BF00178324.

[7]

A. Fink, Almost Periodic Differential Equations, Springer, Berlin, 1974.

[8]

A. B. GumelS. Ruan and T. Day, Modelling strategies for controlling SARS outbreaks, Proceedings of the Royal Society of London Series B Biological Sciences, 271 (2020), 2223-2232.  doi: 10.1098/rspb.2004.2800.

[9]

Z. D. GuoZ. Y. Wang and S. F. Zhang, Aerosol and surface distribution of severe acute respiratory syndrome coronavirus 2 in Hospital Wards, Wuhan, China, Emerg. Infect. Dis., 26 (2020), 1586-1591.  doi: 10.3201/eid2607.200885.

[10]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, Providence, 1988.

[11]

J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer, New York, 1993.

[12]

X. HaoS. Cheng and D. Wu, Reconstruction of the full transmission dynamics of COVID-19 in Wuhan, Nature, 584 (2020), 420-424.  doi: 10.1038/s41586-020-2554-8.

[13]

X. HeE. H. Y. Lau and P. Wu, Temporal dynamics in viral shedding and transmissibility of COVID-19, Nat Med, 26 (2020), 672-675.  doi: 10.1038/s41591-020-0869-5.

[14]

J. HellewellS. Abbott and A. Gimma, Feasibility of controlling COVID-19 outbreaks by isolation of cases and contacts, The Lancet Global Health, 8 (2020), 488-496.  doi: 10.1016/S2214-109X(20)30074-7.

[15]

C. HuangY. Wang and X. Li, Clinical features of patients infected with 2019 novel coronavirus in Wuhan, China, The Lancet, 395 (2020), 497-506.  doi: 10.1016/S0140-6736(20)30183-5.

[16]

G. M. HwangP. J. Mahoney and J. H. James, A model-based tool to predict the propagation of infectious disease via airports, Travel Medicine Infectious Disease, 10 (2020), 32-42. 

[17]

S. M. KisslerC. Tedijanto and E. Goldstein, Projecting the transmission dynamics of SARS-CoV-2 through the post-pandemic period, Science, 368 (2020), 860-868.  doi: 10.1126/science.abb5793.

[18]

J. S. LavineM. Poss and B. T. Grenfell, Directly transmitted viral diseases: modeling the dynamics of transmission, Trends Microbiol, 16 (2020), 165-172.  doi: 10.1016/j.tim.2008.01.007.

[19]

J. LeeG. Chowell and E. Jung, A dynamic compartmental model for the Middle East respiratory syndrome outbreak in the Republic of Korea: A retrospective analysis on control interventions and superspreading events, J. Theor. Biol., 408 (2016), 118-126.  doi: 10.1016/j.jtbi.2016.08.009.

[20]

Y. LiuZ. Ning and Y. Chen, Aerodynamic analysis of SARS-CoV-2 in two Wuhan hospitals, Nature, 582 (2020), 557-560.  doi: 10.1038/s41586-020-2271-3.

[21]

L. MarcC. Ted and C. Ben, Transmission dynamics and control of severe acute respiratory syndrome, Science, 300 (2003), 1966-1970. 

[22]

V. J. MunsterM. Koopmans and N. van Doremalen, A Novel coronavirus emerging in China-key questions for impact assessment, N. Engl. J. Med., 382 (2020), 692-694. 

[23]

R. A. Neher, R. Dyrdak and V. Druelle et al., Potential impact of seasonal forcing on a SARS-CoV-2 pandemic, Schwzerische medizinische Wochenschrift, 150 (2020).

[24]

S. NovoR. Obaya and A. M. Sanz, Topological dynamics for monotone skew-product semiflows with applications, J. Dyn. Differ. Equ., 25 (2013), 1201-1231. 

[25]

A. Pan, L. Liu and C. Wang et al., Association of Public Health Interventions With the Epidemiology of the COVID-19 Outbreak in Wuhan, China, JAMA, (2020), 9 pp.

[26]

P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48. 

[27]

L. QiangB. G. Wang and X. Q. Zhao, Basic reproduction ratios for almost periodic compartmental models with time delay, J. Differ. Equ., 269 (2020), 4440-4476. 

[28]

J. Read, J. Bridgen and D. Cummings et al., Novel coronavirus 2019-nCoV: early estimation of epidemiological parameters and epidemic predictions, Phil. Trans. R. Soc. B, 376 (2021), 9 pp. doi: 10.1098/rstb.2020.0265.

[29]

H. L. Smith, Monotone Dynamical System: An Introduction to the Theory of Competitive and Cooperative Systems, American Mathematical society, Providence, 1995.

[30]

B. Tang, X. Wang and Q. Li et al., Estimation of the Transmission Risk of 2019-nCov and Its Implication for Public Health Interventions. Social ence Electronic Publishing, 2020.

[31]

Z. TongA. Tang and K. Li, Potential presymptomatic transmission of SARS-CoV-2, Zhejiang Province, China, 2020, Emerg. Infect. Dis., 26 (2020), 1052-1054.  doi: 10.3201/eid2605.200198.

[32]

B. G. Wang and X. Q. Zhao, Basic reproduction ratios for almost periodic compartmental epidemic models, J. Dyn. Differ. Equ., 25 (2013), 535-562.  doi: 10.1007/s10884-013-9304-7.

[33]

S. WangY. Liu and M. Liu, Research progress of basic regeneration number in COVID-19, Chinese Science Bulletin, 65 (2020), 2334-2341.  doi: 10.1360/TB-2020-0413.

[34]

WHO, Middle East respiratory syndrome coronavirus (MERS-CoV).

[35]

WHO, Statement on the second meeting of the International Health Regulations (2005) Emergency Committee regarding the outbreak of novel coronavirus (2019-nCoV), 30 January 2020 Statement Geneva, Switzerland.

[36]

W. Wang and X. Q. Zhao, Threshold dynamics for compartmental epidemic models in periodic environments, J. Dyn. Differ. Equ., 20 (2008), 699-717.  doi: 10.1007/s10884-008-9111-8.

[37]

J. T. WuK. Leung and G. M. Leung, Nowcasting and forecasting the potential domestic and international spread of the 2019-nCoV outbreak originating in Wuhan, China: a modelling study, The Lancet, 395 (2020), 689-697.  doi: 10.1016/S0140-6736(20)30260-9.

[38]

Z. YangZ. Zeng and K. Wang, Modified SEIR and AI prediction of the epidemics trend of COVID-19 in China under public health interventions, J. Thorac. Dis., 12 (2020), 165-174.  doi: 10.21037/jtd.2020.02.64.

[39]

X. Q. Zhao, Dynamical Systems in Population Biology, Springer-Verlag, New York, 2017.

[40]

N. ZhuD. Zhang and W. Wang, For the China Novel Coronavairus Investigating and Reaseach Team. A Novel coronavirus from patients with pneumonia in China, 2019., N. Engl. J. Med., 382 (2020), 727-733. 

Figure 3.  The variation of basic reproduction number, effective reproduction number and natural reproduction number
Figure 1.  The variation of model parameters
Figure 2.  Prediction and surveillance of new infected cases and total infected cases
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