# American Institute of Mathematical Sciences

doi: 10.3934/dcdsb.2021294
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## Modeling the second outbreak of COVID-19 with isolation and contact tracing

 1 Complex Systems Research Center, Shanxi University, Taiyuan 030006, China 2 Shanxi Key Laboratory of Mathematical Techniques and Big Data Analysis, on Disease Control and Prevention, Shanxi University, Taiyuan 030006, China 3 Data Science and Technology, North University of China, Taiyuan 030051, China 4 Department of Mathematics, Xinzhou Teachers University, Xinzhou 034000, China 5 College of Arts and Sciences, Shanxi Agricultural University, Taigu 030801, China 6 School of Applied Mathematics, Shanxi University of Finance and Economics, Taiyuan 030006, China 7 Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton T6G 2G1, Canada 8 School of Public Health, Peking University, Beijing 100191, China 9 Centre for Diseases Modeling (CDM), Laboratory of Mathematical Parallel Systems (LAMPS), Department of Mathematics and Statistics, York University, Toronto M3J 1P3, Canada 10 School of Mathematical Sciences, Fudan University, Shanghai 200433, China

* Corresponding author: Zhen Jin(jinzhn@263.net)

Received  July 2021 Early access December 2021

Fund Project: The authors are supported by the National Natural Science Foundation of China (12171291, 61873154), the Fund Program for the Scientific Activities of Selected Returned Overseas Professionals in Shanxi Province (20200001), the Key Research and Development Project in Shanxi Province (202003D31011/GZ), the Program for the Outstanding Innovative Teams (OIT) of Higher Learning Institutions of Shanxi, the Shanxi Scholarship Council of China (HGKY2019004), and the Scientific and Technological Innovation Programs (STIP) of Higher Education Institutions in Shanxi (2019L0082)

The first case of Corona Virus Disease 2019 (COVID-19) was reported in Wuhan, China in December 2019. Since then, COVID-19 has quickly spread out to all provinces in China and over 150 countries or territories in the world. With the first level response to public health emergencies (FLRPHE) launched over the country, the outbreak of COVID-19 in China is achieving under control in China. We develop a mathematical model based on the epidemiology of COVID-19, incorporating the isolation of healthy people, confirmed cases and contact tracing measures. We calculate the basic reproduction numbers 2.5 in China (excluding Hubei province) and 2.9 in Hubei province with the initial time on January 30 which shows the severe infectivity of COVID-19, and verify that the current isolation method effectively contains the transmission of COVID-19. Under the isolation of healthy people, confirmed cases and contact tracing measures, we find a noteworthy phenomenon that is the second epidemic of COVID-19 and estimate the peak time and value and the cumulative number of cases. Simulations show that the contact tracing measures can efficiently contain the transmission of the second epidemic of COVID-19. With the isolation of all susceptible people or all infectious people or both, there is no second epidemic of COVID-19. Furthermore, resumption of work and study can increase the transmission risk of the second epidemic of COVID-19.

Citation: Haitao Song, Fang Liu, Feng Li, Xiaochun Cao, Hao Wang, Zhongwei Jia, Huaiping Zhu, Michael Y. Li, Wei Lin, Hong Yang, Jianghong Hu, Zhen Jin. Modeling the second outbreak of COVID-19 with isolation and contact tracing. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021294
##### References:
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Shan, Modeling the effect of temperature on dengue virus transmission with periodic delay differential equations, Math. Biosci. Eng., 17 (2020), 4147-4164.  doi: 10.3934/mbe.2020230. [30] H. Song, et al., Using traveller-derived cases in Henan Province to quantify the spread of COVID-19 in Wuhan, China, Nonlinear Dynamics, 101 (2020), 1821-1831. [31] H. Song, et al., Estimation of COVID-19 outbreak size in Harbin, China, Nonlinear Dynamics, 106 (2021), 1229-1237. [32] B. Tang, et al., Estimation of the transmission risk of the 2019-nCoV and its implication for public health interventions, Journal of Clinical Medicine, 9 (2020), 462. [33] H. Tian, Y. Liu, Y. Li, et al., An investigation of transmission control measures during the first 50 days of the COVID-19 epidemic in China, Science, 368 (2020), 638-642. [34] J. Tian, J. Wu, Y. Bao, et al., Modeling analysis of COVID-19 based on morbidity data in Anhui, China, Math. Biosci. Eng., 17 (2020), 2842-2852. doi: 10.3934/mbe.2020158. [35] W. Tu, et al., Epidemic update and risk assessment of 2019 novel coronavirus-China, January 28, 2020, China CDC Weekly, 2 (2020), 83-86. doi: 10.3934/mbe.2020148. [36] 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.  doi: 10.1016/S0025-5564(02)00108-6. [37] K. Wang, Z. Lu, X. Wang, et al., Current trends and future prediction of novel coronavirus disease (COVID-19) epidemic in China: A dynamical modeling analysis, Math. Biosci. Eng., 17 (2020), 3052-3061. doi: 10.3934/mbe.2020173. [38] J. Wu, et al., Quantifying the role of social distancing, personal protection and case detection in mitigating COVID-19 outbreak in Ontario, Canada, J. Math. Ind., 10 (2020), Paper No. 15, 12 pp. doi: 10.1186/s13362-020-00083-3. [39] Y. Yang, et al., Epidemiological and clinical features of the 2019 novel coronavirus outbreak in China, MedRxiv, 2020. [40] S. You, H. Wang, M. Zhang, et al., Assessment of monthly economic losses in Wuhan under the lockdown against COVID-19, Humanities and Social Sciences Communications, 7 (2020), 1-12. [41] S. Zhao, et al., Preliminary estimation of the basic reproduction number of novel coronavirus (2019-nCoV) in China, from 2019 to 2020: A data-driven analysis in the early phase of the outbreak, International Journal of Infectious Diseases, 92 (2020), 214-217. [42] N. Zhu, A novel coronavirus from patients with pneumonia in China, 2019, New England Journal of Medicine, 382 (2020), 727-733.

show all references

##### References:
 [1] Centers for Disease Control and Prevention, Coronavirus Disease 2019 (COVID-19), Available from: https://www.cdc.gov/coronavirus/2019-ncov/index.html. [2] Chinese Center for Disease Control and Prevention,, Available from: http://2019ncov.chinacdc.cn/2019-nCoV/. [3] National Health Commission of the People's Republic of China,, Available from: http://en.nhc.gov.cn/. [4] World Health Organization (WHO), Coronavirus Disease (COVID-19) Pandemic, Available from: https://www.who.int/emergencies/diseases/novel-coronavirus-2019. [5] World Health Organization (WHO), Coronavirus Disease (COVID-2019) Situation Reports, Available from: https://www.who.int/emergencies/diseases/novel-coronavirus-2019/situation-reports/. [6] World Health Organization (WHO), Coronavirus Disease (COVID-2019) Outbreak, Report of the WHO-China Joint Mission on COVID-19, Available from: https://www.who.int/docs/default-source/coronaviruse/who-china-joint-mission-on-covid-19-final-report.pdf. [7] T. Chen, J. Rui, Q. Wang, et al., A mathematical model for simulating the phase-based transmissibility of a novel coronavirus, Infectious Diseases of Poverty, 9 (2020), 1-8. [8] O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations, J. Math. Biol., 28 (1990), 365-382.  doi: 10.1007/BF00178324. [9] M. Egger, et al., Developing WHO guidelines: Time to formally include evidence from mathematical modelling studies, F1000Research, 6 (2017). [10] M. Elisabeth, Covid-19: UK starts social distancing after new model points to 260000 potential deaths, BMJ, 368 (2020), m1089. [11] T. Frieden, A strong public health system: Essential for health and economic progress, China CDC Weekly, 2 (2020), 128-130. [12] M. Gilbert, et al., Preparedness and vulnerability of African countries against importations of COVID-19: A modelling study, The Lancet, 395 (2020), 871-877. [13] W. Guan, Clinical characteristics of 2019 novel coronavirus infection in China, MedRxiv, 2020. [14] A. B. Gumel, et al., Modelling strategies for controlling SARS outbreaks, Proceedings of the Royal Society of London. Series B: Biological Sciences, 271 (2004), 2223-2232. [15] H. Haario, et al., DRAM: efficient adaptive MCMC, Stat. Comput., 16 (2006), 339-354. doi: 10.1007/s11222-006-9438-0. [16] W. M. Hirsch, H. Hanisch and J.-P. Gabriel, Differential equation models of some parasitic infections: Methods for the study of asymptotic behavior, Comm. Pure Appl. Math., 38 (1985), 733-753.  doi: 10.1002/cpa.3160380607. [17] B. Huang, J. Wang, J. Cai, et al., Integrated vaccination and physical distancing interventions to prevent future COVID-19 waves in Chinese cities, Nature Human Behaviour, 5 (2021), 695-705. [18] C. Huang, et al., Clinical features of patients infected with 2019 novel coronavirus in Wuhan, China, The Lancet, 395 (2020), 497-506. doi: 10.3934/mbe.2020148. [19] N. Imai, et al, Estimating the potential total number of novel Coronavirus (2019-nCoV) cases in Wuhan City, China, Preprint published by the Imperial College London, 2020. [20] P. R. Krause, T. R. Fleming, I. M. Longini, et al., SARS-CoV-2 variants and vaccines, New England Journal of Medicine, 385 (2021), 179-186. [21] R. Li, et al., Substantial undocumented infection facilitates the rapid dissemination of novel coronavirus (SARS-CoV2), Science, 368 (2020), 489-493. [22] Z. Liu, et al., Predicting the cumulative number of cases for the COVID-19 epidemic in China from early data, Math. Biosci. Eng., 17 (2020), 3040-3051. doi: 10.3934/mbe.2020172. [23] R. Lu, et al., Genomic characterisation and epidemiology of 2019 novel coronavirus: implications for virus origins and receptor binding, The Lancet, 395 (2020), 565-574. [24] M. S. Majumder, et al., Estimation of MERS-coronavirus reproductive number and case fatality rate for the spring 2014 Saudi Arabia outbreak: Insights from publicly available data, PLoS Currents, 6 (2014). [25] J. Read, et al., Novel coronavirus 2019-nCoV: Early estimation of epidemiological parameters and epidemic predictions, MedRxiv, 2020. [26] M. Shen, Z. Peng, Y. Xiao, et al., Modeling the epidemic trend of the 2019 novel coronavirus outbreak in China, The Innovation, 1 (2020), 100048. [27] H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems: An Introduction to the Theory of Competitive and Cooperative Systems, American Mathematical Soc., 2008. [28] H. Song, S. Liu and W. Jiang, Global dynamics of a multistage SIR model with distributed delays and nonlinear incidence rate, Math. Methods Appl. Sci., 40 (2017), 2153-2164.  doi: 10.1002/mma.4130. [29] H. Song, D. Tian and C. Shan, Modeling the effect of temperature on dengue virus transmission with periodic delay differential equations, Math. Biosci. Eng., 17 (2020), 4147-4164.  doi: 10.3934/mbe.2020230. [30] H. Song, et al., Using traveller-derived cases in Henan Province to quantify the spread of COVID-19 in Wuhan, China, Nonlinear Dynamics, 101 (2020), 1821-1831. [31] H. Song, et al., Estimation of COVID-19 outbreak size in Harbin, China, Nonlinear Dynamics, 106 (2021), 1229-1237. [32] B. Tang, et al., Estimation of the transmission risk of the 2019-nCoV and its implication for public health interventions, Journal of Clinical Medicine, 9 (2020), 462. [33] H. Tian, Y. Liu, Y. Li, et al., An investigation of transmission control measures during the first 50 days of the COVID-19 epidemic in China, Science, 368 (2020), 638-642. [34] J. Tian, J. Wu, Y. Bao, et al., Modeling analysis of COVID-19 based on morbidity data in Anhui, China, Math. Biosci. Eng., 17 (2020), 2842-2852. doi: 10.3934/mbe.2020158. [35] W. Tu, et al., Epidemic update and risk assessment of 2019 novel coronavirus-China, January 28, 2020, China CDC Weekly, 2 (2020), 83-86. doi: 10.3934/mbe.2020148. [36] 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.  doi: 10.1016/S0025-5564(02)00108-6. [37] K. Wang, Z. Lu, X. Wang, et al., Current trends and future prediction of novel coronavirus disease (COVID-19) epidemic in China: A dynamical modeling analysis, Math. Biosci. Eng., 17 (2020), 3052-3061. doi: 10.3934/mbe.2020173. [38] J. Wu, et al., Quantifying the role of social distancing, personal protection and case detection in mitigating COVID-19 outbreak in Ontario, Canada, J. Math. Ind., 10 (2020), Paper No. 15, 12 pp. doi: 10.1186/s13362-020-00083-3. [39] Y. Yang, et al., Epidemiological and clinical features of the 2019 novel coronavirus outbreak in China, MedRxiv, 2020. [40] S. You, H. Wang, M. Zhang, et al., Assessment of monthly economic losses in Wuhan under the lockdown against COVID-19, Humanities and Social Sciences Communications, 7 (2020), 1-12. [41] S. Zhao, et al., Preliminary estimation of the basic reproduction number of novel coronavirus (2019-nCoV) in China, from 2019 to 2020: A data-driven analysis in the early phase of the outbreak, International Journal of Infectious Diseases, 92 (2020), 214-217. [42] N. Zhu, A novel coronavirus from patients with pneumonia in China, 2019, New England Journal of Medicine, 382 (2020), 727-733.
The flow diagram of COVID-19 in SIHR model
The cumulative number of confirmed cases and unfound infectious cases in China (excluding Hubei province) and Hubei province are shown under the current intervention methods. Where $CumH$ denotes the cumulative number of confirmed cases, $Cum I$ is the cumulative number of infectious people, $I_1+I_2$ is the unfound infectious people and $H$ represents the confirmed cases. Other parameter values and initial values are defined in Table Table 1
The effect of the isolation of healthy people, confirmed cases and contact tracing measures on the transmission of COVID-19 in China (excluding Hubei province). The top figure shows the cumulative number of confirmed cases and confirmed cases under the current intervention methods. The bottom figure shows the cumulative number of confirmed cases and confirmed cases when the current isolation strategy is not carried out. Where $CumH$ denotes the cumulative number of confirmed cases, $Cum I$ is the cumulative number of infectious people, $S_2+I_2$ is the quarantined susceptible and infectious people, $I_1+I_2$ is the unfound infectious people and $H$ represents the confirmed cases. Other parameter values and initial values are defined in Table 1
The predicted cumulative number of confirmed cases and infectious cases of the potential second epidemic of COVID-19 in China (excluding Hubei province) and Hubei province are shown under the current intervention methods. The embedded figures are the thumbnail of Fig. 2. Where $CumH$ denotes the cumulative number of confirmed cases, $Cum I$ is the cumulative number of infectious people, $I_1+I_2$ is the unfound infectious people and $H$ represents the confirmed cases. Other parameter values and initial values are defined in Table 1
There is no second epidemic in China (excluding Hubei province) when all susceptible people (top) or all infectious people (middle) or all susceptible people and infectious people (bottom) are isolated. Where $CumH$ denotes the cumulative number of confirmed cases, $Cum I$ is the cumulative number of infectious people, $S_2+I_2$ is the quarantined susceptible and infectious people, $I_1+I_2$ is the unfound infectious people and $H$ represents the confirmed cases. Parameter values and initial values are defined in Table 1
Under the current intervention methods, the top figure shows the effect of the average number of contact tracing $b$ on the second epidemic of COVID-19 when the time of isolation is 14 days, and the bottom figure shows the effect of the time of isolation on the second epidemic of COVID-19 when the average number of contact tracing $b$ is 12. Here we assumed that 80% of the susceptible people are isolated in China (excluding Hubei province) on January 30. Where $CumH$ denotes the cumulative number of confirmed cases, $Cum I$ is the cumulative number of infectious people, $S_2+I_2$ is the quarantined susceptible and infectious people, $I_1+I_2$ is the unfound infectious people and $H$ represents the confirmed cases. Parameter values and initial values are defined in Table 1
The effect of resumption of work on the second epidemic of COVID-19 in China (excluding Hubei province) is assessed under the isolation strategy. Where $0.8N$, $0.5N$ and $0.38N$ denote that 80%, 50% and 38% of the susceptible people are isolated on January 30, respectively. Where $CumH$ denotes the cumulative number of confirmed cases, $Cum I$ is the cumulative number of infectious people, $S_2+I_2$ is the quarantined susceptible and infectious people, $I_1+I_2$ is the unfound infectious people and $H$ represents the confirmed cases. Other parameter values and initial values are defined in Table 1
Under the isolation strategy, the effect of resumption of study on the second epidemic of COVID-19 in China (excluding Hubei province) is assessed in case of no school, resumption of study on March 30 and April 20. We assume that 38% of the susceptible people is isolated before school. Here, $CumH$ denotes the cumulative number of confirmed cases, $Cum I$ is the cumulative number of infectious people, $S_2+I_2$ is the quarantined susceptible and infectious people, $I_1+I_2$ is the unfound infectious people and $H$ represents the confirmed cases. Other parameter values and initial values are defined in Table 1
Related parameters and initial values in China (excluding Hubei province) and Hubei province
 Related parameters and initial values in (China excluding Hubei province) Parameter Descriptions Mean value 95% CI Source α1 The time of isolation at home for susceptible people 1/14 – [3] β The transmission rate of COVID-19 0.3567 (0.3291, 0.3815) Estimated b The average number of contact tracing 12 (11.5448, 12.5346) Estimated γ The hospitalization rate of infectious people 0.1429 (0.1306, 0.1538) Estimated δ The discharged rate from hospital 0.0949 – Calculated μ The transfer rate from susceptible to isolated susceptible people 1.76 × 10−4 (1.66 × 10−4, 1.86 × 10−4) Estimated α2 The transfer rate from isolated susceptible to susceptible people 5.05 × 10−6 (4.95 × 10−6, 5.15 × 10−6) Estimated Initial Values Descriptions Mean value 95% CI Source N Total population of China (excluding Hubei province) 1.3362 × 109 – [3] S1(0) The number of initial susceptible people 2.6723 × 108 (2.6723 × 108, 2.6723 × 108) Estimated S2(0) The number of initial quarantined susceptible people 3762 (3754, 3772) Estimated S3(0) The number of initial isolated susceptible people 1.069 × 109 (1.069 × 109, 1.069 × 109) Estimated I1(0) The number of initial unfound infectious people 4101 (4091, 4115) Estimated I2(0) The number of initial quarantined infectious people 700 (695,706) Estimated H(0) The number of initial hospitalized people 3886 – Data R(0) The number of initial removed people 64 – Data Related parameters and initial values (in Hubei province) Parameter Descriptions Mean value 95% CI Source α1 The time of isolation at home for susceptible people 1/14 – [3] β The transmission rate of COVID-19 0.3999 (0.3845, 0.4045) Estimated b The average number of contact tracing 5 (4.968, 5.1203) Estimated γ The hospitalization rate of infectious people 0.1379 (0.1289, 0.1490) Estimated δ The discharged rate from hospital 1/18 – Calculated μ The transfer rate from susceptible to isolated susceptible people 9.983 × 10−6 (9.56 × 10−6, 1.02 × 10-5) Estimated α2 The transfer rate from isolated susceptible to susceptible people 4.825 × 10-5 (4.7606 × 10-5, 4.9615 × 10-5) Estimated Initial Values Descriptions Mean value 95% CI Source N Total population of Hubei province 5.917 × 107 – [3] S1(0) The number of initial susceptible people 1.18 × 107 (1.18 × 107, 1.18 × 107) Estimated S2(0) The number of initial quarantined susceptible people 5367 (5352, 5382) Estimated S3(0) The number of initial isolated susceptible people 4.7336 × 107 (4.7336 × 107, 4.7336 × 107) Estimated I1(0) The number of initial unfound infectious people 12973 (12963, 12985) Estimated I2(0) The number of initial quarantined infectious people 2023 (2014, 2029) Estimated H(0) The number of initial hospitalized people 5806 – Data R(0) The number of initial removed people 320 – Data Notes: 95% CI: 95% highest posterior density interval.
 Related parameters and initial values in (China excluding Hubei province) Parameter Descriptions Mean value 95% CI Source α1 The time of isolation at home for susceptible people 1/14 – [3] β The transmission rate of COVID-19 0.3567 (0.3291, 0.3815) Estimated b The average number of contact tracing 12 (11.5448, 12.5346) Estimated γ The hospitalization rate of infectious people 0.1429 (0.1306, 0.1538) Estimated δ The discharged rate from hospital 0.0949 – Calculated μ The transfer rate from susceptible to isolated susceptible people 1.76 × 10−4 (1.66 × 10−4, 1.86 × 10−4) Estimated α2 The transfer rate from isolated susceptible to susceptible people 5.05 × 10−6 (4.95 × 10−6, 5.15 × 10−6) Estimated Initial Values Descriptions Mean value 95% CI Source N Total population of China (excluding Hubei province) 1.3362 × 109 – [3] S1(0) The number of initial susceptible people 2.6723 × 108 (2.6723 × 108, 2.6723 × 108) Estimated S2(0) The number of initial quarantined susceptible people 3762 (3754, 3772) Estimated S3(0) The number of initial isolated susceptible people 1.069 × 109 (1.069 × 109, 1.069 × 109) Estimated I1(0) The number of initial unfound infectious people 4101 (4091, 4115) Estimated I2(0) The number of initial quarantined infectious people 700 (695,706) Estimated H(0) The number of initial hospitalized people 3886 – Data R(0) The number of initial removed people 64 – Data Related parameters and initial values (in Hubei province) Parameter Descriptions Mean value 95% CI Source α1 The time of isolation at home for susceptible people 1/14 – [3] β The transmission rate of COVID-19 0.3999 (0.3845, 0.4045) Estimated b The average number of contact tracing 5 (4.968, 5.1203) Estimated γ The hospitalization rate of infectious people 0.1379 (0.1289, 0.1490) Estimated δ The discharged rate from hospital 1/18 – Calculated μ The transfer rate from susceptible to isolated susceptible people 9.983 × 10−6 (9.56 × 10−6, 1.02 × 10-5) Estimated α2 The transfer rate from isolated susceptible to susceptible people 4.825 × 10-5 (4.7606 × 10-5, 4.9615 × 10-5) Estimated Initial Values Descriptions Mean value 95% CI Source N Total population of Hubei province 5.917 × 107 – [3] S1(0) The number of initial susceptible people 1.18 × 107 (1.18 × 107, 1.18 × 107) Estimated S2(0) The number of initial quarantined susceptible people 5367 (5352, 5382) Estimated S3(0) The number of initial isolated susceptible people 4.7336 × 107 (4.7336 × 107, 4.7336 × 107) Estimated I1(0) The number of initial unfound infectious people 12973 (12963, 12985) Estimated I2(0) The number of initial quarantined infectious people 2023 (2014, 2029) Estimated H(0) The number of initial hospitalized people 5806 – Data R(0) The number of initial removed people 320 – Data Notes: 95% CI: 95% highest posterior density interval.
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