    doi: 10.3934/cpaa.2021088

## Analysis of COVID-19 epidemic transmission trend based on a time-delayed dynamic model

 1 School of Science, Chang'an University, Xi'an, 710064, China 2 Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John's, NL A1C 5S7, Canada

* Corresponding author

Received  February 2021 Revised  April 2021 Published  May 2021

Fund Project: Supported by China Scholarship Council (201906565049)

Based on the control strategies, transmission mechanism and clinical progression of COVID-19, we propose a compartmental model with time delays. Two time delays are introduced into the model to describe the incubation period of the disease and the quarantine period of uninfected individuals who have contacts with infected people. In order to reveal the spread rule for COVID-19, we study the threshold dynamics for the model. The basic reproduction number $\mathcal R_0$ is obtained. When $\mathcal R_0<1$, the disease-free equilibrium is locally asymptotically stable, when $\mathcal R_0>1$, the disease-free equilibrium is unstable and the disease is uniformly persistent. As the model's applications, we study COVID-19 transmission in the United States. The parameters are chosen to fit public data in the US. The numerical results indicate that an outbreak peak time in the US will appear in the middle of March. Sensitive analysis results show that enhancing the control measures, such as keeping social distance, wearing masks, isolation etc., can significantly contribute to the prevention and control of COVID-19 infection.

Citation: Tailei Zhang, Zhimin Li. Analysis of COVID-19 epidemic transmission trend based on a time-delayed dynamic model. Communications on Pure & Applied Analysis, doi: 10.3934/cpaa.2021088
##### References:
  R. M. Anderson and R. M. May, Infectious Diseases of Humans: Dynamics and Control, Oxford University Press, 1991.   Google Scholar  F. Brauer, C. C. Carlos and Z. Feng, Mathematical Models in Epidemiology, Springer, New York, 2019. doi: 10.1007/978-1-4939-9828-9.  Google Scholar  COVID-19 Dashboard by the Center for Systems Science and Engineering (CSSE) at Johns Hopkins University (JHU), Available from: https://www.arcgis.com/apps/opsdashboard/index.html. Google Scholar  A. Das, A. Dhar, S. Goyal, A. Kundu and S. Pandey, COVID-19: Analytic results for a modified SEIR model and comparison of different intervention strategies, Chaos Soliton. Fract., 144 (2021), 110595. doi: 10.1016/j.chaos.2020.110595.  Google Scholar  O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz, On the defifinition and the computation of the basic reproduction ratio $R_0$, in models for infectious diseases in heterogeneous populations, J. Math. Biol., 28 (1990), 365-382.  doi: 10.1007/BF00178324.  Google Scholar  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.  Google Scholar  T. Ganyani, C. Kremer, D. Chen, et al., Estimating the generation interval for coronavirus disease (COVID-19) based on symptom onset data, March 2020, Euro. Surveill., 25 (2020), 2000257. Google Scholar  M. Gatto, E. Bertuzzo, L. Mari and et al., Spread and dynamics of the COVID-19 epidemic in Italy: effects of emergency containment measures, Proc. Natl. Acad Sci., 117 (2020), 10484-10491.   Google Scholar  H. Haario, M. Laine, A. Mira and et al., DRAM: efficient adaptive MCMC, Stat. Comput., 16 (2006), 339-354.  doi: 10.1007/s11222-006-9438-0.  Google Scholar  J. K. Hale, Theory of Functional Differential Equations, Springer-Verlag, New York, 1977. Google Scholar  J. K. Hale, Asymptotic Behavior of Dissipative Systems, Amer. Math. Soc., Providence, RI, 1988. doi: 10.1090/surv/025.  Google Scholar  J. K. Hale, SM Verduyn Lunel, Introduction to Functional Differential Equations, Springer, New York, 1993. doi: 10.1007/978-1-4612-4342-7.  Google Scholar  Z. Hu, Q. Cui, J. Han, X. Wang, W. E. I. Sha and Z. Teng, Evaluation and prediction of the COVID-19 variations at different input population and quarantine strategies, a case study in Guangdong province, China, Int. J. Infec. Dis., 95 (2020), 231-240.  doi: 10.1002/bimj.202000116.  Google Scholar  Y. Kuang, Delay Differential Equations with Application to Population Dynamics, Academic Press, an Diego, 1993. Google Scholar  A. J. Kucharski, T. W. Russell and C. Diamond et al., Early dynamics of transmission and control of COVID-19: a mathematical modelling study, Lancet Infect. Dis., 20 (2020), 553-558.   Google Scholar  T. Kuniya, Prediction of the epidemic peak of coronavirus disease in Japan, J. Chin. Med., 9 (2020), 789. Google Scholar  S. A. Lauer, K. H. Grantz and et al., The incubation period of coronavirus disease 2019 (COVID-19) from publicly reported confirmed cases: estimation and application, Ann. of Intern. Med., 172 (2020), 577-582.   Google Scholar  Z. Liu, P. Magal, O. Seydi and G. Webb, A COVID-19 epidemic model with latency period, Infect. Dis. Model., 5 (2020), 323-337.   Google Scholar  S. Marino, I. B. Hogue, C. J. Ray and D. E. Kirschner, A methodology for performing global uncertainty and sensitivity analysis in systems biology, J. Theor. Biol., 254 (2008), 178-196.  doi: 10.1016/j.jtbi.2008.04.011.  Google Scholar  C. Munayco, A. Tariq and R. Rothenberg, Early transmission dynamics of COVID-19 in a southern hemisphere setting: Lima-Peru: February 29th-March 30th, 2020, Infec. Dis. Model., 5 (2020), 338-345.   Google Scholar  G. A. Muñoz-Fernández, J. M. Seoane and J. B. Seoane-Sepúlveda, A SIR-type model describing the successive waves of COVID-19, Chaos Soliton. Fract., 144 (2021), 110682. doi: 10.1016/j.chaos.2021.110682.  Google Scholar  H. L. Smith, Monotone dynamical systems: an introduction to the theory of competitive and cooperative systems, Mathematical Surveys and Monographs, Amer. Math. Soc., Providence, 1995. Google Scholar  H. L. Smith and X. Q. Zhao, Robust persistence for semidynamical systems, Nonlinear Anal., 47 (2001), 6169-6179.  doi: 10.1016/S0362-546X(01)00678-2.  Google Scholar  B. Tang, N. L. Bragazzi, Q. Li, S. Tang, Y. Xiao and J. Wu, An updated estimation of the risk of transmission of the novel coronavirus (2019-nCov), Infec. Dis. Model., 5 (2020), 248-255.   Google Scholar  B. Tang, X. Wang, Q. Li, N. L. Bragazzi, S. Tang, Y. Xiao and J. Wu, Estimation of the transmission risk of the 2019-nCoV and its implication for public health interventions, J. Chin. Med., 9 (2020), 462. Google Scholar  R. Verity, L. C. Okell, I. Dorigatti, et al., Estimates of the severity of COVID-19 disease 2019: a model-based analysis, Retour au numéro, 2020. doi: 10.1016/S1473-3099(20)30243-7. Google Scholar  WHO Coronavirus Disease (COVID-19) Dashboard, Available from: https://covid19.who.int/. Google Scholar  World Health Organization, Available from: http://www.who.int/. Google Scholar  Y. Yan, Y. Chen and K. Liu, Modeling and prediction for the trend of outbreak of NCP based on a time-delay dynamic system, Sci. Sin. Math., 50 (2020), 1-8.  doi: 10.3934/mbe.2020153.  Google Scholar  Z. Yang, Z. 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.   Google Scholar  J. Zhang, M. Litvinova and W. Wang, Evolving epidemiology and transmission dynamics of coronavirus disease 2019 outside Hubei province, China: a descriptive and modelling study, Lancet Infect. Dis., 20 (2020), 793-802.  doi: 10.3934/mbe.2020173.  Google Scholar  X. Q. Zhao, Basic reproduction ratios for periodic compartmental models with time delay, J. Dyn. Differ. Equ., 29 (2017), 67-82.  doi: 10.1007/s10884-015-9425-2.  Google Scholar

show all references

##### References:
  R. M. Anderson and R. M. May, Infectious Diseases of Humans: Dynamics and Control, Oxford University Press, 1991.   Google Scholar  F. Brauer, C. C. Carlos and Z. Feng, Mathematical Models in Epidemiology, Springer, New York, 2019. doi: 10.1007/978-1-4939-9828-9.  Google Scholar  COVID-19 Dashboard by the Center for Systems Science and Engineering (CSSE) at Johns Hopkins University (JHU), Available from: https://www.arcgis.com/apps/opsdashboard/index.html. Google Scholar  A. Das, A. Dhar, S. Goyal, A. Kundu and S. Pandey, COVID-19: Analytic results for a modified SEIR model and comparison of different intervention strategies, Chaos Soliton. Fract., 144 (2021), 110595. doi: 10.1016/j.chaos.2020.110595.  Google Scholar  O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz, On the defifinition and the computation of the basic reproduction ratio $R_0$, in models for infectious diseases in heterogeneous populations, J. Math. Biol., 28 (1990), 365-382.  doi: 10.1007/BF00178324.  Google Scholar  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.  Google Scholar  T. Ganyani, C. Kremer, D. Chen, et al., Estimating the generation interval for coronavirus disease (COVID-19) based on symptom onset data, March 2020, Euro. Surveill., 25 (2020), 2000257. Google Scholar  M. Gatto, E. Bertuzzo, L. Mari and et al., Spread and dynamics of the COVID-19 epidemic in Italy: effects of emergency containment measures, Proc. Natl. Acad Sci., 117 (2020), 10484-10491.   Google Scholar  H. Haario, M. Laine, A. Mira and et al., DRAM: efficient adaptive MCMC, Stat. Comput., 16 (2006), 339-354.  doi: 10.1007/s11222-006-9438-0.  Google Scholar  J. K. Hale, Theory of Functional Differential Equations, Springer-Verlag, New York, 1977. Google Scholar  J. K. Hale, Asymptotic Behavior of Dissipative Systems, Amer. Math. Soc., Providence, RI, 1988. doi: 10.1090/surv/025.  Google Scholar  J. K. Hale, SM Verduyn Lunel, Introduction to Functional Differential Equations, Springer, New York, 1993. doi: 10.1007/978-1-4612-4342-7.  Google Scholar  Z. Hu, Q. Cui, J. Han, X. Wang, W. E. I. Sha and Z. Teng, Evaluation and prediction of the COVID-19 variations at different input population and quarantine strategies, a case study in Guangdong province, China, Int. J. Infec. Dis., 95 (2020), 231-240.  doi: 10.1002/bimj.202000116.  Google Scholar  Y. Kuang, Delay Differential Equations with Application to Population Dynamics, Academic Press, an Diego, 1993. Google Scholar  A. J. Kucharski, T. W. Russell and C. Diamond et al., Early dynamics of transmission and control of COVID-19: a mathematical modelling study, Lancet Infect. Dis., 20 (2020), 553-558.   Google Scholar  T. Kuniya, Prediction of the epidemic peak of coronavirus disease in Japan, J. Chin. Med., 9 (2020), 789. Google Scholar  S. A. Lauer, K. H. Grantz and et al., The incubation period of coronavirus disease 2019 (COVID-19) from publicly reported confirmed cases: estimation and application, Ann. of Intern. Med., 172 (2020), 577-582.   Google Scholar  Z. Liu, P. Magal, O. Seydi and G. Webb, A COVID-19 epidemic model with latency period, Infect. Dis. Model., 5 (2020), 323-337.   Google Scholar  S. Marino, I. B. Hogue, C. J. Ray and D. E. Kirschner, A methodology for performing global uncertainty and sensitivity analysis in systems biology, J. Theor. Biol., 254 (2008), 178-196.  doi: 10.1016/j.jtbi.2008.04.011.  Google Scholar  C. Munayco, A. Tariq and R. Rothenberg, Early transmission dynamics of COVID-19 in a southern hemisphere setting: Lima-Peru: February 29th-March 30th, 2020, Infec. Dis. Model., 5 (2020), 338-345.   Google Scholar  G. A. Muñoz-Fernández, J. M. Seoane and J. B. Seoane-Sepúlveda, A SIR-type model describing the successive waves of COVID-19, Chaos Soliton. Fract., 144 (2021), 110682. doi: 10.1016/j.chaos.2021.110682.  Google Scholar  H. L. Smith, Monotone dynamical systems: an introduction to the theory of competitive and cooperative systems, Mathematical Surveys and Monographs, Amer. Math. Soc., Providence, 1995. Google Scholar  H. L. Smith and X. Q. Zhao, Robust persistence for semidynamical systems, Nonlinear Anal., 47 (2001), 6169-6179.  doi: 10.1016/S0362-546X(01)00678-2.  Google Scholar  B. Tang, N. L. Bragazzi, Q. Li, S. Tang, Y. Xiao and J. Wu, An updated estimation of the risk of transmission of the novel coronavirus (2019-nCov), Infec. Dis. Model., 5 (2020), 248-255.   Google Scholar  B. Tang, X. Wang, Q. Li, N. L. Bragazzi, S. Tang, Y. Xiao and J. Wu, Estimation of the transmission risk of the 2019-nCoV and its implication for public health interventions, J. Chin. Med., 9 (2020), 462. Google Scholar  R. Verity, L. C. Okell, I. Dorigatti, et al., Estimates of the severity of COVID-19 disease 2019: a model-based analysis, Retour au numéro, 2020. doi: 10.1016/S1473-3099(20)30243-7. Google Scholar  WHO Coronavirus Disease (COVID-19) Dashboard, Available from: https://covid19.who.int/. Google Scholar  World Health Organization, Available from: http://www.who.int/. Google Scholar  Y. Yan, Y. Chen and K. Liu, Modeling and prediction for the trend of outbreak of NCP based on a time-delay dynamic system, Sci. Sin. Math., 50 (2020), 1-8.  doi: 10.3934/mbe.2020153.  Google Scholar  Z. Yang, Z. 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.   Google Scholar  J. Zhang, M. Litvinova and W. Wang, Evolving epidemiology and transmission dynamics of coronavirus disease 2019 outside Hubei province, China: a descriptive and modelling study, Lancet Infect. Dis., 20 (2020), 793-802.  doi: 10.3934/mbe.2020173.  Google Scholar  X. Q. Zhao, Basic reproduction ratios for periodic compartmental models with time delay, J. Dyn. Differ. Equ., 29 (2017), 67-82.  doi: 10.1007/s10884-015-9425-2.  Google Scholar ]">Figure 2.  Cumulative infection cases of COVID-19 in the US from January 22, 2020 to December 30, 2020, the data is from ">Figure 3.  The model (4.1) is used to fit the cumulative COVID-19 infection cases in the US in Figure 2 This is MCMC analysis of parameters $c$, $\beta$, $q$, $\theta$ and $\tau_2$. The first row is the random sequence of five parameters, and the second row is the histogram of MCMC corresponding to five parameters This is MCMC analysis of parameters $\omega$, $\delta_I$, $\alpha$, $\gamma_I$ and $\gamma_A$. The first row is the random sequence of five parameters, and the second row is the histogram of MCMC corresponding to five parameters are used to substitute the model (4.1), and the model is employed to predict the cumulative number of infected individuals from December 30, 2020 to April 6, 2021. At the same time, the reported data from December 30, 2020 to February 10, 2021 are compared with the predicted data">Figure 6.  The parameter values in Table 2 are used to substitute the model (4.1), and the model is employed to predict the cumulative number of infected individuals from December 30, 2020 to April 6, 2021. At the same time, the reported data from December 30, 2020 to February 10, 2021 are compared with the predicted data The dependence of $\mathcal R_0$ on $\beta$, $c$, $\delta_I$ and $\gamma_I$ Contour plot of $\mathcal R_0$ with respect to $c$ varying from 0 to 10 and $\delta_I$ varying from 0 to 1
Descriptions of parameters in model (2.1)
 Parameters Description $\Lambda$ recruitment rate $c$ contact rate $\beta$ the probability of infection by each time contact $q$ tracing quarantined rate of individuals who have contacts with infected people $\theta$ relative infectivity of an asymptomatic individual to a symptomatic individual $\mu$ natural mortality rate $\tau_1$ incubation period of COVID-19 $\tau_2$ quarantine period of uninfected individuals who have contacts with infected people $\omega$ proportion of the exposed developing infected with symptoms $\delta_I$ diagnostic rate of symptomatic infected individuals $\delta_q$ diagnostic rate of quarantined infected individuals $\alpha$ disease-induced mortality rate $\gamma_I$ recovery rate of symptomatic infected individuals $\gamma_A$ recovery rate of asymptomatic infected individuals $\gamma_q$ recovery rate of quarantined infected individuals $\gamma_c$ recovery rate of confirmed patients
 Parameters Description $\Lambda$ recruitment rate $c$ contact rate $\beta$ the probability of infection by each time contact $q$ tracing quarantined rate of individuals who have contacts with infected people $\theta$ relative infectivity of an asymptomatic individual to a symptomatic individual $\mu$ natural mortality rate $\tau_1$ incubation period of COVID-19 $\tau_2$ quarantine period of uninfected individuals who have contacts with infected people $\omega$ proportion of the exposed developing infected with symptoms $\delta_I$ diagnostic rate of symptomatic infected individuals $\delta_q$ diagnostic rate of quarantined infected individuals $\alpha$ disease-induced mortality rate $\gamma_I$ recovery rate of symptomatic infected individuals $\gamma_A$ recovery rate of asymptomatic infected individuals $\gamma_q$ recovery rate of quarantined infected individuals $\gamma_c$ recovery rate of confirmed patients
Description of some model parameters and their values, as well as the initial value of the warehouse variables in model (4.1). For variable $x$, we define $x_0 = x(0 + \theta ) = x(\theta ),\; - \tau \leqslant \theta \leqslant 0,\;\tau = \max \left\{ {\tau _1 ,\tau _2} \right\}$. We can estimate that the basic reproduction number of COVID-19 in the US is $\mathcal R_0 = 2.9018$
 Parameters Value/range Sources $\Lambda$ 103.5 $day^{-1}$ Estimated $c$ 14.0046 $day^{-1}$ Estimated $\beta$ $1.8 \times {\rm{10}}^{-8}$ $day^{-1}$ Estimated $q$ 0.0057 $day^{-1}$ Estimated $\theta$ 1 $day^{-1}$ Estimated $\mu$ ${1/ {(78.6 \times 365)}}$ $day^{-1}$  $\tau_1$ 5.1 $days$ [3,17] $\tau_2$ 8 $days$ Estimated $\omega$ 0.6834 $day^{-1}$ Estimated $\delta_I$ 0.1404 $day^{-1}$ Estimated $\alpha$ $5.6253 \times {\rm{10}}^{-4}$ $day^{-1}$ Estimated $\gamma_I$ 0.1005 $day^{-1}$ Estimated $\gamma_A$ 0.2959 $day^{-1}$ Estimated $S(\theta)$ $2.97 \times {\rm{10}}^{8}$  $I(\theta)$ 1  $A(\theta)$ 0 
 Parameters Value/range Sources $\Lambda$ 103.5 $day^{-1}$ Estimated $c$ 14.0046 $day^{-1}$ Estimated $\beta$ $1.8 \times {\rm{10}}^{-8}$ $day^{-1}$ Estimated $q$ 0.0057 $day^{-1}$ Estimated $\theta$ 1 $day^{-1}$ Estimated $\mu$ ${1/ {(78.6 \times 365)}}$ $day^{-1}$  $\tau_1$ 5.1 $days$ [3,17] $\tau_2$ 8 $days$ Estimated $\omega$ 0.6834 $day^{-1}$ Estimated $\delta_I$ 0.1404 $day^{-1}$ Estimated $\alpha$ $5.6253 \times {\rm{10}}^{-4}$ $day^{-1}$ Estimated $\gamma_I$ 0.1005 $day^{-1}$ Estimated $\gamma_A$ 0.2959 $day^{-1}$ Estimated $S(\theta)$ $2.97 \times {\rm{10}}^{8}$  $I(\theta)$ 1  $A(\theta)$ 0 
Partial rank correlation coefficients (PRCCs) for the aggregate $\mathcal R_0$ and each input parameter variable
 Parameters PRCCs $p-$value $\beta$ 0.8782 0 $c$ 0.8791 0 $q$ -0.8777 0 $\omega$ 0.6095 0 $\theta$ 0.7081 0 $\delta_I$ -0.2504 0 $\alpha$ -0.2008 0.1441 $\gamma_I$ -0.2460 0 $\gamma_A$ -0.6564 0
 Parameters PRCCs $p-$value $\beta$ 0.8782 0 $c$ 0.8791 0 $q$ -0.8777 0 $\omega$ 0.6095 0 $\theta$ 0.7081 0 $\delta_I$ -0.2504 0 $\alpha$ -0.2008 0.1441 $\gamma_I$ -0.2460 0 $\gamma_A$ -0.6564 0
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