Article Contents
Article Contents

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

• * Corresponding author

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.

Mathematics Subject Classification: 34K13, 37N25, 92D30.

 Citation:

• Figure 1.  Schematic diagram for the transmission of COVID-19

Figure 2.  Cumulative infection cases of COVID-19 in the US from January 22, 2020 to December 30, 2020, the data is from [3]

Figure 3.  The model (4.1) is used to fit the cumulative COVID-19 infection cases in the US in Figure 2

Figure 4.  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

Figure 5.  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

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

Figure 7.  PRCC values for model (4.1)

Figure 8.  The dependence of $\mathcal R_0$ on $\beta$, $c$, $\delta_I$ and $\gamma_I$

Figure 9.  Contour plot of $\mathcal R_0$ with respect to $c$ varying from 0 to 10 and $\delta_I$ varying from 0 to 1

Table 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

Table 2.  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 $\mu$ ${1/ {(78.6 \times 365)}}$ $day^{-1}$ [28] $\tau_1$ 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 $S(\theta)$ $2.97 \times {\rm{10}}^{8}$ [28] $I(\theta)$ 1 [3] $A(\theta)$ 0 [3]

Table 3.  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
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