# A Multistage Time-Delay Control Model for COVID-19 Transmission

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## Abstract

**:**

## 1. Introduction

- (1)
- Based on a survey of the literature on epidemiology, due to the three periods of the development of the epidemic, a multistage time-delay COVID-19 transmission model is established in this paper. The classic SEIR model, the improved SEQIR (susceptible–exposed–quarantine–infected–recovered) model and the SEQIR Ⅱ (susceptible–exposed–quarantine–infected–recovered Ⅱ) model are adopted and used to model the outbreak period, the control period, and the steady period, respectively. These models could better simulate the transmission of three phases of COVID-19, and the results are more actual with the development of the epidemic.
- (2)
- Because COVID-19 has an incubation period, the state shall adopt prevention and control measures designed to provide prevention and control, isolating the close contacts of confirmed patients until they are cured; isolation, the spread of death and medical measures such as process improvement have a time lag. In this paper, the SEQIR model adds the isolation measures Q, considering the effective time of Q, that is, the delay time as a delay factor. In addition, several change rates are improved to better simulate the results.
- (3)
- Population mobility has a significant impact on the spread of COVID-19. The SEQIR II model used in this paper takes population mobility during the steady period, which includes both immigrating and emigrating populations.
- (4)
- In the steady period, this paper’s focus shifts from the cumulative number of confirmed patients to the number of new patients. In addition, the symptoms of patients are classified as asymptomatic and symptomatic. Therefore, at the beginning of the steady period, the new patients could be divided into symptomatic patients and asymptomatic patients, a division that is more consistent with the current transmission situation.

## 2. Related Work

## 3. Model Hypothesis and Parameter Design

#### 3.1. Model Hypothesis

- (1)
- The initial number of virus transmission patients is a constant value, as is the number of new infections.
- (2)
- In the outbreak period and the control period, the effect of the prevention and control measures on the spread of the epidemic is mainly studied without considering the influence of the immigration and emigration population.
- (3)
- Confirmed patients can infect the susceptible population and transform them into latent patients.
- (4)
- Close contacts during the incubation period will be quarantined, assuming that all close contacts can accept the isolation.
- (5)
- There are certain antibodies in recovered patients, and although there is a risk of reinfection, its incidence is greatly reduced compared with normal people; therefore, reinfection in recovered patients was not considered.
- (6)
- It is assumed that all patients infected during the incubation period go to the hospital for testing after onset, become confirmed patients, and are hospitalized.
- (7)
- After recovery, the nucleic acid test of the confirmed patient becomes negative, the patient is kept in the hospital for 5 days for observation, and his or her body temperature is normal. After reexamination, the patient is discharged.

#### 3.2. Parameter Design

## 4. SEIR Model and SEQIR Ⅱ Model

#### 4.1. Classical SEIR Model

#### 4.2. SEQIR Model

#### 4.3. SEQIR II Model

#### 4.4. Assignment of Model Parameters

## 5. Experimental Results and Analysis

#### 5.1. Outbreak Period

#### 5.2. Control Period

#### 5.3. Steady Period

## 6. Conclusions and Discussion

- For the comparison between the predicted data of the classical SEIR model and the simulated data of the improved SEQIR model for the control period, the actual data is not added for discussion, which needs to be improved later.
- Changes in mortality in the improved SEQIR model will be modified to make it more consistent with changes in the number of deaths during the control period.
- This may alter or improve aspects of the SEQIR Ⅱ model, in which the simulation effect of the number of recovered patients is poor.
- The SEQIR Ⅱ model will be modified to better simulate the process of a sudden and rapid decline in the number of recovered patients.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 6.**SEQIR Ⅱ model diagram for the steady period (migration and emigration populations, the confirmed patients who are symptomatic and asymptomatic patients).

Symbol | Meaning | Symbol | Meaning |
---|---|---|---|

$S$ | Susceptible population | ${m}_{0}$ | Number of initial contacts |

$E$ | Patients in incubation period | $m$ | Number of contacts after taking measures |

$I$ | Patients with confirmed infection | ${k}_{0}$ | Contact transmission rate |

$Q$ | Isolated latent population | $T$ | Preventive action delay time |

$R$ | Number of recovered patients | ${T}_{1}$ | Improved infection delay time |

$D$ | Number of deaths by COVID-19 disease | ${T}_{2}$ | Delay in recovery of patients |

${N}_{0}$ | Initial virus spreaders | ${T}_{3}$ | Delay time for improvement of medical measures |

$N$ | Total quantity (Stages 1 and 2) | ${T}_{4}$ | Delay in patient isolation |

${\beta}_{S}$ | Rate of susceptible population infected | ${T}_{5}$ | Incubation period |

${\beta}_{0}$ | Initial rate of infection | ${S}_{in}$ | Immigration of susceptible groups |

$\beta $ | Infection rate after taking measures | ${S}_{out}$ | Emigration of susceptible groups |

$\delta $ | Virus incidence | ${E}_{in}$ | Immigration of patients in the incubation period |

${q}_{0}$ | Initial isolation rate | ${E}_{out}$ | Emigration of patients in the incubation period |

$q$ | Improved isolation rate | $In$ | Current inflow of population in the province |

${\omega}_{0}$ | Initial mortality rate | $Out$ | Current outflow of population in the province |

$\omega $ | Improved mortality rate | ${P}_{{E}_{out}}$ | Probability of emigration of patients who are in the incubation period |

${\omega}_{1}$ | Mortality rate 1 (Mortality in asymptomatic patients) | ${P}_{{S}_{out}}$ | Probability of emigration of susceptible groups |

${\omega}_{2}$ | Mortality rate 2 (Mortality among symptomatic patients) | ${P}_{{E}_{in}}$ | Probability of immigration of patients who are in the incubation period |

${\gamma}_{0}$ | Initial recovery rate | ${P}_{{S}_{in}}$ | Probability of immigration of susceptible groups |

$\gamma $ | Improved recovery rate | ${I}_{1}$ | Number of asymptomatic patients |

${j}_{0}$ | Initial effective contact | ${I}_{2}$ | Number of patients with symptoms |

$j$ | Exposure frequency after taking action | ${I}_{N}$ | New infections |

Outbreak Period | Control Period | ||||
---|---|---|---|---|---|

Symbol | Initial Value | Distribution | Symbol | Initial Value | Distribution |

${N}_{0}$ | 1000 | - | ${N}_{0}$ | 100,000 | - |

${k}_{0}$ | 0.0002 | $N\left(0.0005,{0.0001}^{2}\right)$ | ${k}_{0}$ | 0.0002 | $N\left(0.0005,{0.0001}^{2}\right)$ |

${m}_{0}$ | 9 | $N\left(12,{1}^{2}\right)$ | ${m}_{0}$ | 9 | $N\left(12,{1}^{2}\right)$ |

${j}_{0}$ | 6 | $N\left(14,{1}^{2}\right)$ | $m$ | 1 | $N\left(7,{1}^{2}\right)$ |

${\beta}_{0}$ | 0.02 | $N\left(0.04,{0.01}^{2}\right)$ | ${j}_{0}$ | 6 | $N\left(14,{1}^{2}\right)$ |

${T}_{5}$ | 4 | $N\left(5.2,{1}^{2}\right)$ | $j$ | 1 | $N\left(5,{1}^{2}\right)$ |

${\gamma}_{0}$ | 0.01 | $N\left(0.015,{0.01}^{2}\right)$ | $T$ | 10 | - |

${T}_{2}$ | 5 | - | ${T}_{1}$ | 10 | - |

${\omega}_{0}$ | 0.01 | - | ${T}_{2}$ | 5 | - |

Steady Period | ${T}_{3}$ | 10 | - | ||

Symbol | Initial Value | Distribution | ${T}_{4}$ | 5 | - |

${I}_{N}$ | 1000 | - | ${T}_{5}$ | 4 | $N(5.2,{1}^{2})$ |

${k}_{0}$ | 0.0002 | $N\left(0.0005,{0.0001}^{2}\right)$ | ${q}_{0}$ | 0.15 | $N\left(0.2,{0.01}^{2}\right)$ |

${m}_{0}$ | 9 | $N\left(12,{1}^{2}\right)$ | $q$ | 0.6 | $N\left(0.75,{0.01}^{2}\right)$ |

${j}_{0}$ | 6 | $N\left(14,{1}^{2}\right)$ | ${\beta}_{0}$ | 0.02 | $N\left(0.04,{0.01}^{2}\right)$ |

${\beta}_{0}$ | 0.02 | $N\left(0.04,{0.01}^{2}\right)$ | $\beta $ | 0.01 | $N\left(0.02,{0.01}^{2}\right)$ |

${T}_{5}$ | 4 | $N\left(5.2,{1}^{2}\right)$ | ${\gamma}_{0}$ | 0.01 | $N\left(0.015,{0.01}^{2}\right)$ |

${\gamma}_{0}$ | 0.01 | $N\left(0.015,{0.01}^{2}\right)$ | $\gamma $ | 0.01 | $N\left(0.02,{0.01}^{2}\right)$ |

${T}_{2}$ | 5 | - | ${\omega}_{0}$ | 0.01 | - |

${\omega}_{0}$ | 0.01 | - | $\omega $ | 0.0005 | $N\left(0.0006,{0.001}^{2}\right)$ |

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**MDPI and ACS Style**

Wu, Z.; Wang, Y.; Gao, J.; Song, J.; Zhang, Y.
A Multistage Time-Delay Control Model for COVID-19 Transmission. *Sustainability* **2022**, *14*, 14657.
https://doi.org/10.3390/su142114657

**AMA Style**

Wu Z, Wang Y, Gao J, Song J, Zhang Y.
A Multistage Time-Delay Control Model for COVID-19 Transmission. *Sustainability*. 2022; 14(21):14657.
https://doi.org/10.3390/su142114657

**Chicago/Turabian Style**

Wu, Zhuang, Yuanyuan Wang, Jing Gao, Jiayang Song, and Yi Zhang.
2022. "A Multistage Time-Delay Control Model for COVID-19 Transmission" *Sustainability* 14, no. 21: 14657.
https://doi.org/10.3390/su142114657