# Mathematical Modeling for Prediction Dynamics of the Coronavirus Disease 2019 (COVID-19) Pandemic, Quarantine Control Measures

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Model

## 3. Analysis of the Model

#### 3.1. Invariant Region

#### 3.2. Existence of the Solution

**Lemma**

**1**

**.**Let Ω denote the region

**Theorem**

**1.**

**Proof.**

#### 3.3. Positivity of the Solution

**Theorem**

**2.**

**Proof.**

#### 3.4. Equilibria

- (i)
- The disease-free equilibrium (DFE)$${E}^{0}=\left({\displaystyle \frac{\mathsf{\Lambda}}{\mu}},0,0,0,0,0\right);$$
- (ii)
- The endemic equilibrium$${E}^{*}=\left({S}^{*},{L}^{*},{I}^{*},{Q}^{*},{H}^{*},{R}^{*}\right),$$$$\begin{array}{cc}\hfill {S}^{*}& =\frac{AB}{{a}_{1}B+{a}_{2}{a}_{3}},\hfill \\ \hfill {L}^{*}& =\frac{\mathsf{\Lambda}}{A}-\frac{\mu B}{{a}_{1}B+{a}_{2}{a}_{3}},\hfill \\ \hfill {I}^{*}& =\frac{{a}_{3}{L}^{*}}{B},\hfill \\ \hfill {Q}^{*}& =\frac{{a}_{4}{L}^{*}+k\beta {I}^{*}}{{a}_{5}+{a}_{6}+\mu},\hfill \\ \hfill {H}^{*}& =\frac{k\alpha {I}^{*}+{a}_{6}{Q}^{*}}{{a}_{7}+\mu},\hfill \\ \hfill {R}^{*}& =\frac{k(1-\alpha -\beta ){I}^{*}+{a}_{5}{Q}^{*}+{a}_{7}{H}^{*}}{\mu},\hfill \end{array}$$

#### 3.5. The Basic Reproduction Number (${R}_{0}$)

#### 3.6. Stability of Disease-Free Equilibrium (DFE)

**Theorem**

**3.**

**Proof.**

#### 3.7. Stability of the Endemic Equilibrium

**Theorem**

**4.**

**Proof.**

## 4. Numerical Simulations

## 5. Sensitivity Analysis

## 6. The Case Study of Thailand

## 7. Conclusions and Discussion

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Flowchart of the $SLIQHR$ (Susceptible-Latent-Infectious-Quarantine-Hospitalized-Recovery) model.

**Figure 2.**Simulation results of system (1). The time series of susceptible population $\left(S\right)$, latent population $\left(L\right)$, infectious population $\left(I\right)$, quarantine population $\left(Q\right)$, hospitalized population $\left(H\right)$, and recovery population $\left(R\right)$, respectively. The solution trajectory tends toward the disease-free equilibrium (DFE) when ${R}_{0}<1$.

**Figure 3.**Simulation results of system (1). The time series of susceptible population $\left(S\right)$, latent population $\left(L\right)$, infectious population $\left(I\right)$, quarantine population $\left(Q\right)$, hospitalized population $\left(H\right)$, and recovery population $\left(R\right)$, respectively. The solution trajectories tend toward the endemic equilibrium $\left({E}^{*}\right)$ when ${R}_{0}>1$.

**Figure 4.**Simulation results of system (1) focused on changes in the transition rate (per unit time) from latent compartment L to infectious compartment I, i.e., ${a}_{3}$.

**Figure 5.**Simulation results of system (1) focused on changes in the transition rate (per unit time) from latent compartment L to quarantine compartment Q, i.e., ${a}_{4}$.

Parameters Symbol | Sensitivity Indices |
---|---|

$\mathsf{\Lambda}$ | 1 |

${a}_{1}$ | 0.8236331570 |

${a}_{2}$ | 0.1763668430 |

${a}_{3}$ | −0.07342499706 |

${a}_{4}$ | −0.7493755204 |

k | −0.06294319877 |

$\mu $ | −1.000958526 |

$\epsilon $ | −0.1798043729 |

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**MDPI and ACS Style**

Prathumwan, D.; Trachoo, K.; Chaiya, I.
Mathematical Modeling for Prediction Dynamics of the Coronavirus Disease 2019 (COVID-19) Pandemic, Quarantine Control Measures. *Symmetry* **2020**, *12*, 1404.
https://doi.org/10.3390/sym12091404

**AMA Style**

Prathumwan D, Trachoo K, Chaiya I.
Mathematical Modeling for Prediction Dynamics of the Coronavirus Disease 2019 (COVID-19) Pandemic, Quarantine Control Measures. *Symmetry*. 2020; 12(9):1404.
https://doi.org/10.3390/sym12091404

**Chicago/Turabian Style**

Prathumwan, Din, Kamonchat Trachoo, and Inthira Chaiya.
2020. "Mathematical Modeling for Prediction Dynamics of the Coronavirus Disease 2019 (COVID-19) Pandemic, Quarantine Control Measures" *Symmetry* 12, no. 9: 1404.
https://doi.org/10.3390/sym12091404