# Dynamics of a COVID-19 Model with a Nonlinear Incidence Rate, Quarantine, Media Effects, and Number of Hospital Beds

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## Abstract

**:**

## 1. Introduction

## 2. The Dimensional Model

## 3. Analysis of the Model

#### Positivity

**Theorem**

**1.**

**Proof.**

## 4. Equilibria Existence and Classification

- 1.
- possesses one steady state if Cases 2, 4, 8, and 16 are satisfied;
- 2.
- can possess more than one steady state if Cases 6, 10, and 12 are satisfied;
- 3.
- can possess two steady states if Cases 3, 5, 7, 9, 11, 13, and 15 are satisfied.

#### Local Stability of the Disease-Free Solution

**Lemma**

**1.**

## 5. Backward Bifurcation

**Theorem**

**3.**

**Proof.**

## 6. Numerical Simulations

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Results of fitting the model to COVID-19 cases in Saudi Arabia for a period of 6 months starting from 24 March 2020. Blue solid line, model predictions; red points, actual data.

**Figure 2.**Bifurcation diagram for $b=6.32\times {10}^{-7}$ showing backward bifurcation; solid line, stable branch; dashed line, unstable branch; circles, unstable periodic branches; LP, static limit point; HB, Hopf point.

**Figure 3.**Bifurcation diagram for $b=1.4\times {10}^{-6}$ showing forward bifurcation; solid line, stable branch; dashed line, unstable branch.

**Figure 4.**Location of the Hopf point of Figure 2 in terms of the model parameters.

Case | ${\mathit{a}}_{4}$ | ${\mathit{a}}_{3}$ | ${\mathit{a}}_{2}$ | ${\mathit{a}}_{1}$ | ${\mathit{a}}_{0}$ | ${\mathcal{R}}_{0}$ | Number of Sign Changes | Number of Positive Roots |
---|---|---|---|---|---|---|---|---|

1 | + | + | + | + | + | $<1$ | 0 | 0 |

2 | + | + | + | + | − | $>1$ | 1 | 1 |

3 | + | + | + | − | + | $<1$ | 2 | 0, 2 |

4 | + | + | + | − | − | $>1$ | 1 | 1 |

5 | + | + | − | + | + | $<1$ | 2 | 0, 2 |

6 | + | + | − | + | − | $>1$ | 3 | 1, 3 |

7 | + | + | − | − | + | $<1$ | 2 | 0,2 |

8 | + | + | − | − | − | $>1$ | 1 | 1 |

9 | + | − | + | + | + | $<1$ | 2 | 0, 2 |

10 | + | − | + | + | − | $>1$ | 3 | 1, 3 |

11 | + | − | + | − | + | $<1$ | 4 | 2, 4 |

12 | + | − | + | − | − | $>1$ | 3 | 1, 3 |

13 | + | − | − | + | + | $<1$ | 2 | 0, 2 |

14 | + | − | − | + | − | $>1$ | 3 | 1, 3 |

15 | + | − | − | − | + | $<1$ | 2 | 0,2 |

16 | + | − | − | − | − | $>1$ | 1 | 1 |

Parameter | Definition | Value | Source |
---|---|---|---|

b | Dimensionless hospital-bed-to-population ratio | $6.32\times {10}^{-7}$ | [21] |

${m}_{1}$ | Half-saturation of incidence rate | $7.2965\times {10}^{5}$ | fitted |

${m}_{2}$ | Half-saturation of media effects | $2.8736\times {10}^{-8}$ | fitted |

${\beta}_{2}$ | Awareness rate | ${10}^{-5}$ | fitted |

${{\gamma}_{1}}_{0}$ | Minimum recovery rate | 0.0026 | fitted |

${{\gamma}_{1}}_{1}$ | Maximum recovery rate | 0.750 | fitted |

${\gamma}_{2}$ | Recovery rate of quarantine at home | $7.14\times {10}^{-2}$ | [27] |

$\mu $ | Natural death rate | $4.21\times {10}^{-5}$ | [26] |

$\u03f5$ | Quarantine rate | $0.0143$ | fitted |

$\sigma $ | Inverse of mean latent period | 0.333 | [27] |

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**MDPI and ACS Style**

Ajbar, A.; Alqahtani, R.T.; Boumaza, M.
Dynamics of a COVID-19 Model with a Nonlinear Incidence Rate, Quarantine, Media Effects, and Number of Hospital Beds. *Symmetry* **2021**, *13*, 947.
https://doi.org/10.3390/sym13060947

**AMA Style**

Ajbar A, Alqahtani RT, Boumaza M.
Dynamics of a COVID-19 Model with a Nonlinear Incidence Rate, Quarantine, Media Effects, and Number of Hospital Beds. *Symmetry*. 2021; 13(6):947.
https://doi.org/10.3390/sym13060947

**Chicago/Turabian Style**

Ajbar, Abdelhamid, Rubayyi T. Alqahtani, and Mourad Boumaza.
2021. "Dynamics of a COVID-19 Model with a Nonlinear Incidence Rate, Quarantine, Media Effects, and Number of Hospital Beds" *Symmetry* 13, no. 6: 947.
https://doi.org/10.3390/sym13060947