# Modeling and Forecasting of COVID-19 Spreading by Delayed Stochastic Differential Equations

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Models Formulation and Well-Posedness

- (1)
- all coefficients involved in the model are positive constants;
- (2)
- natural birth and death rate are not factors;
- (3)
- true asymptomatic patients will stay asymptomatic until recovery and do not spread the virus;
- (4)
- patients who are temporarily asymptomatic are included on symptomatic ones;
- (5)
- the second infection is not considered in the model;
- (6)
- the Moroccan health system is not overwhelmed.

**Remark**

**1.**

**Remark**

**2.**

**Remark**

**3.**

**Remark**

**4.**

**Theorem**

**1.**

**Proof.**

## 3. Qualitative Analysis of the Models

**Theorem**

**2.**

**Theorem**

**3.**

**Proof.**

## 4. Assessment of Parameters

## 5. Numerical Simulation of Moroccan COVID-19 Evolution

## 6. 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.**Comparison of the deterministic and the stochastic dynamical behavior with the daily reported cases of COVID-19 in Morocco.

**Figure 4.**Cumulative diagnosed cases, severe forms, critical forms and deaths 240 days from the start of the COVID-19 pandemic in Morocco.

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

$\beta $ | $0.4517$ | Estimated | u | [0–1] | Varied |

$\u03f5$ | $0.794$ | [26] | ${\gamma}_{b}$ | $0.8$ | [25] |

${\gamma}_{g}$ | $0.15$ | [25] | ${\gamma}_{c}$ | $0.05$ | [25] |

$\alpha $ | 0.06 | Assumed | ${\eta}_{a}$ | $1/21$ | Calculated |

${\eta}_{s}$ | $0.8/21$ | Calculated | ${\mu}_{s}$ | $0.01/21$ | Calculated |

${\mu}_{b}$ | 0 | Assumed | ${\mu}_{g}$ | 0 | Assumed |

${\mu}_{c}$ | $0.4/13.5$ | Calculated | ${r}_{b}$ | $1/13.5$ | Calculated |

${r}_{g}$ | $1/13.5$ | Calculated | ${r}_{c}$ | $0.6/13.5$ | Calculated |

${\tau}_{1}$ | $5.5$ | [27,28] | ${\tau}_{2}$ | $7.5$ | [29,30,31] |

${\tau}_{3}$ | 21 | Assumed | ${\tau}_{4}$ | $13.5$ | Assumed |

${\sigma}_{1}$ | $1.03$ | Calculated | ${\sigma}_{2}$ | $0.1$ | Assumed |

**Table 2.**Estimated peaks and cumulative of diagnosed cases, severe forms, critical forms and deaths.

Compartments | Peak | Cumulative |
---|---|---|

Diagnosed | Around 190 | 18,890 |

Severe forms | Around 28 | 2233 |

Critical forms | Around 10 | 997 |

Deaths | Around 5 | 468 |

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**MDPI and ACS Style**

Mahrouf, M.; Boukhouima, A.; Zine, H.; Lotfi, E.M.; Torres, D.F.M.; Yousfi, N.
Modeling and Forecasting of COVID-19 Spreading by Delayed Stochastic Differential Equations. *Axioms* **2021**, *10*, 18.
https://doi.org/10.3390/axioms10010018

**AMA Style**

Mahrouf M, Boukhouima A, Zine H, Lotfi EM, Torres DFM, Yousfi N.
Modeling and Forecasting of COVID-19 Spreading by Delayed Stochastic Differential Equations. *Axioms*. 2021; 10(1):18.
https://doi.org/10.3390/axioms10010018

**Chicago/Turabian Style**

Mahrouf, Marouane, Adnane Boukhouima, Houssine Zine, El Mehdi Lotfi, Delfim F. M. Torres, and Noura Yousfi.
2021. "Modeling and Forecasting of COVID-19 Spreading by Delayed Stochastic Differential Equations" *Axioms* 10, no. 1: 18.
https://doi.org/10.3390/axioms10010018