# A Mechanistic Model for Long COVID Dynamics

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Model Formulation

## 3. Mathematical Analysis

**Theorem**

**1.**

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

**Theorem**

**4.**

**Proof.**

## 4. Numerical Simulation

#### 4.1. Simulation for the Tennessee State in the US

#### 4.2. Simulation for the UK

## 5. Discussion

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Proof of Theorem 3**

**.**

## References

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**Figure 1.**A flow chart showing movement between the compartments of the system (1).

**Figure 2.**Fitting result for the number of cumulative COVID-19 cases in the US state of Tennessee beginning from 31 August 2021. The horizontal axis represents the number of days since 31 August 2021, and the vertical axis represents the number of cumulative cases.

**Figure 3.**Simulation results for the number of active long COVID cases in Tennessee using different values of $\rho $. The horizontal axis represents the number of days since 31 August 2021, and the vertical axis represents the number of active long COVID cases. On each simulation curve, the square marks the peak and the circle marks the lowest point of the curve.

**Figure 4.**Relative sensitivities for the five parameters $\gamma $, $\rho $, ${\gamma}_{L}$, ${\omega}_{L}$, and $\mu $ that are related to the long COVID compartment L.

**Figure 5.**Fitting and prediction results for the long COVID cases in the UK from 6 February 2021 to 6 December 2022. The vertical dashed line in purple separates the fitting and prediction periods. The red circles represent the reported data. The blue solid line represents the fitting result and the green solid line represents the prediction result.

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

$\Lambda $ | Population influx rate | 252.780 persons per day | [32,33] |

$\mu $ | Natural death rate | $3.624\times {10}^{-5}$ per day | [33] |

$\theta $ | Breakthrough infection ratio | 10% | [36] |

${\gamma}^{-1}$ | Acute infection recovery period | 9.5 days | [34] |

$\omega $ | Death rate for acute infection | 0.012 per day | [35] |

$\beta $ | Transmission rate | Found via data fitting | - |

$\varphi $ | Vaccination rate | Found via data fitting | - |

$\rho $ | Proportion of long COVID cases | Varied | - |

${\gamma}_{L}^{-1}$ | Long COVID recovery period | 90 days | Assumed |

${\omega}_{L}$ | Death rate for long COVID | 0.0012 per day | Assumed |

Parameter | Value | 95% Confidence Interval |
---|---|---|

$\beta $ | $4.046\times {10}^{-8}$/person/day | $(3.969\times {10}^{-8},\phantom{\rule{0.166667em}{0ex}}4.123\times {10}^{-8})$ |

$\varphi $ | $0.00474$/day | $(0.00326,\phantom{\rule{0.166667em}{0ex}}0.00622)$ |

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

$\mu $ | Natural death rate | $3.91\times {10}^{-5}$ per day | [38] |

${\gamma}^{-1}$ | Acute infection recovery period | 10 days | [39] |

${\omega}_{L}$ | Death rate for long COVID | 0.0012 per day | Assumed |

${\gamma}_{L}^{-1}$ | Long COVID recovery period | Found via data fitting | - |

$\rho $ | Proportion of long COVID cases | Found via data fitting | - |

Parameter | Value | 95% Confidence Interval |
---|---|---|

${\gamma}_{L}$ | $0.0112$ per day | $(0.00,\phantom{\rule{0.277778em}{0ex}}0.169)$ |

$\rho $ | $0.316$ | $(0.280,\phantom{\rule{0.277778em}{0ex}}1.00)$ |

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**MDPI and ACS Style**

Derrick, J.; Patterson, B.; Bai, J.; Wang, J.
A Mechanistic Model for Long COVID Dynamics. *Mathematics* **2023**, *11*, 4541.
https://doi.org/10.3390/math11214541

**AMA Style**

Derrick J, Patterson B, Bai J, Wang J.
A Mechanistic Model for Long COVID Dynamics. *Mathematics*. 2023; 11(21):4541.
https://doi.org/10.3390/math11214541

**Chicago/Turabian Style**

Derrick, Jacob, Ben Patterson, Jie Bai, and Jin Wang.
2023. "A Mechanistic Model for Long COVID Dynamics" *Mathematics* 11, no. 21: 4541.
https://doi.org/10.3390/math11214541