# Dynamic Analysis of a Delayed Fractional Infectious Disease Model with Saturated Incidence

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1**

**Lemma**

**1**

**Lemma**

**2**

## 3. Model Description

## 4. Basic Properties

#### 4.1. Invariant Region and Boundedness

**Theorem**

**1.**

**Proof.**

#### 4.2. Solution Nonnegativity

**Theorem**

**2.**

**Proof.**

**Part 1.**Prove that $S\left(t\right)>0$, for $\forall t>{t}_{0}$.

**Part 2.**Prove that $E\left(t\right)\ge 0$, for $\forall t>{t}_{0}$.

**Part 3.**Combining part 1 and the part 2 of the proof process, we can get $Q\left(t\right)\ge 0,{I}_{A}\left(t\right)\ge 0,{I}_{S}\left(t\right)\ge 0,R\left(t\right)\ge 0,$ for $\forall t>{t}_{0}$.

#### 4.3. Disease-Free Equilibrium (DFE)

#### 4.4. Existence of Endemic Equilibrium Point

**Theorem**

**3.**

**Proof.**

## 5. Stability Analysis

#### 5.1. Local Stability of DFE

**Theorem**

**4.**

**Proof.**

**Remark**

**1.**

**the system tends to the point**${E}_{0}$ in the feasible region, so DFE of the system (7) is global progressive stability when $\tau =0$.

#### 5.2. Local Stability Analysis of the Endemic Equilibrium

**Theorem**

**5.**

## 6. Numerical Simulations

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

SIR | susceptible infectious recovered |

SEIR | susceptible exposed infectious recovered |

HIV | human immunodeficiency virus |

HRSV | human respiratory syncytial virus |

SIRC | susceptible infectious recovered cleaared |

DFE | disease-free equilibrium |

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**Figure 2.**Local asymptotic stability of the disease-free equilibrium point when $\tau $ is different.

Parameters | $\mathbf{\Lambda}$ | $\mathit{\mu}$ | $\mathit{\beta}$ | m | $\mathit{\sigma}$ | $\mathit{\gamma}$ |
---|---|---|---|---|---|---|

1 | 0.02537 | 0.0106 | 0.0805 | 0.12 | 0.0668 | 2.0138 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ |

2 | 0.5 | 0.018 | 0.3 | 0.12 | 0.25 | 0.8 |

Parameters | $\mathbf{\eta}$ | $\mathbf{\theta}$ | $\mathit{v}$ | $\mathbf{\delta}$ | ${\mathit{r}}_{\mathbf{1}}$ | ${\mathit{r}}_{\mathbf{2}}$ |

1 | 0.4478 | 0.0101 | 3.2084$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | 1.6728$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 5.7341$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 1.6728$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ |

2 | 0.2 | 0.08 | 0.2 | 0.018 | 0.05 | 0.05 |

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**MDPI and ACS Style**

Ding, P.; Wang, Z.
Dynamic Analysis of a Delayed Fractional Infectious Disease Model with Saturated Incidence. *Fractal Fract.* **2022**, *6*, 138.
https://doi.org/10.3390/fractalfract6030138

**AMA Style**

Ding P, Wang Z.
Dynamic Analysis of a Delayed Fractional Infectious Disease Model with Saturated Incidence. *Fractal and Fractional*. 2022; 6(3):138.
https://doi.org/10.3390/fractalfract6030138

**Chicago/Turabian Style**

Ding, Peng, and Zhen Wang.
2022. "Dynamic Analysis of a Delayed Fractional Infectious Disease Model with Saturated Incidence" *Fractal and Fractional* 6, no. 3: 138.
https://doi.org/10.3390/fractalfract6030138