#
Global Stability Condition for the Disease-Free Equilibrium Point of Fractional Epidemiological Models^{ †}

^{*}

^{†}

^{‡}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

**Remark**

**1.**

- if $A=\mathrm{diag}({a}_{11},\dots ,{a}_{nn})$, then ${E}_{\alpha ,\beta}\left(A\right)=\mathrm{diag}({E}_{\alpha ,\beta}\left({a}_{11}\right),\dots ,{E}_{\alpha ,\beta}\left({a}_{nn}\right))$;
- if there exists a non-singular matrix P such that $A=PD{P}^{-1}$, then ${E}_{\alpha ,\beta}\left(A\right)=P{E}_{\alpha ,\beta}\left(D\right){P}^{-1}$.

**Definition**

**5.**

**Lemma**

**1**

**Theorem**

**1**

**.**Let $A\in {\mathbb{R}}^{n\times n}$. If the spectrum of A satisfies the relation

## 3. Main Results

**Lemma**

**2.**

**Proof.**

**Lemma**

**3.**

**Proof.**

**Lemma**

**4.**

**Proof.**

**Lemma**

**5.**

**Proof.**

**Theorem**

**2.**

- 1.
- For ${}^{C}{\mathbb{D}}_{0+}^{\alpha}X\left(t\right)=F(X,0)$, the vector ${X}^{\u2605}$ is globally asymptotically stable;
- 2.
- Function G can be written as $G(X,I)=A\xb7I-\widehat{G}(X,I)$, where $\widehat{G}(X,I)\ge 0$ for all $(X,I)$, and $A=\frac{\partial G}{\partial I}({X}^{\u2605},0)$ can be written as $A=M-D$, where $M\ge 0$ and $D>0$ is a diagonal matrix;
- 3.
- Matrix A is diagonalizable, the real parts of the eigenvalues of A are nonpositive, and ${E}_{\alpha ,\alpha}\left(A\right)\ge 0$;
- 4.
- $I\left(t\right)\ge 0$ for all $t>0$ (nonnegativity of solutions).

**Proof.**

**Remark**

**2.**

## 4. Examples

#### 4.1. A Fractional SEIR Model with Traditional Incidence Rate

#### 4.2. A Fractional SIRS Model with General Incidence Rate

#### 4.3. A Modified Fractional SICA Model for HIV/AIDS

## 5. 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.**Stability of the disease free equilibrium ${\Sigma}_{0}=(1,0,0,0)$, for the SEIR fractional model (3), considering different values of $\alpha \in \{0.8,0.85,0.9,0.95,1.0\}$. On the left: S. On the right: $E+I+R$.

**Figure 3.**Stability of the disease free equilibrium ${\Sigma}_{0}=(\frac{\Lambda}{\mu}=8,0,0)$, for the SIRS fractional model (6). In the left: $I+R$, considering different values of $\alpha \in \{0.8,0.85,0.9,0.95,1.0\}$ and initial condition ${y}_{1}$ from (8). On the right: S and $I+R$, considering the initial conditions ${y}_{i}$, $i=0,\dots ,4$ from (8) and fixed $\alpha =0.9$.

**Figure 4.**Stability of the disease free equilibrium ${\Sigma}_{0}=(1,0,0,0)$, for the SICA fractional model (9), considering different values of $\alpha \in \{0.8,0.85,0.9,0.95,1.0\}$. On the left: S. On the right: I + C + A.

Symbol | Description | Value |
---|---|---|

μ | Recruitment rate/natural death rate | 1/69.54 |

β | HIV transmission rate | 0.05 |

η_{A} | Modification parameter | 1.3 |

ϕ | HIV treatment rate for I individuals | 1 |

ρ | Default treatment rate for I individuals | 0.1 |

γ | AIDS treatment rate | 0.33 |

ω | Default treatment rate for C individuals | 0.09 |

d | AIDS induced death rate | 1 |

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**MDPI and ACS Style**

Almeida, R.; Martins, N.; Silva, C.J.
Global Stability Condition for the Disease-Free Equilibrium Point of Fractional Epidemiological Models. *Axioms* **2021**, *10*, 238.
https://doi.org/10.3390/axioms10040238

**AMA Style**

Almeida R, Martins N, Silva CJ.
Global Stability Condition for the Disease-Free Equilibrium Point of Fractional Epidemiological Models. *Axioms*. 2021; 10(4):238.
https://doi.org/10.3390/axioms10040238

**Chicago/Turabian Style**

Almeida, Ricardo, Natália Martins, and Cristiana J. Silva.
2021. "Global Stability Condition for the Disease-Free Equilibrium Point of Fractional Epidemiological Models" *Axioms* 10, no. 4: 238.
https://doi.org/10.3390/axioms10040238