# On a Novel Dynamics of SEIR Epidemic Models with a Potential Application to COVID-19

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## Abstract

**:**

## 1. Introduction

## 2. Background

#### 2.1. SEIR Epidemic Model

#### 2.2. Contextualization of the New Epidemic Model

#### 2.3. Preliminaries on Fractional Differential Equations

**Definition**

**1**

**.**The ABC fractional derivative of function f supported on $[0,t]$ is stated as

**Definition**

**2**

**.**The Laplace transform of the ABC derivative of Definition 1 is given by

**Definition**

**3**

**.**The Atangana–Baleanu integral of f on $[0,t]$ with $\alpha >0$ is expressed as

## 3. Mathematical Analysis

#### 3.1. Basic Reproduction Number

#### 3.2. Equilibrium Points

#### 3.3. Feasibility Region Analysis

**Lemma**

**1.**

**Proof.**

**Theorem**

**1.**

- (i)
- The initial populations.
- (ii)
- All possible disease-free equilibrium points.
- (iii)
- The disease-dependent equilibrium points.

**Proof.**

## 4. Positivity in Solutions and Sensitivity Analysis

#### 4.1. Existence of Positivity in Solutions

**Theorem**

**2.**

- (i)
- The solution of the SEIR model is positive for any non-negative initial conditions and for all t.
- (ii)
- The total population remains unvaried even for a long time, which holds for any given arbitrary non-negative initial conditions.
- (iii)
- All the solutions are positively bounded.

**Proof.**

#### 4.2. Sensitivity Analysis

## 5. Numerical Results

#### 5.1. Graphical Representation of $S\left(t\right)$, $E\left(t\right)$, $I\left(t\right)$, and $R\left(t\right)$

`SIMULINK`of

`Matlab`, and executed with

`SOLVER`of

`SIMULINK`, using the fourth-order Runge–Kutta method. Note that this block diagram is a graphical representation of a first-order vector initial value problem. We consider the system given in (1), which is the vector initial value problem proposed by us.

#### 5.2. Laplace Adomian Decomposition Method

`Mathematica`software.

## 6. Conclusions and Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**Plot of real and imaginary values of the eigenvalues of $\mathit{J}\left({F}^{\left(0\right)}\right)$, for ${F}^{\left(0\right)}=(0,0,0,0)$.

**Figure 4.**Plot of real and imaginary values of the eigenvalues of $J\left({F}^{\left(0\right)}\right)$, for ${F}^{\left(0\right)}=(-333.\overline{333},0,0,0)$.

**Figure 5.**Plot of real and imaginary values of the eigenvalues of $J\left({F}^{\ast}\right)$, for ${F}^{\ast}=(-333.333,20,20,10)$.

**Figure 6.**Plot of the number of susceptible, exposed, infected, and recovered people over time t (in days) to show positivity of solutions on $S\left(t\right)$, $E\left(t\right)$, $I\left(t\right)$, and $R\left(t\right)$.

**Figure 8.**Block diagram of the first-order vector initial value problem given in (1).

**Figure 9.**Plot of the number of susceptible, exposed, infected, and recovered people, $S\left(t\right)$, $E\left(t\right)$, $I\left(t\right)$, and $R\left(t\right)$, over time t (in days) of the SEIR model for solutions based on first-order derivatives ${\alpha}_{i}=1$ for all $i\in \{1,2,3,4\}$.

**Figure 10.**Plots of the number of susceptible $S\left(t\right)$ (

**a**), exposed $E\left(t\right)$ (

**b**), infected $I\left(t\right)$ (

**c**), and recovered $R\left(t\right)$ (

**d**), cases over time $t\in [0,1]$ (in days) for 11 solutions each based on the ${\alpha}_{i}$th fractional derivative with ${\alpha}_{i}\in \{0.0,0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1.0\}$ for all $i\in \{1,2,3,4\}$.

Notations/Symbols | Definition |
---|---|

t | Time instant |

$S\left(t\right)$ | Susceptible population at t |

$E\left(t\right)$ | Exposed population at t |

$I\left(t\right)$ | Infected population at t |

$R\left(t\right)$ | Recovered population at t |

$N\left(t\right)$ | Total population at t |

$S\left(0\right)$ | Initial susceptible population, $S\left(0\right)=40$ (fixed) |

$E\left(0\right)$ | Initial exposed population, $E\left(0\right)=30$ (fixed) |

$I\left(0\right)$ | Initial infected population, $I\left(0\right)=20$ (fixed) |

$R\left(0\right)$ | Initial recovered population, $R\left(0\right)=10$ (fixed) |

$N\left(0\right)$ | Initial total population |

${\mathcal{R}}_{0}$ | Basic reproduction number |

a | Rate at which susceptible become exposed, $a=0.00003$ (fixed) |

b | Rate at which exposed become infected, $b=0.02$ (fixed) |

c | Rate at which exposed become recovered, $c=0.03$ (fixed) |

r | Rate at which infected become recovered, $r=0.01$ (fixed) |

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**MDPI and ACS Style**

Rangasamy, M.; Chesneau, C.; Martin-Barreiro, C.; Leiva, V.
On a Novel Dynamics of SEIR Epidemic Models with a Potential Application to COVID-19. *Symmetry* **2022**, *14*, 1436.
https://doi.org/10.3390/sym14071436

**AMA Style**

Rangasamy M, Chesneau C, Martin-Barreiro C, Leiva V.
On a Novel Dynamics of SEIR Epidemic Models with a Potential Application to COVID-19. *Symmetry*. 2022; 14(7):1436.
https://doi.org/10.3390/sym14071436

**Chicago/Turabian Style**

Rangasamy, Maheswari, Christophe Chesneau, Carlos Martin-Barreiro, and Víctor Leiva.
2022. "On a Novel Dynamics of SEIR Epidemic Models with a Potential Application to COVID-19" *Symmetry* 14, no. 7: 1436.
https://doi.org/10.3390/sym14071436