# Front Propagation of Exponentially Truncated Fractional-Order Epidemics

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## Abstract

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## 1. Introduction

## 2. Preliminaries for Fractional-Order Operators

## 3. Spatial Propagation of an Epidemic

#### 3.1. Brownian Motion

#### 3.2. Pure (Untruncated) Lévy Flights

#### 3.3. Truncated Lévy Flights

#### 3.3.1. Theoretical Analysis (Right-Propagating Front)

#### 3.3.2. Theoretical Analysis (Left-Propagating Front)

## 4. Numerical Examples

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Profiles of the susceptible and infective waves moving to the right side at different times obtained by solving Equations (19) and (20) with $\alpha =1.2$ and $\mu =5\times {10}^{-6}<\nu $. The arrow shows the direction of the front propagation. (

**b**,

**c**) Highlighting an algebraic decaying tail for the susceptible and infective waves, i.e., $1-S$ and $I\sim {y}_{1}={x}^{-(\alpha +1)}$ shown by the black dashed curves. The duration of the simulation and the time interval between curves equal 30 and 6, respectively.

**Figure 2.**(

**a**) Time evolution of the Lagrangian trajectory ${x}_{L}\left(t\right)$ at the leading edge of the infective waves such that $I({x}_{L}\left(t\right),t)={I}_{0}$, where ${I}_{0}=0.01$. The value of $\mu $ is equal to $5\times {10}^{-6}$. The dashed curve corresponds to the asymptotic expansion of the Lagrangian trajectory, i.e., ${x}_{L}\left(t\right)\sim {t}^{1/(\alpha +1)}{e}^{\left(\right(1-\theta )/(\alpha +1\left)\right)t}$. (

**b**) Time evolution of the instantaneous velocity of the right-propagating infective waves obtained by $c\left(t\right)=d{x}_{L}\left(t\right)/dt$. The dashed curve corresponds to the asymptotic expansion of the Lagrangian velocity, i.e., $c\left(t\right)\sim (1-\theta ){e}^{\left(\right(1-\theta )/(\alpha +1\left)\right)t}$, highlighting the exponential speed of the epidemic and also the agreement of the numerical result with the analytical velocity.

**Figure 3.**(

**a**) Profiles of the susceptible and infective waves moving to the right side at different times obtained by solving Equations (19) and (20) with $\alpha =1.2$ and $\mu ={10}^{-4}<\nu $. The arrow shows the direction of the front propagation. (

**b**,

**c**) Highlighting an exponentially tempered algebraic decaying tail for the susceptible and infective waves, i.e., $1-S$ and $I\sim {y}_{2}={e}^{-\mu x}{x}^{-(\alpha +1)}$ shown by the black dashed curves. The duration of the simulation and the time interval between curves equal 90 and 9, respectively.

**Figure 4.**(

**a**) Time evolution of the Lagrangian trajectory ${x}_{L}\left(t\right)$ at the leading edge of the infective waves such that $I({x}_{L}\left(t\right),t)={I}_{0}$, where ${I}_{0}=0.01$. The value of $\mu $ is equal to ${10}^{-4}$. The dashed curve corresponds to the analytical Lagrangian trajectory satisfies in the equation $lnt+(1-\theta )t-\mu {x}_{L}\left(t\right)-(\alpha +1)ln{x}_{L}\left(t\right)=ln{I}_{0}$. (

**b**) Time evolution of the instantaneous velocity of the right-propagating infective waves obtained by $c\left(t\right)=d{x}_{L}\left(t\right)/dt$. The dashed curve corresponds to the analytical Lagrangian velocity obtained by $c\left(t\right)=((1-\theta )+1/t)/(\mu +(\alpha +1)/{x}_{L}\left(t\right))$, highlighting the agreement of the numerical result with the analytical velocity and also the convergence of the epidemic speed towards the maximum epidemic speed value ${c}_{max}\approx 5\times {10}^{3}$ (see the black dashed line).

**Figure 5.**(

**a**) Profiles of the susceptible and infective waves moving to the right side at different times obtained by solving Equations (19) and (20) with $\alpha =1.2$ and $\mu ={10}^{-3}=\nu $. The arrow shows the direction of the front propagation. (

**b**,

**c**) Highlighting an exponential decaying tail for the susceptible and infective waves, i.e., $1-S$ and $I\sim {y}_{3}={e}^{-\nu x}$ shown by the back dashed curves. The duration of the simulation and the time interval between curves equal 90 and 9, respectively.

**Figure 6.**(

**a**) Time evolution of the Lagrangian trajectory ${x}_{L}\left(t\right)$ at the leading edge of the infective waves such that $I({x}_{L}\left(t\right),t)={I}_{0}$, where ${I}_{0}=0.01$. The value of $\mu $ is equal to ${10}^{-3}$. The dashed curve corresponds to the asymptotic expansion of the Lagrangian trajectory, i.e., ${x}_{L}\left(t\right)=(-ln{I}_{0}+(1-\theta )t)/\nu $. (

**b**) Time evolution of the instantaneous velocity of the right-propagating infective waves obtained by $c\left(t\right)=d{x}_{L}\left(t\right)/dt$. The dashed curve corresponds to the constant Lagrangian velocity, i.e., $c\left(t\right)=\overline{c}=(1-\theta )/\nu $, highlighting the agreement of the numerical result with the analytical velocity.

**Figure 7.**(

**a**) Profiles of the susceptible and infective waves moving to the left side at different times obtained by solving Equations (19) and (20) with $\alpha =1.2$ and different values of the truncation parameter $\mu =5\times {10}^{-6},{10}^{-4}$ and ${10}^{-3}$. The arrow shows the direction of the front propagation. (

**b**,

**c**) Highlighting an exponential decaying tail for the susceptible and infective waves, i.e., $1-S$ and $I\sim y={e}^{\nu x}$ shown by the black dashed curves. The duration of the simulation and the time interval between curves equal 90 and 9, respectively.

**Figure 8.**(

**a**) Time evolution of the Lagrangian trajectory for fractional-order derivative $\alpha =1.2$ and different values of the truncation parameter $\mu $. The blue, red and black curves correspond to $\mu =5\times {10}^{-6}$, $\mu ={10}^{-4}$, and $\mu ={10}^{-3}$, respectively. (

**b**) Time evolution of the Lagrangian trajectory for different values of the fractional-order derivative $\alpha $ and the truncation parameter $\mu ={10}^{-4}$. The blue, red and black curves correspond to $\alpha =1.2$, $\alpha =1.6$, and $\alpha =1.99$, respectively.

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**MDPI and ACS Style**

Farhadi, A.; Hanert, E.
Front Propagation of Exponentially Truncated Fractional-Order Epidemics. *Fractal Fract.* **2022**, *6*, 53.
https://doi.org/10.3390/fractalfract6020053

**AMA Style**

Farhadi A, Hanert E.
Front Propagation of Exponentially Truncated Fractional-Order Epidemics. *Fractal and Fractional*. 2022; 6(2):53.
https://doi.org/10.3390/fractalfract6020053

**Chicago/Turabian Style**

Farhadi, Afshin, and Emmanuel Hanert.
2022. "Front Propagation of Exponentially Truncated Fractional-Order Epidemics" *Fractal and Fractional* 6, no. 2: 53.
https://doi.org/10.3390/fractalfract6020053