# A Comparative Numerical Study and Stability Analysis for a Fractional-Order SIR Model of Childhood Diseases

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## Abstract

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## 1. Introduction

- Susceptible group (S): this group is not infected but perhaps becomes infected due to the spread of the virus or stay susceptible.
- Infected group (I): This group has already been infected by the virus and may spread it to the susceptible group. An infected people may stay infected or may be removed from this group because of recovering or death.
- Removed group (R): This group has been vaccinated against the virus as well as recovered individuals with permanent immunity. The SIR model considers that the efficiency of the vaccination is one hundred percentage. The normalized dynamical system that describes the SIR model is given below [3]:

_{0}is the start time, n = ⌊γ⌋ + 1 and ⌊γ⌋ is the integer part of the fraction γ. The Caputo fractional-order derivative has advantages for solving initial value problems only when it applied with analytical or semi-analytical approaches. The Caputo derivative can be approximated by the Grünwald–Letnikov (GL) method using finite differences of the fractional order, similar to the Euler method, to handle numerical solutions of initial value problems. The GL method is proceeding iteratively but the sum in the scheme becomes longer and longer, which reflects the memory effect.

## 2. Explanation of the Grünwald–Letnikov Fractional Derivatives

## 3. Grünwald–Letnikov Discretization of Fractional-Order SIR Model

## 4. Stability Analysis of the Fractional-Order SIR Model

_{r}can be defined as,

_{r}is higher than a limit value, an infectious disease can propagate in a susceptible group. Moreover, the condition in Equation (15) shows that a critical vaccination fraction p

_{c}can defined as,

_{c}. Therefore, the vaccination fraction must be large enough in order to effectively avoid surge period of the disease.

_{r}> 1. At this point, the eigenvalues relating to the matrix J are obtained as,

_{e}is asymptotically stable if the following condition is fulfilled,

## 5. Numerical Results and Discussion

_{c}= 0.4625. Therefore, this case falls under the disease-free equilibrium case at which the steady state solution asymptotically approaches to the following fixed point,

_{c}= 0.4625. The results of the RK4 method at γ = 1 are plotted by circle points (∘).

**Case 1**

_{1}= 1, k

_{2}= 0, k

_{3}= 0 and p = 0.9. Here, we have a stable disease-free equilibrium case with a vaccination reproduction number V

_{r}= 0.186047.

**Case 2**

_{1}= 0.8, k

_{2}= 0.2, k

_{3}= 0 and p = 0.9. This case is also a stable disease-free equilibrium one, with a vaccination reproduction number V

_{r}= 0.186047, in which the disease will be eradicated.

_{c}(i.e., p > p

_{c}) as occurred in Case 1. The impact of the fractional order γ on the dynamic of the solutions is the same as its effect in Case 1.

**Case 3**

_{1}= 0.8, k

_{2}= 0.2, k

_{3}= 0 and p = 0.3. This case explains the impact of low-level vaccination exposure on the initial individuals with low-level infective population. Figure 4 shows a graphical representation of numerical results for the susceptible, infective, and removed population versus time for various fractional order γ, alongside the results obtained in [3]. As expected, when γ = 1, an excellent agreement between the obtained results and the results displayed in [3] is achieved.

_{c}). In this case, the endemic equilibrium stays stable because a vaccination reproduction number V

_{r}= 1.302326 satisfies the stability condition in Equation (19). In such an endemic-equilibrium case, the fractional order γ plays an influential role. The fractional order affects the susceptible, infective, and removed individuals by a different manner from disease-free equilibrium cases. When γ < 1, we found that as the fractional order γ decreases, the susceptible and infective groups rapidly decrease in the transient stage and then begin to increase once again in the steady state stage. In addition, for γ < 1, the susceptible individuals will be more than the amount for γ = 1 in the transient stage. While the infective individuals remain less than the amount for γ = 1 in both transient and steady state stages.

**Case 4**

_{1}= 0.8, k

_{2}= 0.2, k

_{3}= 0 and p = 0. This case describes the influence of low-level vaccination exposure on the initial individuals with no vaccinated people. Figure 5 shows the results visualization for the susceptible, infective, and removed population with time for various values of γ, beside the results obtained in [3] for γ = 1.

_{r}= 1.860465. Here, the number of susceptible individuals decreases while the infective individuals increase by the time for the steady state stage. The only role of the removed individuals is the very little fraction of the recovered population with long-lasting immunity. Here, the disease is quickly transported to the bulk of people. The impact of the fractional order γ on the dynamic of the solutions is similar in its impact in Case 3.

## 6. Conclusions

_{c}. The dynamics of the model is strongly dependent on the value of the fractional order γ. For the endemic equilibrium cases, the fraction of the infected population decreases by the reduction of the fractional order γ as time passes.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Behavior of various individual fractions (S in black, I in red and R in blue) versus time for Case 1 at different fractional order values (

**a**) till T = 10, (

**b**) results of [3] and (

**c**) till T = 250.

**Figure 3.**Behavior of various individual fractions (S in black, I in red and R in blue) versus time for Case 2 at different fractional order values (

**a**) till T = 10, (

**b**) results of [3] and (

**c**) till T = 250.

**Figure 4.**Behavior of various individual fractions (S in black, I in red and R in blue) versus time for Case 3 at different fractional order values (

**a**) till T = 10, (

**b**) results of [3] and (

**c**) till T = 250.

**Figure 5.**Behavior of various individual fractions (S in black, I in red and R in blue) versus time for Case 4 at different fractional order values (

**a**) till T = 10, (

**b**) results of [3] and (

**c**) till T = 250.

Case | V_{r} | γ | Fixed Point of the Stability Analysis | Steady State Equilibrium of the Numerical Solution at T = 250 | Comment | ||||
---|---|---|---|---|---|---|---|---|---|

S | I | R | S | I | R | ||||

1 | 0.186 | 1 | 0.1 | 0.0 | 0.9 | 0.1000 | 0.0 | 0.8999 | Removal of the disease |

0.75 | 0.0989 | 0.0 | 0.8897 | ||||||

0.5 | 0.0914 | 0.0 | 0.8207 | ||||||

2 | 0.186 | 1 | 0.1 | 0.0 | 0.9 | 0.0999 | 0.0 | 0.9000 | |

0.75 | 0.0989 | 0.0 | 0.8897 | ||||||

0.5 | 0.0913 | 0.0 | 0.8208 | ||||||

3 | 1.302 | 1 | 0.5375 | 0.1512 | 0.3113 | 0.5375 | 0.1512 | 0.3113 | No removal of the disease |

0.75 | 0.5457 | 0.1363 | 0.3066 | ||||||

0.5 | 0.6277 | 0.0102 | 0.2742 | ||||||

4 | 1.860 | 1 | 0.5375 | 0.4302 | 0.0323 | 0.5375 | 0.4302 | 0.0323 | |

0.75 | 0.5440 | 0.4140 | 0.0306 | ||||||

0.5 | 0.6063 | 0.2870 | 0.0188 |

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Mousa, M.M.; Alsharari, F.
A Comparative Numerical Study and Stability Analysis for a Fractional-Order SIR Model of Childhood Diseases. *Mathematics* **2021**, *9*, 2847.
https://doi.org/10.3390/math9222847

**AMA Style**

Mousa MM, Alsharari F.
A Comparative Numerical Study and Stability Analysis for a Fractional-Order SIR Model of Childhood Diseases. *Mathematics*. 2021; 9(22):2847.
https://doi.org/10.3390/math9222847

**Chicago/Turabian Style**

Mousa, Mohamed M., and Fahad Alsharari.
2021. "A Comparative Numerical Study and Stability Analysis for a Fractional-Order SIR Model of Childhood Diseases" *Mathematics* 9, no. 22: 2847.
https://doi.org/10.3390/math9222847