International Journal of Differential Equations

Volume 2020, Issue 1 4325640

Research Article

Open Access

Stability Analysis of a Fractional-Order Model for Abstinence Behavior of Registration on the Electoral Lists

O. Balatif,

Corresponding Author

O. Balatif

[email protected]

orcid.org/0000-0003-1887-5350

Laboratory of Dynamical Systems, Mathematical Engineering Team, Department of Mathematics, Faculty of Sciences El Jadida, Chouaib Doukkali University, El Jadida, Morocco ucd.ac.ma

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L. Boujallal,

L. Boujallal

Department of Mathematics, Faculty of Sciences Ain Chock, University Hassan II, Casablanca, Morocco uh2c.ac.ma

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A. Labzai,

A. Labzai

orcid.org/0000-0002-4417-8893

Laboratory of Analysis, Modeling and Simulation, Department of Mathematics, Computer Science, Faculty of Sciences Ben M’Sik, Hassan II University Mohammedia, Casablanca, Morocco univh2c.ma

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M. Rachik,

M. Rachik

orcid.org/0000-0002-5118-2786

Laboratory of Analysis, Modeling and Simulation, Department of Mathematics, Computer Science, Faculty of Sciences Ben M’Sik, Hassan II University Mohammedia, Casablanca, Morocco univh2c.ma

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O. Balatif,

Corresponding Author

O. Balatif

[email protected]

orcid.org/0000-0003-1887-5350

Laboratory of Dynamical Systems, Mathematical Engineering Team, Department of Mathematics, Faculty of Sciences El Jadida, Chouaib Doukkali University, El Jadida, Morocco ucd.ac.ma

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L. Boujallal,

L. Boujallal

Department of Mathematics, Faculty of Sciences Ain Chock, University Hassan II, Casablanca, Morocco uh2c.ac.ma

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A. Labzai,

A. Labzai

orcid.org/0000-0002-4417-8893

Laboratory of Analysis, Modeling and Simulation, Department of Mathematics, Computer Science, Faculty of Sciences Ben M’Sik, Hassan II University Mohammedia, Casablanca, Morocco univh2c.ma

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M. Rachik,

M. Rachik

orcid.org/0000-0002-5118-2786

Laboratory of Analysis, Modeling and Simulation, Department of Mathematics, Computer Science, Faculty of Sciences Ben M’Sik, Hassan II University Mohammedia, Casablanca, Morocco univh2c.ma

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First published: 22 June 2020

https://doi.org/10.1155/2020/4325640

Citations: 2

Academic Editor: Peiguang Wang

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Abstract

In this work, we propose a fractional-order model that describes the dynamics of citizens who have the right to register on the electoral lists and the negative influence of abstainers on the potential electors. By using Routh–Hurwitz criteria and constructing Lyapunov functions, the local and the global stability of abstaining-free equilibrium and abstaining equilibrium are obtained. Finally, some numerical simulations are performed to verify the theoretical analysis, and they are given for different parameter setting of the order of derivative α.

1. Introduction

The first appearance of the fractional calculus was in a letter written to Guillaume de l’Hôpital by Gottfried Wilhelm Leibniz in 1695 [1–3]. It has become an important field of mathematics due to its immense application in different areas such as chemistry, physics, engineering, psychology, finance, and control theory [4–13, 14].

Mathematical modeling by differential fractional equations has more advantages for modeling and describing the dynamics of memory phenomena that have hereditary properties because fractional derivatives depend not only on local conditions but also on the past and the history of the phenomenon studied. This is, precisely, more suitable and reasonable when modeling sociological phenomena and description of real-world problems.

In this work, we introduce a fractional order for the model developed by Balatif et al. in [15], which describes the dynamics of citizens who have the right to register on the electoral lists and the negative influence of abstainers, who abstain registration on the electoral lists and the electoral process, on the potential electors. The population N is divided into three compartments: the potential electors (P) are entitled to participate in the elections, and they are not yet registered on the electoral lists; the abstainers (A) who have an attitude of abstaining the elections and the registration on the electoral lists; and the registered (R) who are listed on the electoral lists and wish to vote in the elections:

()

where D^αP, D^αA, and D^αR are the derivatives of P(t), A(t), and R(t), respectively, of an arbitrary order α (where

) in the sense of Caputo. Note that when α = 1, the fractional-order model (1) represents the classical PAR model studied in [15]. P(0) ≥ 0, A(0) ≥ 0, and R(0) ≥ 0 are the given initial states.

We note that all parameters are nonnegative. They are defined in Table 1.

Table 1. Parameters description.

Parameter	Definition
Λ	Recruitment rate
β	Effective contact rate
δ	Registration rate of the potential electors on the electoral lists
γ	Registration rate of the abstainers on the electoral lists
μ	Natural mortality rate

There exist several definitions of the fractional derivative operator: Riemann–Liouville, Caputo, Grunwald–Letnikov, etc [16–19]. The reasons to use the Caputo fractional derivative in this work are firstly, the fractional derivative of a constant is zero, and the second reason is that the initial conditions for the fractional-order differential equations with Caputo’s derivatives are in the same form as for the integer-order differential equations [19–21].

This paper is organized as follows. In Section 2, we present some preliminaries about the fractional calculus. In Section 3, we give some basic properties of the model. Section 4 is devoted to analyze the local and global stability of the proposed fractional-order model. Numerical simulations are given and discussed in Section 5. Finally, we conclude the paper in Section 6.

2. Preliminaries

Firstly, we introduce the definition of the Caputo fractional derivative, and we present some functions and useful properties that are used throughout this work [17, 19, 21]:

(1)
The Caputo fractional derivative of order α > 0 of a continuous function f : ℝ₊⟶ℝ is given by

()

where D = d/dt,

, and Γ(.) is the gamma function.

In particular, when

, we have

()

(2)
The Laplace transform of the Caputo fractional derivative is given by

()

with F(λ) the Laplace transform of f(t).

(3)
Let α, β > 0. The Mittag-Leffler function E_α,β of parameters α and β is defined as follows:

()

(4)
The Laplace transform of the Mittag-Leffler functions is

()

(5)
Let α, β > 0 and z ∈ ℂ, then the Mittag-Leffler function satisfies the equality given by

()

(6)
Let f : ℝⁿ⟶ℝⁿ with n ≥ 1. Consider the following fractional-order system:

()

with

, t₀ ∈ ℝ, and x₀ ∈ ℝⁿ. For the global existence of a solution of system (9), we need the following lemma.

Lemma 1 (see [22].)Assume that f satisfies the following conditions:

(i)
f(x) and (∂f/∂x)(x) are continuous for all x ∈ ℝⁿ
(ii)
‖f(x)‖ ≤ ω + λ‖x‖ for all x ∈ ℝⁿ, where ω and λ are the two positive constants

Then, system (9) has a unique solution on [t₀, +∞).

3. Basic Properties of the Model

System (1) describes human population, and therefore it is necessary to prove that all solutions of system (1) with positive initial data will remain positive for all times t > 0 and are bounded. This will be established by the following lemma and theorems.

Lemma 2. The feasible region Ω defined by

()

with initial conditions P(0) ≥ 0, A(0) ≥ 0, and R(0) ≥ 0, is a positive invariant for system (1), where N(0) represents the initial values of the total population.

Proof. The fractional derivative of the total population, obtained by adding all the equations of model (1), is given by

()

Applying the Laplace transform in the previous inequality, we obtain

()

then

()

from (6), we deduce

()

using (7), we have

()

since

()

therefore,

()

It implies that the region Ω is a positive invariant set for system (1).

Theorem 3. The fractional-order initial value problem (1) has a unique solution.

Proof. Let

()

where F is the right side of system (1).

Firstly, it is easy to see that F(X) and (∂F/∂X)(X) satisfy the first condition of Lemma 1.

Secondly, system (1) can be rewritten as follows:

()

where

()

Since

()

where

()

Then, system (1) has a unique solution on [0, +∞).

Theorem 4. If P(0) ≥ 0, A(0) ≥ 0, and R(0) ≥ 0, then the solutions P(t), A(t), and R(t) of system (1) are positive for all t ≥ 0.

Proof. It follows from the first equation of system (1) that

()

Since A(t) is bounded by a constant Λ₀, we have

()

where c is a constant equal to β(Λ₀/N) + (δ + μ).

Applying the Laplace transform in the previous inequality, we obtain

()

from (6), we deduce

()

then

()

Since E_α,1(−ct^α) ≥ 0, therefore the solution P(t) is positive.

Similarly, from the second and third equations of (1), we can easily prove that A(t) and R(t) are positive for all t ≥ 0.

4. Stability Analysis

In this section, we will study the stability behavior of system (1) at an abstaining-free equilibrium point and an abstaining equilibrium point. System (1) has two equilibrium points:

(i)
Abstaining-free equilibrium given by E⁰ = ((Λ/δ + μ), 0, (αΛ/μ(δ + μ))).

This equilibrium corresponds to the case when there are no abstainers in the population.

(ii)
Abstaining equilibrium point, if ℜ₀≻1, given by E^∗ = (P^∗, A^∗, R^∗), where P^∗ = (Λ(γ + μ)/μβ), A^∗ = ((Λ(δ + μ)(ℜ₀ − 1))/μβ), and R^∗ = ((δΛ²(γ + μ)² + γΛ(δ + μ)(γ + μ)(ℜ₀ − 1))/μβ(γ + μ)).

This equilibrium corresponds to the case when the behavior of abstaining the registration on the electoral lists is able to invade the population.

Where ℜ₀ is the basic reproduction number given by

()

In epidemiology, the basic reproduction number ℜ₀ is defined as the average number of secondary infections produced by an infected individual in a completely susceptible population.

In the context of our work, this threshold indicates the average number of persons that an abstainer will “infect” during his “infection” period within the potential elector population, so that the infected individuals will enter to the compartment of abstainers. This number can be obtained by using the next-generation matrix method formulated in [15, 23, 24].

4.1. Local Stability Analysis

In this section, we analyze the local stability of the abstaining-free equilibrium and the abstaining equilibrium.

Theorem 5. The abstaining-free equilibrium E⁰ is locally asymptotically stable if ℜ₀ < 1, whereas E⁰ is unstable if ℜ₀ > 1.

Proof. The Jacobian matrix at E⁰ is given by

()

Therefore, eigenvalues of the characteristic equation of are

()

Therefore, all the eigenvalues of the characteristic equation are negative if ℜ₀ < 1. Thus, |arg(λ_i)| = π≻(απ/2) for i = 1,2,3.

Hence, the equilibrium point E⁰ is locally asymptotically stable if ℜ₀ < 1 and unstable if ℜ₀ > 1.

Now, we study the local stability of the abstaining equilibrium E^∗. The Jacobian matrix at E^∗ is given by

()

Its characteristic equation is

()

where

()

If ℜ⁰≻1, then a₁ > 0, a₂ > 0, a₃ > 0, and a₁a₂ > a₃. So, the Routh–Hurwitz conditions are satisfied. Let D(Q) denote the discriminant of the polynomial Q(λ) given by (32), then

()

From [25], we have the following theorem.

Theorem 6. We assume that ℜ⁰≻1:

(1)
If D(Q)≻0 and , then E^∗ is locally asymptotically stable
(2)
If D(Q)≺0 and α≺2/3, then E^∗ is locally asymptotically stable

4.2. Global Stability Analysis

Now, we are concerned with the global asymptotic stability of abstaining-free equilibrium E⁰ and abstaining equilibrium E^∗ of model (1), respectively, using the results in [26–29].

Theorem 7. If ℜ₀ ≤ 1, then the free equilibrium E⁰ of system (1) is globally asymptotically stable on Ω.

Proof. To prove the global stability of the free equilibrium E⁰, we construct the following Lyapunov function V : Ω⟶R:

()

Then, the time derivative of V is

()

Thus, D^αV(P, A) ≤ 0 for ℜ⁰ ≤ 1.

In addition, if ℜ⁰ ≤ 1, then D^αV(P, A) = 0⇔P = P⁰ and A = 0.

Hence, by LaSalle’s invariance principle [30, 31], the free equilibrium point E⁰ is globally asymptotically stable on Ω.

Theorem 8. If ℜ₀≻1, then the abstaining equilibrium E^∗ of the system is globally asymptotically stable on Ω.

Proof. For the global stability of the abstaining equilibrium E^∗, we construct the Lyapunov function V : Ω⟶ℝ given by

()

Then, the time derivative of the Lyapunov function is given by

()

Using (1) and the expressions of the coordinates of the equilibrium point E^∗, we get

()

Furthermore, it is clear that the largest invariant set of {(P, A) ∈ Ω : D^αV(P, A) = 0} is the singleton {E^∗}. Hence, by LaSalle’s invariance principle [30, 31], the abstaining equilibrium point E^∗ is globally asymptotically stable on Ω.

5. Numerical Simulation

In this section, we present some numerical simulations of system (1) to illustrate our results. By choosing Λ = 73.8 × 10⁴, α = 0.01, β = 0.1, γ = 0.06, μ = 0.04, t_f = 90, and different values of α, we have the abstaining-free equilibrium E⁰ = (14.76 × 10⁶, 0,3.7 × 10⁶) and ℜ₀ = 0.8≺1. In this case, according to theorem (7), the abstaining-free equilibrium E⁰ of system (1) is globally asymptotically stable on Ω (see Figure 1).

Details are in the caption following the image — **Figure 1 (a)**
Open in figure viewer PowerPoint

Stability of the free equilibrium E⁰ for different values of α.

For a fixed value of α and for different initial values for each variable of state, we show in Figure 2 that the solutions converge to the abstaining-free equilibrium E⁰ = (14.76 × 10⁶, 0,3.7 × 10⁶) when ℜ₀ = 0.8≺1, which implies that the abstaining-free equilibrium E⁰ of system (1) is globally asymptotically stable on Ω.

Concerning the stability analysis of the abstaining equilibrium E^∗, we have E^∗ = (2.85 × 10⁶, 6.1433 × 10⁶, 10.007 × 10⁶) and ℜ₀ = 1.79≻1 for Λ = 10.45 × 10⁵, α = 0.15, β = 0.5, γ = 0.02, μ = 0.055, and t_f = 90. In this case, according to theorem (9), the abstaining equilibrium E^∗ of system (1) is globally asymptotically stable on Ω (see Figure 3).

In addition, we show in Figure 4 that for a fixed value of α and for different initial values for each variable of state, the solutions converge to the abstaining equilibrium E^∗ = (2.85 × 10⁶, 6.1433 × 10⁶, 10.007 × 10⁶) and ℜ₀ = 1.79≻1, which implies that the abstaining equilibrium E^∗ of system (1) is globally asymptotically stable on Ω.

From all these figures, we show that the equilibrium points E⁰ and E^∗ of system (1) are globally asymptotically stable on Ω if the conditions of theorems (7) and (9) are satisfied. Also, all solutions of model (1) converge to the equilibrium points E⁰ and E^∗ for different values of α. In addition, the solutions converge rapidly to their steady state when the value of α is very small.

6. Conclusion

In this work, we presented a fractional-order model that describes the dynamics of citizens who have the right to register on the electoral lists and the negative influence of abstainers on the potential electors. By using the Routh–Hurwitz criteria and constructing Lyapunov functions, the local and the global stability of abstaining-free equilibrium and abstaining equilibrium are obtained. The numerical simulation was carried out using Matlab.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Open Research

Data Availability

The disciplinary data used to support the findings of this study have been deposited in the Network Repository (http://networkrepository.com/proÖle.php).

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Stability Analysis of a Fractional-Order Model for Abstinence Behavior of Registration on the Electoral Lists

Abstract

1. Introduction

2. Preliminaries

3. Basic Properties of the Model

4. Stability Analysis

4.1. Local Stability Analysis

4.2. Global Stability Analysis

5. Numerical Simulation

6. Conclusion

Conflicts of Interest

Open Research

Data Availability

References

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Stability Analysis of a Fractional-Order Model for Abstinence Behavior of Registration on the Electoral Lists

Abstract

1. Introduction

2. Preliminaries

3. Basic Properties of the Model

4. Stability Analysis

4.1. Local Stability Analysis

4.2. Global Stability Analysis

5. Numerical Simulation

6. Conclusion

Conflicts of Interest

Open Research

Data Availability

References

Citing Literature

Figures

References

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