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

Laboratory of Dynamical Systems, Mathematical Engineering Team, Department of Mathematics, Faculty of Sciences El Jadida, Chouaib Doukkali University, El Jadida, Morocco Department of Mathematics, Faculty of Sciences Ain Chock, University Hassan II, Casablanca, Morocco Laboratory of Analysis, Modeling and Simulation, Department of Mathematics, Computer Science, Faculty of Sciences Ben M’Sik, Hassan II University Mohammedia, Casablanca, Morocco

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. is 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. e 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 α P, D α A, and D α R are the derivatives of P(t), A(t), and R(t), respectively, of an arbitrary order α (where 0 ≺ α ≺ 1) 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. ey are defined in Table 1. ere exist several definitions of the fractional derivative operator: Riemann-Liouville, Caputo, Grunwald-Letnikov, etc [16][17][18][19]. e 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][20][21].
is 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.

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) e Caputo fractional derivative of order α > 0 of a continuous function f: R + ⟶ R is given by where D � d/dt, n − 1 ≺ α ≺ n, n ∈ N, and Γ(.) is the gamma function.
In particular, when 0 ≺ α ≺ 1, we have (2) e Laplace transform of the Caputo fractional derivative is given by with F(λ) the Laplace transform of f(t).
(3) Let α, β > 0. e Mittag-Leffler function E α,β of parameters α and β is defined as follows: (4) e 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: R n ⟶ R n with n ≥ 1. Consider the following fractional-order system: with 0 ≺ α ≺ 1, t 0 ∈ R, and x 0 ∈ R n . 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: where ω and λ are the two positive constants en, system (9) has a unique solution on [t 0 , +∞).

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. is will be established by the following lemma and theorems.
Proof. e 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.
e 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 (zF/zX)(X) satisfy the first condition of Lemma 1.
Applying the Laplace transform in the previous inequality, we obtain so 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.
is equilibrium corresponds to the case when there are no abstainers in the population.
(ii) Abstaining equilibrium point, if R 0 ≻ 1, given by is equilibrium corresponds to the case when the behavior of abstaining the registration on the electoral lists is able to invade the population.
Where R 0 is the basic reproduction number given by In epidemiology, the basic reproduction number R 0 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. is number can be obtained by using the next-generation matrix method formulated in [15,23,24].

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

Theorem 5.
e abstaining-free equilibrium E 0 is locally asymptotically stable if R 0 < 1, whereas E 0 is unstable if R 0 > 1.
Theorem 7. If R 0 ≤ 1, then the free equilibrium E 0 of system (1) is globally asymptotically stable on Ω.
Proof. To prove the global stability of the free equilibrium E 0 , we construct the following Lyapunov function V: Ω ⟶ R: us, D α V(P, A) ≤ 0 for R 0 ≤ 1. In addition, if R 0 ≤ 1, then D α V(P, A) � 0⇔P � P 0 and A � 0.
Hence, by LaSalle's invariance principle [30,31], the free equilibrium point E 0 is globally asymptotically stable on Ω. □ Theorem 8. If R 0 ≻ 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 International Journal of Differential Equations 5 en, 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: Hence, by LaSalle's invariance principle [30,31], the abstaining equilibrium point E * is globally asymptotically stable on Ω. □
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 × 10 6 , 6.1433 × 10 6 , 10.007 × 10 6 ) and R 0 � 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 E 0 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 E 0 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.

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. e numerical simulation was carried out using Matlab.

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