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

1. Introduction

The first appearance of the fractional calculus was in a letter written to Guillaume de l'Hopital 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:

[mathematical expression not reproducible], (1)

where [D.sup.[alpha]]P, [D.sup.[alpha]]A, and [D.sup.[alpha]]R are the derivatives of P(t), A(t), and R(t), respectively, of an arbitrary order [alpha] (where 0 < [alpha] < 1) in the sense of Caputo. Note that when [alpha] = 1, the fractional-order model (1) represents the classical PAR model studied in [15]. P(0) [greater than or equal to] 0, A(0) [greater than or equal to] 0, and R(0) [greater than or equal to] 0 are the given initial states.

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

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 [alpha] > 0 of a continuous function f: [R.sub.+] [right arrow] R is given by

[mathematical expression not reproducible], (2)

where D = d/dt, n - 1 < [alpha] < n, n [member of] N, and [GAMMA] (.) is the gamma function.

In particular, when 0 < [alpha] < 1, we have

[mathematical expression not reproducible]. (3)

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

[mathematical expression not reproducible], (4)

with F([lambda]) the Laplace transform of f(t).

(3) Let [alpha], [beta] > 0. The Mittag-Leffler function [E.sub.[alpha],[beta]] of parameters [alpha] and [beta] is defined as follows:

[E.sub.[alpha],[beta]] (z) = [[infinity].summation over (k=0)] [z.sup.k]/[GAMMA]([alpha]k+[beta]). (5)

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

L [[t.sup.[beta]-1][E.sub.[alpha],[beta]] ([+ or -] [at.sup.[alpha]] = [[lambda].sup.[alpha]-[beta]]/ [[lambda].sup.[alpha][- or +] a, (6)

(5) Let [alpha], [beta] > 0 and z [member of] C, then the Mittag-Leffler function satisfies the equality given by

[E.sub.[alpha],[beta]] (z) = z[E.sub.[alpha],[alpha]+[beta]] (z) + 1/[GAMMA]([beta]). (7)

(6) Let f: [R.sup.n] [right arrow] [R.sup.n] with n [greater than or equal to] 1. Consider the following fractional-order system:

[mathematical expression not reproducible], (8)

with 0 < [alpha] < 1, [t.sub.0] [member of] R, and [x.sub.0] [member of] [R.sup.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:

(i) f(x) and ([partial derivative]f/[partial derivative]x)(x) are continuous for all * [member of] [R.sup.n]

(ii) [parallel]f (x)[parallel] [less than or equal to] [omega] + [parallel]x[parallel] for all * [member of] [R.sup.n], where [omega] and [lambda] are the two positive constants

Then, system (9) has a unique solution on [[t.sub.0], +[infinity]).

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 Q defined by

[OMEGA] = {(P(t), A(t), R(t)) [member of] [R.sup.4.sub.+], P(t) + A(t) + R(t) = N(t) [less than or equal to] N(0) [LAMBDA]/[mu]}, (9)

with initial conditions P(0) [greater than or equal to] 0, A(0) [greater than or equal to] 0, and R(0) [greater than or equal to] 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

[D.sup.[alpha]] N(t) [less than or equal to] [LAMBDA] - [mu]N(t). (10)

Applying the Laplace transform in the previous inequality, we obtain

[[lambda].sup.[alpha]] L(N (t)) - [[lambda].sup.[alpha]-1] N(0) [less than or equal to] [LAMBDA]/[lambda] - [mu] L(N(t)), (11)

then

[mathematical expression not reproducible], (12)

from (6), we deduce

[mathematical expression not reproducible], (13)

using (7), we have

[mathematical expression not reproducible], (14)

since

0 [less than or equal to] [E.sub.[alpha],1] (-[mu][t.sup.[alpha]]) [less than or equal to] 1, (15)

therefore,

N (t) [less than or equal to] N(0) + [LAMBDA]/[mu]. (16)

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

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

Proof. Let

X(t) = [[PAR].sup.T], [D.sup.[alpha]]X(t) = F(X(t)), (17)

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

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

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

F(X(t))= [bar.[LAMBDA]] + ([AM.sub.1] + [M.sub.2]) X(t), (18)

where

[mathematical expression not reproducible]. (19)

Since

[mathematical expression not reproducible], (20)

where

[omega] = [parallel][bar.[LAMBDA]][parallel] [lambda] = c[parallel][M.sub.1][parallel] + [parallel][M.sub.2][parallel]. (21)

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

Theorem 4. If P(0) [greater than or equal to] 0, A(0) [greater than or equal to] 0, and R(0) [greater than or equal to] 0, then the solutions P (t), A (t), and R (t) of system (') are positive for all t [greater than or equal to] 0.

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

[D.sup.[alpha]] P [greater than or equal to] ([beta] A(t)/N) + ([delta] + [mu])P(t). (22)

Since A(t) is bounded by a constant [[LAMBDA].sub.0], we have

[D.sup.[alpha]]P [greater than or equal to] - cP(t), (23)

where c is a constant equal to [beta]([[LAMBDA].sub.0]/N) + ([delta] + [mu]).

Applying the Laplace transform in the previous inequality, we obtain

[[lambda].sub.[alpha]] L(P(t))- [[lambda].sup.[alpha]-1] P(0) [greater than or equal to] - cL(P(t)), (24)

so

L (P(t)) [greater than or equal to] [[lambda].sup.[alpha]-1]/[[lambda].sup.[alpha]+c] P(0), (25)

from (6), we deduce

L (P(t)) [greater than or equal to] L([E.sub.[alpha],1] (-[ct.sup.[alpha]])) P(0), (26)

then

P(t) [greater than or equal to] [E.sub.[alpha],1] (-[ct.sup.[alpha]])P(0). (27)

Since [E.sub.[alpha],1] (-[ct.sup.[alpha]]) [greater than or equal to] 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 [greater than or equal to] 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 [E.sup.0] = (([LAMBDA]/[delta]+ [mu]), 0, ([alpha][LAMBDA]/[mu]([delta] + [mu]))).

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

(ii) Abstaining equilibrium point, if [R.sub.0] > 1, given by [mathematical expression not reproducible].

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

Where [R.sub.0] is the basic reproduction number given by

[R.sub.0] = [mu][beta]/([delta] + [mu]]([gamma] + [mu]]). (28)

In epidemiology, the basic reproduction number [R.sub.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. 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 [E.sup.0] is locally asymptotically stable if [R.sub.0] < 1, whereas [E.sup.0] is unstable if

[R.sub.0] > 1.

Proof. The Jacobian matrix at [E.sup.0] is given by

[mathematical expression not reproducible]. (29)

Therefore, eigenvalues of the characteristic equation of [mathematical expression not reproducible] are

[[lambda].sub.1] = -[mu], [[lambda].sub.2] = -([delta] + [mu]), (30) [[lambda].sub.3] = ([gamma] + [mu])([R.sub.0] - 1). (30)

Therefore, all the eigenvalues of the characteristic equation are negative if [R.sub.0] < 1. Thus, [absolute value of (arg([[lambda].sub.i]))] = [pi] > ([alpha][pi]/2) for i = 1, 2, 3.

Hence, the equilibrium point [E.sup.0] is locally asymptotically stable if [R.sub.0] < 1 and unstable if [R.sub.0] > 1.

Now, we study the local stability of the abstaining equilibrium [E.sup.*]. The Jacobian matrix at [E.sup.*] is given by

[mathematical expression not reproducible]. (31)

Its characteristic equation is

[[lambda].sup.3] + [a.sub.1][[lambda].sup.2] + [a.sub.2][lambda] + [a.sub.3] = 0, (32)

where

[mathematical expression not reproducible]. (33)

If [R.sup.0] > 1, then [a.sub.1] > 0, [a.sub.2] > 0, [a.sub.3] > 0, and [a.sub.1][a.sub.2] > [a.sub.3]. So, the Routh-Hurwitz conditions are satisfied. Let D(Q) denote the discriminant of the polynomial Q ([lambda]) given by (32), then

[mathematical expression not reproducible]. (34)

From [25], we have the following theorem.

Theorem 6. We assume that [R.sup.0] > 1:

(1) If D(Q) > 0 and 0 < [alpha] < 1, then [E.sup.*] is locally asymptotically stable

(2) If D(Q) < 0 and [alpha] < 2/3, then [E.sup.*] 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.sup.*] of model (1), respectively, using the results in [26-29].

Theorem 7. If [R.sub.0] [less than or equal to] 1, then the free equilibrium [E.sup.0] of system (1) is globally asymptotically stable on [OMEGA].

Proof. To prove the global stability of the free equilibrium [E.sup.0], we construct the following Lyapunov function V: [OMEGA] [right arrow] R:

[mathematical expression not reproducible]. (35)

Then, the time derivative of V is

[mathematical expression not reproducible]. (36)

Thus, [D.sup.[alpha]]V (P, A) [less than or equal to] 0 for [R.sup.0] [less than or equal to] 1.

In addition, if [R.sup.0] [less than or equal to] 1, then [D.sup.[alpha]]V(P, A) = 0 [??] P = [P.sup.0] and A = 0.

Hence, by LaSalle's invariance principle [30, 31], the free equilibrium point [E.sup.0] is globally asymptotically stable on [OMEGA].

Theorem 8. If [R.sub.0] > 1, then the abstaining equilibrium [E.sup.*] of the system is globally asymptotically stable on [OMEGA].

Proof. For the global stability of the abstaining equilibrium [E.sup.*], we construct the Lyapunov function V: [OMEGA] [right arrow] R given by

V(P,A) = P - [P.sup.*] ln(P/[P.sup.*]) + A - [A.sup.*] ln(A/[A.sup.*]). (37)

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

[mathematical expression not reproducible]. (38)

Using (1) and the expressions of the coordinates of the equilibrium point [E.sup.*], we get

[D.sup.[alpha]]V (P,A) [less than or equal to] - [LAMBDA][(P - [P.sup.*]).sup.2]/[PP.sup.*] [less than or equal to] 0. (39)

Furthermore, it is clear that the largest invariant set of {(P,A) [member of] [OMEGA]: [D.sup.[alpha]]V(P, A) = 0} is the singleton {[E.sup.*]}. Hence, by LaSalle's invariance principle [30, 31], the abstaining equilibrium point [E.sup.*] is globally asymptotically stable on [OMEGA].

5. Numerical Simulation

In this section, we present some numerical simulations of system (1) to illustrate our results. By choosing [LAMBDA] = 73.8 x 104, [alpha] = 0.01, [beta] = 0.1, [gamma] = 0.06, [mu] = 0.04, [t.sub.f] = 90, and different values of [alpha], we have the abstaining-free equilibrium [E.sup.0] =(14.76 * [10.sup.6], 0,3.7 * [10.sup.6]) and [R.sub.0] = 0.8 < 1. In this case, according to theorem (7), the abstaining-free equilibrium [E.sup.0] of system (1) is globally asymptotically stable on [OMEGA] (see Figure 1).

For a fixed value of [alpha] and for different initial values for each variable of state, we show in Figure 2 that the solutions converge to the abstaining-free equilibrium [E.sup.0] = (14.76 x [10.sup.6], 0,3.7 * [10.sup.6]) when [R.sub.0] = 0.8 < 1, which implies that the abstaining-free equilibrium [E.sup.0] of system (1) is globally asymptotically stable on [OMEGA].

Concerning the stability analysis of the abstaining equilibrium [E.sup.*], we have [E.sup.*] = (2.85 * [10.sup.6], 6.1433x [10.sup.6], 10.007 * [10.sup.6]) and [R.sub.0] = 1.79 > 1 for [LAMBDA] = 10.45 * [10.sup.5], [alpha] = 0.15, [beta] = 0.5, [gamma] = 0.02, [mu] = 0.055, and [t.sub.f] = 90. In this case, according to theorem (9), the abstaining equilibrium [E.sup.*] of system (1) is globally asymptotically stable on Q (see Figure 3).

In addition, we show in Figure 4 that for a fixed value of a and for different initial values for each variable of state, the solutions converge to the abstaining equilibrium [E.sup.*] = (2.85 * [10.sup.6], 6.1433 * [10.sup.6], 10.007 * [10.sup.6]) and [R.sub.0] = 1.79 > 1, which implies that the abstaining equilibrium [E.sup.*] of system (1) is globally asymptotically stable on [OMEGA].

From all these figures, we show that the equilibrium points [E.sup.0] and [E.sup.*] of system (1) are globally asymptotically stable on [OMEGA] if the conditions of theorems (7) and (9) are satisfied. Also, all solutions of model (1) converge to the equilibrium points [E.sup.0] and [E.sup.*] for different values of [alpha]. In addition, the solutions converge rapidly to their steady state when the value of [alpha]a 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.

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

Data Availability

The disciplinary data used to support the findings of this study have been deposited in the Network Repository (http://networkrepository.com/proC0le.php).

Conflicts of Interest

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

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O. Balatif [ID], (1) L. Boujallal, (2) A. Labzai [ID], (3) and M. Rachik [ID] (3)

(1) Laboratory of Dynamical Systems, Mathematical Engineering Team, Department of Mathematics, Faculty of Sciences El Jadida, Chouaib Doukkali University, El Jadida, Morocco

(2) Department of Mathematics, Faculty of Sciences Ain Chock, University Hassan II, Casablanca, Morocco

(3) Laboratory of Analysis, Modeling and Simulation, Department of Mathematics, Computer Science, Faculty of Sciences Ben M'Sik, Hassan II University Mohammedia, Casablanca, Morocco

Correspondence should be addressed to O. Balatif; balatif.maths@gmail.com

Received 18 March 2020; Accepted 27 May 2020; Published 22 June 2020

Caption: FIGURE 1: Stability of the free equilibrium [E.sup.0] for different values of [alpha].

Caption: FIGURE 2: Stability of [E.sup.0] for different initial values for each variable of state.

Caption: FIGURE 3: Stability of the abstaining equilibrium [E.sup.*] for different values of [alpha].

Caption: FIGURE 4: Stability of [E.sup.*] for different initial values for each variable of state.
```Table 1: Parameters description.

Parameter                           Definition

[LAMBDA]                         Recruitment rate
[beta]                        Effective contact rate
[delta]        Registration rate of the potential electors on the
electoral lists
[gamma]     Registration rate of the abstainers on the electoral lists
[mu]                          Natural mortality rate
```