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

Authors: O. Balatif; L. Boujallal; A. Labzai; M. Rachik
Journal: International Journal of Differential Equations
 (2020)

---

## 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 derivativeα.


---

## Body

## 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:(1)DαP=Λ−βAPN−δ+μP,DαA=βAPN−γ+μA,DαR=δP+γA−μR,where DαP, DαA, and DαR are the derivatives of Pt, At, and Rt, 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]. P0≥0, A0≥0, and R0≥0 are the given initial states.We note that all parameters are nonnegative. They are defined in Table1.Table 1
Parameters description.
ParameterDefinitionΛRecruitment rateβEffective contact rateδRegistration rate of the potential electors on the electoral listsγRegistration rate of the abstainers on the electoral listsμNatural mortality rateThere 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 Section2, 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 orderα>0 of a continuous function f:ℝ+⟶ℝ is given by(2)Dαft=1Γn−α∫0tt−xn−α−1fnxdx,where D=d/dt, n−1≺α≺¯n,n∈ℕ, and Γ. is the gamma function.In particular, when0≺α≺¯1, we have(3)Dαft=1Γ1−α∫0tf′xt−xαdx.(2)
The Laplace transform of the Caputo fractional derivative is given by(4)ℒDαft=λαFλ−∑k=0n−1fk0λα−k−1,with Fλ the Laplace transform of ft.(3)
Letα,β>0. The Mittag-Leffler function Eα,β of parameters α and β is defined as follows:(5)Eα,βz=∑k=0∞zkΓαk+β.(4)
The Laplace transform of the Mittag-Leffler functions is(6)ℒtβ−1Eα,β±atα=λα−βλα∓a.(5)
Letα,β>0 and z∈ℂ, then the Mittag-Leffler function satisfies the equality given by(7)Eα,βz=zEα,α+βz+1Γβ.(6)
Letf:ℝn⟶ℝn with n≥1. Consider the following fractional-order system:(8)Dαxt=fx,x0=x0,with 0≺α≺¯1, 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 thatf satisfies the following conditions:(i)
fx and ∂f/∂xx are continuous for all x∈ℝn(ii)
fx≤ω+λx 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 regionΩ defined by(9)Ω=Pt,At,Rt∈ℝ+4,Pt+At+Rt=Nt≤N0+Λμ,with initial conditions P0≥0, A0≥0, and R0≥0, is a positive invariant for system (1), where N0 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(10)DαNt≤Λ−μNt.
Applying the Laplace transform in the previous inequality, we obtain(11)λαℒNt−λα−1N0≤Λλ−μℒNt,then(12)ℒNt≤λα−1λα+μN0+λα−α+1λα+μΛ,from (6), we deduce(13)Nt≤Eα,1−μtαN0+tαEα,α+1−μtαΛ,using (7), we have(14)Nt≤Eα,1−μtαN0+Λμ1−Eα,1−μtα,since(15)0≤Eα,1−μtα≤1,therefore,(16)Nt≤N0+Λμ.
It implies that the regionΩ is a positive invariant set for system (1).Theorem 3.
The fractional-order initial value problem (1) has a unique solution.Proof.
Let(17)Xt=PART,DαXt=FXt,where F is the right side of system (1).
Firstly, it is easy to see thatFX and ∂F/∂XX satisfy the first condition of Lemma 1.
Secondly, system (1) can be rewritten as follows:(18)FXt=Λ¯+AM1+M2Xt,where(19)Λ¯=Λ00,M1=−δ+μ000−γ+μ0δγ−μ,M2=−βN00βN00000.
Since(20)FXt≤Λ¯+AM1+M2Xt,≤Λ¯cM1+M2Xt,≤ω+λXt,where(21)ω=Λ¯,λ=cM1+M2.
Then, system (1) has a unique solution on 0,+∞.Theorem 4.
IfP0≥0, A0≥0, and R0≥0, then the solutions Pt, At, and Rt of system (1) are positive for all t≥0.Proof.
It follows from the first equation of system (1) that(22)DαP≥−βAtN+δ+μPt.
SinceAt is bounded by a constant Λ0, we have(23)DαP≥−cPt,where c is a constant equal to βΛ0/N+δ+μ.
Applying the Laplace transform in the previous inequality, we obtain(24)λαℒPt−λα−1P0≥−cℒPt,so(25)ℒPt≥λα−1λα+cP0,from (6), we deduce(26)ℒPt≥ℒEα,1−ctαP0,then(27)Pt≥Eα,1−ctαP0.
SinceEα,1−ctα≥0, therefore the solution Pt is positive.
Similarly, from the second and third equations of (1), we can easily prove that At and Rt 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 byE0=Λ/δ+μ,0,αΛ/μδ+μ.This equilibrium corresponds to the case when there are no abstainers in the population.(ii)
Abstaining equilibrium point, ifℜ0≻1, given by E∗=P∗,A∗,R∗, where P∗=Λγ+μ/μβ, A∗=Λδ+μℜ0−1/μβ, and R∗=δΛ2γ+μ2+γΛδ+μγ+μℜ0−1/μβγ+μ.This equilibrium corresponds to the case when the behavior of abstaining the registration on the electoral lists is able to invade the population.Whereℜ0 is the basic reproduction number given by(28)ℜ0=μβδ+μγ+μ.In epidemiology, the basic reproduction numberℜ0 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 equilibriumE0 is locally asymptotically stable if ℜ0<1, whereas E0 is unstable if ℜ0>1.Proof.
The Jacobian matrix atE0 is given by(29)JE0=−δ+μ−μβδ+μ00μβδ+μ−γ+μ0δγ−μ.
Therefore, eigenvalues of the characteristic equation ofJE0 are(30)λ1=−μ,λ2=−δ+μ,λ3=γ+μℜ0−1.
Therefore, all the eigenvalues of the characteristic equation are negative ifℜ0<1. Thus, argλi=π≻απ/2 for i=1,2,3.
Hence, the equilibrium pointE0 is locally asymptotically stable if ℜ0<1 and unstable if ℜ0>1.
Now, we study the local stability of the abstaining equilibriumE∗. The Jacobian matrix at E∗ is given by(31)JE∗=−βA∗N∗−δ+μ−βP∗N∗0βA∗N∗βP∗N∗−γ+μ0δγ−μ.
Its characteristic equation is(32)λ3+a1λ2+a2λ+a3=0,where(33)a1=μβδ+μ+μ,a2=μ2βδ+μ+δ+μγ+μℜ0−1,a3=μ3βδ+μ+μδ+μγ+μℜ0−1,a1a2−a3=μ3β2δ+μ2+μβγ+μℜ0−1.
Ifℜ0≻1, then a1>0, a2>0, a3>0, and a1a2>a3. So, the Routh–Hurwitz conditions are satisfied. Let DQ denote the discriminant of the polynomial Qλ given by (32), then(34)DQ=−1a1a2a3001a1a2a332a1a200032a1a200032a1a2=4a13a3−a12a22−18a1a2a3+4a23+27a32.
From [25], we have the following theorem.Theorem 6.
We assume thatℜ0≻1:(1)
IfDQ≻0 and 0≺α≺¯1, then E∗ is locally asymptotically stable(2)
IfDQ≺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 equilibriumE0 and abstaining equilibrium E∗ of model (1), respectively, using the results in [26–29].Theorem 7.
Ifℜ0≤1, then the free equilibrium E0 of system (1) is globally asymptotically stable on Ω.Proof.
To prove the global stability of the free equilibriumE0, we construct the following Lyapunov function V:Ω⟶R:(35)VP,A=12P−P0+A2+Λμβδ+γ+2μA.
Then, the time derivative ofV is(36)DαVP,A=−δ+μP−P02−γ+μA2−Λμβδ+γ+2μγ+μA1−ℜ0.
Thus,DαVP,A≤0 for ℜ0≤1.
In addition, ifℜ0≤1, then DαVP,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.
Ifℜ0≻1, then the abstaining equilibrium E∗ of the system is globally asymptotically stable on Ω.Proof.
For the global stability of the abstaining equilibriumE∗, we construct the Lyapunov function V:Ω⟶ℝ given by(37)VP,A=P−P∗lnPP∗+A−A∗lnAA∗.
Then, the time derivative of the Lyapunov function is given by(38)DαVP,A=P∗DαPP∗−lnPP∗+A∗DαAA∗−lnAA∗≤P∗1−P∗PDαPP∗+A∗1−A∗ADαAA∗≤1−P∗PDαP+1−A∗ADαA.
Using (1) and the expressions of the coordinates of the equilibrium point E∗, we get(39)DαVP,A≤−ΛP−P∗2PP∗≤0.
Furthermore, it is clear that the largest invariant set ofP,A∈Ω:DαVP,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.


## 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 equilibriumE0 is locally asymptotically stable if ℜ0<1, whereas E0 is unstable if ℜ0>1.Proof.
The Jacobian matrix atE0 is given by(29)JE0=−δ+μ−μβδ+μ00μβδ+μ−γ+μ0δγ−μ.
Therefore, eigenvalues of the characteristic equation ofJE0 are(30)λ1=−μ,λ2=−δ+μ,λ3=γ+μℜ0−1.
Therefore, all the eigenvalues of the characteristic equation are negative ifℜ0<1. Thus, argλi=π≻απ/2 for i=1,2,3.
Hence, the equilibrium pointE0 is locally asymptotically stable if ℜ0<1 and unstable if ℜ0>1.
Now, we study the local stability of the abstaining equilibriumE∗. The Jacobian matrix at E∗ is given by(31)JE∗=−βA∗N∗−δ+μ−βP∗N∗0βA∗N∗βP∗N∗−γ+μ0δγ−μ.
Its characteristic equation is(32)λ3+a1λ2+a2λ+a3=0,where(33)a1=μβδ+μ+μ,a2=μ2βδ+μ+δ+μγ+μℜ0−1,a3=μ3βδ+μ+μδ+μγ+μℜ0−1,a1a2−a3=μ3β2δ+μ2+μβγ+μℜ0−1.
Ifℜ0≻1, then a1>0, a2>0, a3>0, and a1a2>a3. So, the Routh–Hurwitz conditions are satisfied. Let DQ denote the discriminant of the polynomial Qλ given by (32), then(34)DQ=−1a1a2a3001a1a2a332a1a200032a1a200032a1a2=4a13a3−a12a22−18a1a2a3+4a23+27a32.
From [25], we have the following theorem.Theorem 6.
We assume thatℜ0≻1:(1)
IfDQ≻0 and 0≺α≺¯1, then E∗ is locally asymptotically stable(2)
IfDQ≺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 equilibriumE0 and abstaining equilibrium E∗ of model (1), respectively, using the results in [26–29].Theorem 7.
Ifℜ0≤1, then the free equilibrium E0 of system (1) is globally asymptotically stable on Ω.Proof.
To prove the global stability of the free equilibriumE0, we construct the following Lyapunov function V:Ω⟶R:(35)VP,A=12P−P0+A2+Λμβδ+γ+2μA.
Then, the time derivative ofV is(36)DαVP,A=−δ+μP−P02−γ+μA2−Λμβδ+γ+2μγ+μA1−ℜ0.
Thus,DαVP,A≤0 for ℜ0≤1.
In addition, ifℜ0≤1, then DαVP,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.
Ifℜ0≻1, then the abstaining equilibrium E∗ of the system is globally asymptotically stable on Ω.Proof.
For the global stability of the abstaining equilibriumE∗, we construct the Lyapunov function V:Ω⟶ℝ given by(37)VP,A=P−P∗lnPP∗+A−A∗lnAA∗.
Then, the time derivative of the Lyapunov function is given by(38)DαVP,A=P∗DαPP∗−lnPP∗+A∗DαAA∗−lnAA∗≤P∗1−P∗PDαPP∗+A∗1−A∗ADαAA∗≤1−P∗PDαP+1−A∗ADαA.
Using (1) and the expressions of the coordinates of the equilibrium point E∗, we get(39)DαVP,A≤−ΛP−P∗2PP∗≤0.
Furthermore, it is clear that the largest invariant set ofP,A∈Ω:DαVP,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,tf=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).Figure 1
Stability of the free equilibriumE0 for different values of α.
(a)(b)(c)For a fixed value ofα 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.Figure 2
Stability ofE0 for different initial values for each variable of state.
(a)(b)(c)Concerning the stability analysis of the abstaining equilibriumE∗, 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,andtf=90. In this case, according to theorem (9), the abstaining equilibrium E∗ of system (1) is globally asymptotically stable on Ω (see Figure 3).Figure 3
Stability of the abstaining equilibriumE∗ for different values of α.
(a)(b)(c)In addition, we show in Figure4 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.Figure 4
Stability ofE∗ for different initial values for each variable of state.
(a)(b)(c)From all these figures, we show that the equilibrium pointsE0 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.



---

*Source: 4325640-2020-06-22.xml*