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 .
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 is divided into three compartments: the potential electors are entitled to participate in the elections, and they are not yet registered on the electoral lists; the abstainers who have an attitude of abstaining the elections and the registration on the electoral lists; and the registered who are listed on the electoral lists and wish to vote in the elections:where , , and are the derivatives of , , and , respectively, of an arbitrary order (where ) in the sense of Caputo. Note that when , the fractional-order model (1) represents the classical model studied in [15]. , , and 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 of a continuous function is given bywhere , , and is the gamma function.
In particular, when , we have(2)The Laplace transform of the Caputo fractional derivative is given bywith the Laplace transform of .(3)Let . The Mittag-Leffler function of parameters and is defined as follows:(4)The Laplace transform of the Mittag-Leffler functions is(5)Let and , then the Mittag-Leffler function satisfies the equality given by(6)Let with . Consider the following fractional-order system:with , , and . For the global existence of a solution of system (9), we need the following lemma.
Lemma 1 (see [22]). Assume that satisfies the following conditions:(i) and are continuous for all (ii) for all , where and are the two positive constantsThen, system (9) has a unique solution on .
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 and are bounded. This will be established by the following lemma and theorems.
Lemma 2. The feasible region defined bywith initial conditions , , and , is a positive invariant for system (1), where 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 byApplying the Laplace transform in the previous inequality, we obtainthenfrom (6), we deduceusing (7), we havesincetherefore,It implies that the region is a positive invariant set for system (1).
Theorem 3. The fractional-order initial value problem (1) has a unique solution.
Proof. Letwhere is the right side of system (1).
Firstly, it is easy to see that and satisfy the first condition of Lemma 1.
Secondly, system (1) can be rewritten as follows:whereSincewhereThen, system (1) has a unique solution on .
Theorem 4. If , , and , then the solutions , , and of system (1) are positive for all .
Proof. It follows from the first equation of system (1) thatSince is bounded by a constant , we havewhere is a constant equal to .
Applying the Laplace transform in the previous inequality, we obtainsofrom (6), we deducethenSince , therefore the solution is positive.
Similarly, from the second and third equations of (1), we can easily prove that and are positive for all .
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 .
This equilibrium corresponds to the case when there are no abstainers in the population.(ii)Abstaining equilibrium point, if , given by , where , , and .
This equilibrium corresponds to the case when the behavior of abstaining the registration on the electoral lists is able to invade the population.
Where is the basic reproduction number given by
In epidemiology, the basic reproduction number 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 is locally asymptotically stable if , whereas is unstable if .
Proof. The Jacobian matrix at is given byTherefore, eigenvalues of the characteristic equation of areTherefore, all the eigenvalues of the characteristic equation are negative if . Thus, for .
Hence, the equilibrium point is locally asymptotically stable if and unstable if .
Now, we study the local stability of the abstaining equilibrium . The Jacobian matrix at is given byIts characteristic equation iswhereIf , then , , , and . So, the Routh–Hurwitz conditions are satisfied. Let denote the discriminant of the polynomial given by (32), thenFrom [25], we have the following theorem.
Theorem 6. We assume that :(1)If and , then is locally asymptotically stable(2)If and , then is locally asymptotically stable
4.2. Global Stability Analysis
Now, we are concerned with the global asymptotic stability of abstaining-free equilibrium and abstaining equilibrium of model (1), respectively, using the results in [26–29].
Theorem 7. If , then the free equilibrium of system (1) is globally asymptotically stable on .
Proof. To prove the global stability of the free equilibrium , we construct the following Lyapunov function :Then, the time derivative of isThus, for .
In addition, if , then and .
Hence, by LaSalle’s invariance principle [30, 31], the free equilibrium point is globally asymptotically stable on .
Theorem 8. If , then the abstaining equilibrium of the system is globally asymptotically stable on .
Proof. For the global stability of the abstaining equilibrium , we construct the Lyapunov function given byThen, the time derivative of the Lyapunov function is given byUsing (1) and the expressions of the coordinates of the equilibrium point , we getFurthermore, it is clear that the largest invariant set of is the singleton . Hence, by LaSalle’s invariance principle [30, 31], the abstaining equilibrium point is globally asymptotically stable on .
5. Numerical Simulation
In this section, we present some numerical simulations of system (1) to illustrate our results. By choosing , , , , and different values of , we have the abstaining-free equilibrium and . In this case, according to theorem (7), the abstaining-free equilibrium of system (1) is globally asymptotically stable on (see Figure 1).
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For a fixed value of and for different initial values for each variable of state, we show in Figure 2 that the solutions converge to the abstaining-free equilibrium when , which implies that the abstaining-free equilibrium of system (1) is globally asymptotically stable on .
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Concerning the stability analysis of the abstaining equilibrium , we have and for , , , . In this case, according to theorem (9), the abstaining equilibrium of system (1) is globally asymptotically stable on (see Figure 3).
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In addition, we show in Figure 4 that for a fixed value of and for different initial values for each variable of state, the solutions converge to the abstaining equilibrium and , which implies that the abstaining equilibrium of system (1) is globally asymptotically stable on .
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From all these figures, we show that the equilibrium points and of system (1) are globally asymptotically stable on if the conditions of theorems (7) and (9) are satisfied. Also, all solutions of model (1) converge to the equilibrium points and for different values of . In addition, the solutions converge rapidly to their steady state when the value of 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.
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).
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.