Abstract
We propose a mathematical model that describes the dynamics of citizens who have the right to register on the electoral lists and participate in the political process and the negative influence of abstainers, who abstain from registration on the electoral lists, on the potential electors. By using Routh–Hurwitz criteria and constructing Lyapunov functions, the local stability and the global stability of abstaining-free equilibrium and abstaining equilibrium are obtained. We also study the sensitivity analysis of the model parameters to know the parameters that have a high impact on the reproduction number . In addition, we propose an optimal strategy for an awareness program that helps politicians and officials to increase the rate of citizens registered on the electoral lists with an optimal effort. Pontryagin’s maximum principle is used to characterize the optimal controls, and the optimality system is solved by an iterative method. Finally, some numerical simulations are performed to verify the theoretical analysis using Matlab.
1. Introduction
Participation in political life is one of the most important indicators of democracy in a country’s political system. Political participation takes different forms: joining political parties, participating in elections as well as the rate of registration on the electoral lists, which all aim at contributing to the different elections for electing people’s representatives in the different political positions.
The higher rate of registered citizens on the electoral lists means that the majority of people are willing to participate in the electoral process. It also indicates that a large group of people is convinced of the possibility of change and reform through political participation. But, when these rates are low, it often means there are some administrative or cultural impediments that make registration on the electoral lists difficult. These low rates can also denote the existence of a behavior of political boycotting and nonregistration on the electoral lists due to electoral and political corruption or the citizens’ lack of confidence in the political class.
Thus, countries seeking to improve their image and political position are in urgent need of improving many democratic indicators including increasing the rate of registration on the electoral lists. For example, many Arab and African countries suffer from a decline in the registration rate on the electoral lists (see Table 1 (data from The International Institute for Democracy and Electoral Assistance. https://www.idea.int/data-tools/2019) below).
According to these data and other data, these countries and others need to conduct more studies and research using a variety of scientific approaches for understanding the reasons and predicting the evolvement of these rates in the future as well as formulating strategies and programs for improving registration rates on the electoral lists.
Many studies and research in social sciences have focused on this topic and other related topics ([1–6] and the references cited therein). But, the mathematical studies and research on this topic are still limited and most of them have focused on the statistical aspect of the phenomenon ([7–10]).
In this work, we propose an epidemiological approach (see [11–14]) to describe and study the boycotting behavior to registration on the electoral lists. In epidemiology, we generally use compartment model to describe the spread of an infectious disease. In these epidemiological models, the population is divided into different classes according to people’s status versus the disease (susceptible to catch the disease, infected, or removed) and the infection process depends on the contact with infectious individuals. Similarly, during the election process, the population can be divided into several classes (potential electors, voters for the opposition parties, voters for the majority parties, etc). Furthermore, the interaction between people is a key factor in the electoral process; it is very similar to the contagion phenomenon since abstainers can affect the potential registrants in their network (family, friends, and coworkers) to abstain from the electoral process. For this reason, the epidemic approach is more appropriate to model the boycotting behavior in an election process.
We propose a mathematical model that describes the dynamics of citizens who have the right to register on the electoral lists and the negative influence of abstainers who abstain from registration on the electoral lists and the electoral process. By using Routh–Hurwitz criteria and constructing Lyapunov functions, the local stability and the global stability of abstaining-free equilibrium and abstaining equilibrium are obtained. Since there are usually errors in data collection and assumed parameter values, we also study the sensitivity analysis of the model parameters to know the parameters that have a high impact on the reproduction number . In addition, we propose an optimal strategy for an awareness program that helps politicians and officials to increase the rate of the registered citizens on the electoral lists with an optimal effort. To achieve this objective, we use some theoretical results of optimal control theory.
The paper is organized as follows. In Section 1, we present the proposed mathematical model and we give some basic properties of the model. In Section 2, we analyze the local stability and global stability and the problem of parameters’ sensitivity and some numerical simulations. In Section 3, we present the optimal control problem for the proposed model where we give some results concerning the existence of the optimal controls and we characterize these optimal controls using Pontryagin’s maximum principle. Also, in this section, numerical simulations are given. Finally, we conclude the paper in Section 4.
1.1. Mathematical Model and Basic Properties
1.1.1. Mathematical Model
We consider a mathematical model that describes the dynamics of citizens who have the right to register on the electoral lists and the negative influence of abstainers, who abstain from registration on the electoral lists and the electoral process, on the potential electors.
We divide the population denoted by N into three compartments: The potential electors are entitled to participate in the elections and they are not yet registered on the electoral lists. The class of potential electors is increased by the recruitment of individuals into the compartment P at a rate and it is decreased when potential electors register on the electoral lists at rate α. It is assumed that potential electors can acquire abstainer behavior (and become the abstainers of the elections) via effective contact with abstainers of the elections at a rate β. In other words, it is assumed that the acquisition of an abstainer behavior is analogous to acquiring disease infection. Finally, potential electors suffer natural death (at a rate μ). The abstainers have a position of abstaining from the elections and the registration on the electoral lists. The population of abstainers is increased when the potential electors abstain to register on the electoral lists via effective contact with abstainers at a rate β. It is decreased by natural death (at the rate μ) and when some abstainers withdraw their position of boycotting the registration on the electoral lists and turn to participate in the electoral process then become registrants at a rate . The registered are listed on the electoral lists and wish to vote in the elections. The compartment of the registered is increased at a rate α when potential electors register on the electoral lists and when the abstainers become registered individuals at a rate γ. This compartment is decreased by natural death (at the rate μ).
The variables , , and are the numbers of the individuals in the three classes at time t, respectively. The unit of time can correspond to periods, years, months, or days; it depends on the frequency of the survey studies as needed.
The graphical representation of the proposed model is shown in Figure 1.
The total population size at time t is denoted by with The dynamics of this model are governed by the following nonlinear system of differential equations:where , and are the given initial states.
1.2. Basic Properties
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 theorem and lemma.
1.2.1. Positivity of the Model’s Solutions
Theorem 1. If , , and , the solutions , and of system (1) are positive for all .
Proof. It follows from the first equation of system (1) thatMultiplying the inequality (2) by we haveSo,Integrating (4) givesSo, the solution is positive.
Similarly, from the second and third equations of (1), we haveandTherefore, we can see that the solutions , and of system (1) are positive for all .
1.2.2. Invariant Region
Lemma 1. The feasible region defined bywith initial conditions, , and are positive invariants for system (1).
Proof. By adding the equations of system (1), we obtainSo,By using a Gronwall lemma, we havewhere represents the initial values of the total population. Thus, . It implies that the region is a positively invariant set for system (1). So, we only need to consider the dynamics of the system on the set .
2. Stability Analysis and Sensitivity of the Model Parameters
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 the following 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 from the registration on the electoral lists is able to invade the population.
Here, 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 the compartment of abstainers. We obtained this number by using the next-generation matrix method formulated in [15, 16].
Indeed, letting , then system (1) can be written aswhere
The Jacobian matrices of and at the free equilibrium are, respectively,where
Finally, we have
2.1. Local Stability Analysis
In this section, we analyze the local stability of the abstaining-free equilibrium and the abstaining equilibrium.
Theorem 2. The abstaining-free equilibrium is locally asymptotically stable if , whereas is unstable if .
Proof. The Jacobian matrix at is given byTherefore, Eigen values of the characteristic equation of areTherefore, all the Eigen values of the characteristic equation are negative if . Hence, the equilibrium point is locally asymptotically stable if and unstable if .
Theorem 3. If , the abstaining equilibrium is locally asymptotically stable.
Proof. The Jacobian matrix at is given byand its characteristic equation iswhereBy Routh-Hurwitz Criterion, system (1) is locally asymptotically stable if , , and .
Thus, is locally asymptotically stable if .
2.2. Global Stability Analysis
Now, we are concerned with the global asymptotic stability of abstaining-free equilibrium and abstaining equilibrium of model (1), respectively.
Theorem 4. 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 V: Then, the time derivative of V isSince (24) becomesThus, for
In addition,
If , then and
Hence, by LaSalle’s invariance principle [17], the free equilibrium point is globally asymptotically stable on .
Theorem 5. 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 bywhere and are positive constants to be chosen latter.
Then, the time derivative of the Lyapunov function is given byFor , we haveAlso, we obtainHence, by LaSalle’s invariance principle [17], the free equilibrium point is globally asymptotically stable on .
2.3. Numerical Simulation
In this section, we present some numerical simulations of system (1) to illustrate our results. By choosing , and different initial values for each variable of state, we have the abstaining-free equilibrium and In this case, according to theorem (5), the abstaining-free equilibrium of system (1) is globally asymptotically stable on (see Figure 2).
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Also, for , we have the abstaining equilibrium and In this case, according to theorem (6), the abstaining equilibrium of system (1) is globally asymptotically stable on (see Figure 3).
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2.4. Sensitivity Analysis of Model the Parameters
Sensitivity analysis is commonly used to determine the model robustness to parameter values, that is, to help us know the parameters that have a high impact on the reproduction number (because there are usually errors in data collection and assumed parameter values). A sensitivity analysis of the model (1) is carried out in the sense of [18, 19].
Definition 1. The normalized forward sensitivity index of a variable that depends differentiably on a parameter s is defined asIn particular, sensitivity indices of the basic reproduction number with respect to the model parameters are computed as follows:The positive sign of sensitivity index (S.I) of the basic reproduction number to the model parameters shows that an increase (or decrease) in the value of each of the parameters in this case will lead to an increase (or decrease) in the basic reproduction number of the disease. For example, suggests that increasing (or decreasing) the effective contact rate β by 10% increases (or decreases) the basic reproduction number, , by 10%. On the other hand, the negative sign of S.I of the basic reproduction number to the model parameters implies that an increase (or decrease) in the value of each of the parameters in this case leads to a corresponding decrease (or increase) in the basic reproduction number of the disease. For example, means that increasing (or decreasing) the coefficient α, by 10%, decreases (or increases) the by . In Table 2, we present the sensitivity indices of all model parameters to . The parameters are arranged from the most sensitive to the least sensitive.
Hence, with sensitivity analysis, one can get insight into the appropriate intervention strategies to prevent and control the spread of the abstention behavior of the registration on the electoral lists described by model (1).
3. The Optimal Control Problem
3.1. Problem Statement
Any country seeking to improve its image and political position is in an urgent need of improving many democratic indicators including increasing the registration rate on the electoral lists. To achieve this objective, it must develop some optimal strategies for an awareness program that helps officials of that country to increase the rate of citizens registered on the electoral lists with an optimal effort, which are always time, money, and human resources.
So, our objective in this proposed strategy of control is to minimize the number of abstainers and maximize the number of registrants during the time interval and also to minimize the cost spent in an awareness program.
In the model (1), we include two controls and for : the control represents the awareness campaign effort and administrative and legal facilities effort to motivate the potential electors to register on the electoral lists. The control represents efforts, i.e., dialogues, debates, and media coverage that are made to influence the abstainers to participate in the electoral process. So, the controlled mathematical system is given by the following system of differential equations:where , and are the given initial states.
Then, the problem is to minimize the objective functional.where the parameters M and N are the strictly positive cost coefficients; they are selected to weigh the relative importance of and at time t; is the final time.
In other words, we seek the optimal controls and such thatwhere is the set of admissible controls defined by
3.2. Existence of Optimal Controls
The existence of the optimal controls can be obtained using a result by Fleming and Rishel [20] (see Corollary 4.1).
Theorem 6. Consider the control problem with system (32). There exists an optimal control such thatIf the following conditions are met:(1)The set of controls and the corresponding state variables is nonempty.(2)The control set is convex and closed.(3)The right-hand side of the state system is bounded by a linear function in the state and control variables.(4)The integrand of the objective functional is convex on and there exist constants and such that:
Proof.
Condition 1. To prove that the set of controls and the corresponding state variables is nonempty, we will use a simplified version of an existence result ([21], Theorem 7.1.1).Let and where , and form the right-hand side of the system of equations (32). Let for for some constants and since all parameters are constants and , and R are continuous, then , and are also continuous. Additionally, the partial derivatives , , , , , , and , , and are all continuous. Therefore, there exists a unique solution () that satisfies the initial conditions. Therefore, the set of controls and the corresponding state variables is nonempty and condition 1 is satisfied.
Condition 2. By definition, is closed. Take any controls and . Then
Additionally, we observe that and then
Hence,Therefore, is convex and condition 2 is satisfied.
Condition 3. From the system of differential equations (32), we haveThen,Therefore, all solutions of the model (32) are bounded. So, there exist positive constants , and such that We consider,So, we can rewrite system (32) in matrix form aswhereIt gives a linear function of control vector and state variable vector. Therefore, we can writewhere and .
Hence, we see the right-hand side is bounded by a sum of state and control vectors. Therefore, condition 3 is satisfied.
Condition 4: The integrand in the objective functional (33) is convex on . It rests to show that there exist constants and such that the integrand of the objective functional satisfiesThe state variables being bounded, let , and , then it follows that
3.3. Characterization of the Optimal Controls
In this section, we apply the Pontryagin’s maximum principle [9]. The key idea is introducing the adjoint function to attach the system of differential equations to the objective functional resulting in the formation of a function called the Hamiltonian. This principle converts the problem of finding the control to optimize the objective functional subject to the state of differential equations with initial condition to find the control to optimize Hamiltonian pointwise (with respect to the control).
Now, we have the Hamiltonian H in time t, defined bywhere is the right side of the system of differential equations (32) of the state variable.
Theorem 7. Given an optimal control and solutions , , and of the corresponding state system (32), there exist adjoint functions , , and satisfyingwith the transversality conditions at time
Furthermore, for , the optimal controls and are given by
Proof. The Hamiltonian in time t is given byFor , the adjoint equations and transversality conditions can be obtained by using Pontryagin’s maximum principle given in [22] such thatFor , the optimal controls and can be solved from the optimality conditionthat is,By the bounds in of the controls, it is easy to obtain and in the form of ((51) and (52)).
3.4. Numerical Simulation
In this section, we present the results obtained by numerically solving the optimality system. In our control problem, we have initial conditions for the state variables and terminal conditions for the adjoints. That is, the optimality system is a two-point boundary value problem with separated boundary conditions at times step and . We solve the optimality system by an iterative method with forward solving of the state system followed by backward solving of the adjoint system. We start with an initial guess for the controls at the first iteration and then before the next iteration, we update the controls by using the characterization. We continue until convergence of successive iterates is achieved. A code is written and compiled in Matlab using the following data: , and the initial values , , and .
The proposed control strategy in this work helps to improve an important political indicator of a democratic election that is the rate of registration on the electoral lists: Case 1: we use only the optimal control In this strategy, we focus the effort of the awareness campaign on potential electors in order to sensitize them of the importance of political and electoral participation and to protect them from the negative impact of abstainers. From these figures, we observe that the number of registered individuals on the electoral lists has increased from to at the end of this awareness campaign (Figure 4(a)). Also, the number of abstainers has changed, such that the number of abstainers has decreased from to at the end of the campaign (Figure 4(b)). Consequently, we observe that the rate of registration abstinence on the electoral lists has decreased by (Figure 5(a)). On the other hand, in (Figure 5(b)), we show that the registration rate on the lists is increased by . These changes are important but not sufficient; it is for this reason that we must also target by this strategy of control the abstainers. Case 2: we combine the optimal control and In this strategy, the two optimal controls and are activated at the same time in order to improve the numerical results of case 1. The optimal control represents the effort to convince the abstainers to change their opinion and reduce their negative effect on the potential electors. From Figure 6(a), we can see that the number of registered individuals increases significantly from to . Also, Figure 6(b) demonstrates that the number of abstainers decreased from to at the end of this political campaign. Therefore, we observe that the rate of registration abstinence on the electoral lists has decreased by (Figure 7(a)) and the registration rate on the lists is increased by (Figure 7(b)). Finally, we conclude that the proposed strategy becomes more effective when we combined two optimal controls and .
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4. Conclusion
In this work, we formulated a mathematical model that describes the dynamics of citizens who have the right to register on the electoral lists and the negative influence of abstainers, who abstain from registration on the electoral lists and the electoral process, on potential electors. By using Routh–Hurwitz criteria and constructing Lyapunov functions, the local stability and the global stability of abstaining-free equilibrium and abstaining equilibrium are obtained. We also studied the sensitivity analysis of the model parameters to know the parameters that have a high impact on the reproduction number . In addition, we proposed an optimal strategy for an awareness program that helps politicians and officials to increase the rate of citizens registered on the electoral lists with an optimal effort. In this work, we introduced two controls: the first control represented the awareness campaign effort and administrative and legal facilities effort to motivate the potential electors to register on the electoral lists. The second control measured the efforts directed to dialogues, debates, and media coverage that are made to make an impact on the abstainers to participate in the electoral process. Pontryagin’s maximum principle is used to characterize the optimal controls and the optimality system is solved by an iterative method.
The numerical simulation was carried out using Matlab. We proposed an algorithm based on the forward and backward difference approximation and we show that the optimal strategy becomes more effective when we combined two optimal controls.
Data Availability
The disciplinary data used to support the findings of this study have been deposited in the Network Repository (http://www.networkrepository.com).
Conflicts of Interest
The authors declare that there is no conflict of interest regarding the publication of this paper.