# Mathematical Modeling, Analysis, and Optimal Control of Abstinence Behavior of Registration on the Electoral Lists

Authors: Omar Balatif; Bouchaib Khajji; Mostafa Rachik
Journal: Discrete Dynamics in Nature and Society
 (2020)

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## 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 numberℜ0. 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.


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## Body

## 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 Table1 (data from The International Institute for Democracy and Electoral Assistance. https://www.idea.int/data-tools/2019) below).Table 1
Country
Election type
Year
Registration rate with respect to voting age (%)
Registration rate with respect to the total population (%)
Bahrain
Parliamentary
2018
34.12
25.33
Burkina Faso
Parliamentary
2015
60.63
29.14
Cote d’Ivoire
Parliamentary
2016
47.37
26.40
Djibouti
Parliamentary
2018
34.76
21.94
Kuwait
Parliamentary
2016
25.41
17.06
Libya
Parliamentary
2014
37.45
24.17
Morocco
Parliamentary
2016
67.89
46.66
Oman
Parliamentary
2015
27.39
15.99
Sudan
Parliamentary
2015
66.74
34.15According 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 numberℜ0. 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 Section1, 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 modelPAR 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 byN 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 class of potential electors is increased by the recruitment of individuals into the compartment P at a rate Λ 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 A 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 γ.The registered R 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 variablesPt, At, and Rt 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 Figure1.Figure 1The total population size at timet is denoted by Nt with Nt=Pt+At+Rt. The dynamics of this model are governed by the following nonlinear system of differential equations:(1)P˙=Λ−βAPN−α+μP,A˙=βAPN−γ+μA,R˙=αP+γA−μR,where P0≥0, A0≥0, and R0≥0 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 t>0 and are bounded. This will be established by the following theorem and lemma.


#### 1.2.1. Positivity of the Model’s Solutions

Theorem 1.
IfP0≥0, A0≥0, and R0≥0, 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(2)dPdt+βAtN+α+μPt≥0.
Multiplying the inequality (2) by exp∫0tβAs/N+α+μds, we have(3)dPdtexp∫0tβAsN+α+μds+βAtN+α+μPtexp∫0tβAsN+α+μds≥0.
So,(4)ddtPtexp∫0tβAsN+α+μds≥0.
Integrating (4) gives(5)Pt≥P0exp∫0t−βAsN−α+μds.
So, the solutionPt is positive.
Similarly, from the second and third equations of (1), we have(6)At≥A0exp−γ+μt≥0,and(7)Rt≥R0exp−μt≥0.
Therefore, we can see that the solutionsPt, At, and Rt of system (1) are positive for all t≥0.


#### 1.2.2. Invariant Region

Lemma 1.
The feasible regionΩ defined by(8)Ω=Pt,Rt,Rt∈ℝ+3,Pt+At+Rt≤Λμ,with initial conditions, P0≥0, A0≥0, and R0≥0 are positive invariants for system (1).Proof.
By adding the equations of system (1), we obtain(9)dNdt≤Λ−α+μN.So,(10)Nt≤N0+Λt+∫0t−μNsds.
By using a Gronwall lemma, we have(11)Nt≤N0exp−μt−Λμ1−exp−μt,where N0 represents the initial values of the total population. Thus, limt⟶∞supNt=Λ/μ. It implies that the region Ω is a positively invariant set for system (1). So, we only need to consider the dynamics of the system on the set Ω.


## 1.1. Mathematical Model and Basic Properties

### 1.1.1. Mathematical Model

We consider a mathematical modelPAR 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 byN 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 class of potential electors is increased by the recruitment of individuals into the compartment P at a rate Λ 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 A 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 γ.The registered R 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 variablesPt, At, and Rt 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 Figure1.Figure 1The total population size at timet is denoted by Nt with Nt=Pt+At+Rt. The dynamics of this model are governed by the following nonlinear system of differential equations:(1)P˙=Λ−βAPN−α+μP,A˙=βAPN−γ+μA,R˙=αP+γA−μR,where P0≥0, A0≥0, and R0≥0 are the given initial states.


## 1.1.1. Mathematical Model

We consider a mathematical modelPAR 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 byN 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 class of potential electors is increased by the recruitment of individuals into the compartment P at a rate Λ 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 A 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 γ.The registered R 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 variablesPt, At, and Rt 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 Figure1.Figure 1The total population size at timet is denoted by Nt with Nt=Pt+At+Rt. The dynamics of this model are governed by the following nonlinear system of differential equations:(1)P˙=Λ−βAPN−α+μP,A˙=βAPN−γ+μA,R˙=αP+γA−μR,where P0≥0, A0≥0, and R0≥0 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 t>0 and are bounded. This will be established by the following theorem and lemma.


### 1.2.1. Positivity of the Model’s Solutions

Theorem 1.
IfP0≥0, A0≥0, and R0≥0, 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(2)dPdt+βAtN+α+μPt≥0.
Multiplying the inequality (2) by exp∫0tβAs/N+α+μds, we have(3)dPdtexp∫0tβAsN+α+μds+βAtN+α+μPtexp∫0tβAsN+α+μds≥0.
So,(4)ddtPtexp∫0tβAsN+α+μds≥0.
Integrating (4) gives(5)Pt≥P0exp∫0t−βAsN−α+μds.
So, the solutionPt is positive.
Similarly, from the second and third equations of (1), we have(6)At≥A0exp−γ+μt≥0,and(7)Rt≥R0exp−μt≥0.
Therefore, we can see that the solutionsPt, At, and Rt of system (1) are positive for all t≥0.


### 1.2.2. Invariant Region

Lemma 1.
The feasible regionΩ defined by(8)Ω=Pt,Rt,Rt∈ℝ+3,Pt+At+Rt≤Λμ,with initial conditions, P0≥0, A0≥0, and R0≥0 are positive invariants for system (1).Proof.
By adding the equations of system (1), we obtain(9)dNdt≤Λ−α+μN.So,(10)Nt≤N0+Λt+∫0t−μNsds.
By using a Gronwall lemma, we have(11)Nt≤N0exp−μt−Λμ1−exp−μt,where N0 represents the initial values of the total population. Thus, limt⟶∞supNt=Λ/μ. It implies that the region Ω is a positively invariant set for system (1). So, we only need to consider the dynamics of the system on the set Ω.


## 1.2.1. Positivity of the Model’s Solutions

Theorem 1.
IfP0≥0, A0≥0, and R0≥0, 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(2)dPdt+βAtN+α+μPt≥0.
Multiplying the inequality (2) by exp∫0tβAs/N+α+μds, we have(3)dPdtexp∫0tβAsN+α+μds+βAtN+α+μPtexp∫0tβAsN+α+μds≥0.
So,(4)ddtPtexp∫0tβAsN+α+μds≥0.
Integrating (4) gives(5)Pt≥P0exp∫0t−βAsN−α+μds.
So, the solutionPt is positive.
Similarly, from the second and third equations of (1), we have(6)At≥A0exp−γ+μt≥0,and(7)Rt≥R0exp−μt≥0.
Therefore, we can see that the solutionsPt, At, and Rt of system (1) are positive for all t≥0.


## 1.2.2. Invariant Region

Lemma 1.
The feasible regionΩ defined by(8)Ω=Pt,Rt,Rt∈ℝ+3,Pt+At+Rt≤Λμ,with initial conditions, P0≥0, A0≥0, and R0≥0 are positive invariants for system (1).Proof.
By adding the equations of system (1), we obtain(9)dNdt≤Λ−α+μN.So,(10)Nt≤N0+Λt+∫0t−μNsds.
By using a Gronwall lemma, we have(11)Nt≤N0exp−μt−Λμ1−exp−μt,where N0 represents the initial values of the total population. Thus, limt⟶∞supNt=Λ/μ. It implies that the region Ω is a positively invariant set for system (1). So, we only need to consider the dynamics of the system on the set Ω.


## 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 byE0=Λ/α+μ,0,αΛ/μα+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∗=Λα+μℜ0−1/μβ, and R∗=αΛ2γ+μ2+γΛα+μγ+μℜ0−1/μβγ+μ.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,ℜ0 is the basic reproduction number given by:(12)ℜ0=μβα+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 the compartment of abstainers. We obtained this number by using the next-generation matrix method formulated in [15, 16].Indeed, lettingx=A,R,P, then system (1) can be written as(13)dxdt=ℱx−ϑx,where(14)ℱx=βAPN00,ϑx=γ+μA−αP−γA+μR−Λ+βAPN+α+μP.The Jacobian matrices ofℱx and ϑx at the free equilibrium E0 are, respectively,(15)DℱE0=F2x200000,DϑE0=V2x20−αμβα+μ00,where(16)F=μβα+μ000,V=γ+μ0−γμ.Finally, we have(17)ℜ0=ρFV−1=μβα+μγ+μ.


### 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 equilibriumE0 is locally asymptotically stable if ℜ0<1, whereas E0 is unstable if ℜ0>1.Proof.
The Jacobian matrix atE0 is given by(18)JE0=−α+μ−μβα+μ00μβα+μ−γ+μ0αγ−μ.
Therefore, Eigen values of the characteristic equation ofJE0 are(19)λ1=−μ,λ2=−α+μ,λ3=γ+μℜ0−1.
Therefore, all the Eigen values of the characteristic equation are negative ifℜ0<1. Hence, the equilibrium point E0 is locally asymptotically stable if ℜ0<1 and unstable if ℜ0>1.Theorem 3.
Ifℜ0≥1, the abstaining equilibrium E∗ is locally asymptotically stable.Proof.
The Jacobian matrix atE∗ is given by(20)JE∗=−βA∗N∗−α+μ−βP∗N∗0βA∗N∗βP∗N∗−γ+μ0αγ−μ,and its characteristic equation is(21)λ3+a1λ2+a2λ+a3=0,where(22)a1=μβα+μ+μ,a2=μ2βα+μ+α+μγ+μℜ0−1,a3=μ3βα+μN+μα+μγ+μℜ0−1.
By Routh-Hurwitz Criterion, system (1) is locally asymptotically stable if a1>0, a3>0, and a1a2>a3.
Thus,E∗ is locally asymptotically stable if ℜ0≥1.


### 2.2. Global Stability Analysis

Now, we are concerned with the global asymptotic stability of abstaining-free equilibriumE0 and abstaining equilibrium E∗ of model (1), respectively.Theorem 4.
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(23)VP,A=12P−P0+A2+α+γ+2μNβA.
Then, the time derivative ofV is(24)V˙P,A=P−P0+AΛ−α+μP−γ+μA+α+γ+2μNβA˙.
SinceΛ=P0α+μ, (24) becomes(25)V˙P,A=P−P0+A−α+μP−P0−γ+μA+α+γ+2μNβA˙=−α+μP−P02−γ+μA2−α+γ+2μγ+μNβ1−ℜ0A.
Thus,V˙P,A≤0 for ℜ0≤1.
In addition,
Ifℜ0≤1, then V˙P,A=0⟺P=P0 and A=0.
Hence, by LaSalle’s invariance principle [17], the free equilibrium point E0 is globally asymptotically stable on Ω.Theorem 5.
Ifℜ0≻1, then the abstaining equilibrium E∗ of the system is globally asymptotically stable on Ω.Proof.
For the global stability of the abstaining equilibriumE0, we construct the Lyapunov function V:Ω⟶R given by(26)VP,A=X1P−P∗lnPP∗+X2A−A∗lnAA∗,where X1 and X2 are positive constants to be chosen latter.
Then, the time derivative of the Lyapunov function is given by(27)V˙P,A=βNX2−X1A−A∗P−P∗−ΛX1P−P∗2PP∗.
ForX1=X2=1, we have(28)V˙P,A=−ΛP−P∗2PP∗≤0.
Also, we obtain(29)V˙P,A=0⟺P=P∗.
Hence, by LaSalle’s invariance principle [17], the free equilibrium point E∗ is globally asymptotically stable on Ω.


### 2.3. Numerical Simulation

In this section, we present some numerical simulations of system (1) to illustrate our results. By choosing Λ=9×105, α=0.01, β=0.1,γ=0.06, μ=0.04,tf=365, and different initial values for each variable of state, we have the abstaining-free equilibrium E0=18×106,0,4.5×105 and ℜ0=0.8≺1. In this case, according to theorem (5), the abstaining-free equilibrium E0 of system (1) is globally asymptotically stable on Ω (see Figure 2).Figure 2
(a)
(b)
(c)Also, forΛ=9×105, α=0.04, β=0.3,γ=0.08, μ=0.04,tf=365, we have the abstaining equilibrium E∗=9×106,1.5×106,1.2×107 and ℜ0=1.25≻1. In this case, according to theorem (6), the abstaining equilibrium E∗ of system (1) is globally asymptotically stable on Ω (see Figure 3).Figure 3
(a)
(b)
(c)


### 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 numberℜ0 (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 variablev that depends differentiably on a parameter s is defined as(30)ϒsv=∂v∂s⋅sv.
In particular, sensitivity indices of the basic reproduction numberℜ0 with respect to the model parameters are computed as follows:(31)ϒβℜ0=∂ℜ0∂β⋅βℜ0=1,ϒαℜ0=∂ℜ0∂α⋅αℜ0=−αα+μ,ϒγℜ0=∂ℜ0∂γ⋅γℜ0=−γγ+μ,ϒμℜ0=∂ℜ0∂μ⋅μℜ0=αγ−μ2α+μγ+μ.
The positive sign of sensitivity index (S.I) of the basic reproduction numberℜ0 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, ϒβℜ0=1 suggests that increasing (or decreasing) the effective contact rate β by 10% increases (or decreases) the basic reproduction number, ℜ0, 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, ϒαℜ0=−0.5 means that increasing (or decreasing) the coefficient α, by 10%, decreases (or increases) the ℜ0 by 5%. In Table 2, we present the sensitivity indices of all model parameters to ℜ0. 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).Table 2
Parameter
Description
Value
Sensitivity index
β
The effective contact rate
0.3
+1.00
α
The registration rate of potential electors
0.04
−0.50
γ
The registration rate of individuals who were abstainers
0.08
−0.67
μ
The natural death rate
0.04
−0.17


## 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 equilibriumE0 is locally asymptotically stable if ℜ0<1, whereas E0 is unstable if ℜ0>1.Proof.
The Jacobian matrix atE0 is given by(18)JE0=−α+μ−μβα+μ00μβα+μ−γ+μ0αγ−μ.
Therefore, Eigen values of the characteristic equation ofJE0 are(19)λ1=−μ,λ2=−α+μ,λ3=γ+μℜ0−1.
Therefore, all the Eigen values of the characteristic equation are negative ifℜ0<1. Hence, the equilibrium point E0 is locally asymptotically stable if ℜ0<1 and unstable if ℜ0>1.Theorem 3.
Ifℜ0≥1, the abstaining equilibrium E∗ is locally asymptotically stable.Proof.
The Jacobian matrix atE∗ is given by(20)JE∗=−βA∗N∗−α+μ−βP∗N∗0βA∗N∗βP∗N∗−γ+μ0αγ−μ,and its characteristic equation is(21)λ3+a1λ2+a2λ+a3=0,where(22)a1=μβα+μ+μ,a2=μ2βα+μ+α+μγ+μℜ0−1,a3=μ3βα+μN+μα+μγ+μℜ0−1.
By Routh-Hurwitz Criterion, system (1) is locally asymptotically stable if a1>0, a3>0, and a1a2>a3.
Thus,E∗ is locally asymptotically stable if ℜ0≥1.


## 2.2. Global Stability Analysis

Now, we are concerned with the global asymptotic stability of abstaining-free equilibriumE0 and abstaining equilibrium E∗ of model (1), respectively.Theorem 4.
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(23)VP,A=12P−P0+A2+α+γ+2μNβA.
Then, the time derivative ofV is(24)V˙P,A=P−P0+AΛ−α+μP−γ+μA+α+γ+2μNβA˙.
SinceΛ=P0α+μ, (24) becomes(25)V˙P,A=P−P0+A−α+μP−P0−γ+μA+α+γ+2μNβA˙=−α+μP−P02−γ+μA2−α+γ+2μγ+μNβ1−ℜ0A.
Thus,V˙P,A≤0 for ℜ0≤1.
In addition,
Ifℜ0≤1, then V˙P,A=0⟺P=P0 and A=0.
Hence, by LaSalle’s invariance principle [17], the free equilibrium point E0 is globally asymptotically stable on Ω.Theorem 5.
Ifℜ0≻1, then the abstaining equilibrium E∗ of the system is globally asymptotically stable on Ω.Proof.
For the global stability of the abstaining equilibriumE0, we construct the Lyapunov function V:Ω⟶R given by(26)VP,A=X1P−P∗lnPP∗+X2A−A∗lnAA∗,where X1 and X2 are positive constants to be chosen latter.
Then, the time derivative of the Lyapunov function is given by(27)V˙P,A=βNX2−X1A−A∗P−P∗−ΛX1P−P∗2PP∗.
ForX1=X2=1, we have(28)V˙P,A=−ΛP−P∗2PP∗≤0.
Also, we obtain(29)V˙P,A=0⟺P=P∗.
Hence, by LaSalle’s invariance principle [17], the free equilibrium point E∗ is globally asymptotically stable on Ω.


## 2.3. Numerical Simulation

In this section, we present some numerical simulations of system (1) to illustrate our results. By choosing Λ=9×105, α=0.01, β=0.1,γ=0.06, μ=0.04,tf=365, and different initial values for each variable of state, we have the abstaining-free equilibrium E0=18×106,0,4.5×105 and ℜ0=0.8≺1. In this case, according to theorem (5), the abstaining-free equilibrium E0 of system (1) is globally asymptotically stable on Ω (see Figure 2).Figure 2
(a)
(b)
(c)Also, forΛ=9×105, α=0.04, β=0.3,γ=0.08, μ=0.04,tf=365, we have the abstaining equilibrium E∗=9×106,1.5×106,1.2×107 and ℜ0=1.25≻1. In this case, according to theorem (6), the abstaining equilibrium E∗ of system (1) is globally asymptotically stable on Ω (see Figure 3).Figure 3
(a)
(b)
(c)


## 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 numberℜ0 (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 variablev that depends differentiably on a parameter s is defined as(30)ϒsv=∂v∂s⋅sv.
In particular, sensitivity indices of the basic reproduction numberℜ0 with respect to the model parameters are computed as follows:(31)ϒβℜ0=∂ℜ0∂β⋅βℜ0=1,ϒαℜ0=∂ℜ0∂α⋅αℜ0=−αα+μ,ϒγℜ0=∂ℜ0∂γ⋅γℜ0=−γγ+μ,ϒμℜ0=∂ℜ0∂μ⋅μℜ0=αγ−μ2α+μγ+μ.
The positive sign of sensitivity index (S.I) of the basic reproduction numberℜ0 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, ϒβℜ0=1 suggests that increasing (or decreasing) the effective contact rate β by 10% increases (or decreases) the basic reproduction number, ℜ0, 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, ϒαℜ0=−0.5 means that increasing (or decreasing) the coefficient α, by 10%, decreases (or increases) the ℜ0 by 5%. In Table 2, we present the sensitivity indices of all model parameters to ℜ0. 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).Table 2
Parameter
Description
Value
Sensitivity index
β
The effective contact rate
0.3
+1.00
α
The registration rate of potential electors
0.04
−0.50
γ
The registration rate of individuals who were abstainers
0.08
−0.67
μ
The natural death rate
0.04
−0.17


## 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 abstainersAtt and maximize the number of registrants Rt during the time interval t0,tf and also to minimize the cost spent in an awareness program.In the model (1), we include two controls u1t and u2t for t∈t0,tf: the control u1 represents the awareness campaign effort and administrative and legal facilities effort to motivate the potential electors to register on the electoral lists. The control u2 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:(32)P⋅t=Λ−βAtPtN−α+μPt−u1tPt,A˙t=βAtPtN−γ+μAt−u2tAt,R⋅t=αPt+γAt−μRt+u1tPt+u2tAt,where P0≥0, R0≥0, A0t≥0, U0≥0, V0f≥0, and V0a≥0 are the given initial states.Then, the problem is to minimize the objective functional.(33)Ju1,u2,u3=Atf−Rtf+∫t0tfAs−Rs+M2u12s+N2u22sds,where the parameters M and N are the strictly positive cost coefficients; they are selected to weigh the relative importance of u1 and u2 at time t; tf is the final time.In other words, we seek the optimal controlsu1 and u2 such that(34)Ju1∗,u2∗=minu1,u2∈Uad3Ju1,u2,where Uad is the set of admissible controls defined by(35)Uad=uit:0≤ui≤1,fori=1,2,andt∈t0,tf.


### 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 u1∗,u2∗∈Uad2 such that(36)Ju1∗,u2∗=minu1,u2∈Uad2Ju1,u2.
If the following conditions are met:(1)
The set of controls and the corresponding state variables is nonempty.(2)
The control setUad 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 integrandLP,A,R,u1,u2 of the objective functional is convex on Uad and there exist constants c1,c2≻0 and β≻1 such that:(37)LP,A,R,u1,u2≥−c1+c2u12+u22β/2.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 P˙=FPt;P,A,R,A˙=FAt;P,A,R and R˙=FRt;P,A,R where FP,FA, and FR form the right-hand side of the system of equations (32). Let uit=ci for i=1,2 for some constants and since all parameters are constants and P,A, and R are continuous, then FP,FA, and FR are also continuous. Additionally, the partial derivatives ∂FP/∂P, ∂FP/∂A, ∂FP/∂R, ∂FA/∂A, ∂FA/∂A, ∂FA/∂R, and ∂FR/∂P, ∂FR/∂A, and ∂FR/∂R are all continuous. Therefore, there exists a unique solution (P,A,R) 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,Uad is closed.
Take any controlsu,v∈Uad and λ∈0,1. Then 0≤λu+1−λv
Additionally, we observe thatλu≤λ and 1−λv≤1−λ, then λu+1−λv≤λ+1−λ=1
Hence,(38)0≤λu+1−λv≤1,for allu,v∈Uadandλ∈0,1.
Therefore,Uad is convex and condition 2 is satisfied.Condition 3.
From the system of differential equations (32), we have(39)dNdt≤Λ−μN.
Then,(40)limsupt⟶∞Nt≤Λμ.
Therefore, all solutions of the model (32) are bounded. So, there exist positive constants B1,B2, and B3 such that ∀t∈t0,tf:(41)Pt≤B1,At≤B2,Rt≤B3.
We consider,(42)FP=P˙t≤Λ,FA=A˙t≤βPt−u2tAt,FR=R˙t≤αPt+γAt+u1tB1+u2tB2.
So, we can rewrite system (32) in matrix form as(43)Ft;P,A,R≤Λ¯+AXt−BUt,where(44)Ft;P,A,R=FPFAFRT,Λ¯=Λ00T,Xt=PART,Ut=u1u2T,A=000β00αγ0,B=P00APA.
It gives a linear function of control vector and state variable vector. Therefore, we can write(45)Ft;P,A,R≤Λ¯+AXt+BUt≤φ+ϕXt+Ut,where φ=Λ¯ and ϕ=maxA,B.
Hence, we see the right-hand side is bounded by a sum of state and control vectors. Therefore, condition3 is satisfied.Condition 4: The integrand in the objective functional (33) is convex on Uad.
It rests to show that there exist constantsc1,c2≻0 and β≻1 such that the integrand LP,A,R,u1,u2 of the objective functional satisfies(46)LP,A,R,u1,u2=At−Rt+M2u12+N2u22≥−c1+c2u12+u22β/2.
The state variables being bounded, letc1=2supt∈t0,tfA,R, c2=infM/2,N/2, and β=2, then it follows that(47)LP,A,R,u1,u2≥−c1+c2u12+u22β/2.


### 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 HamiltonianH in time t, defined by(48)Ht=At−Rt+M2u12t+N2u22t+∑i=13λifi,where fi is the right side of the system of differential equations (32) of the ith state variable.Theorem 7.
Given an optimal controlu∗=u1∗,u2∗∈Uad2 and solutions P∗, A∗, and R∗ of the corresponding state system (32), there exist adjoint functions λ1, λ2, and λ3 satisfying(49)λ˙1=λ1βAtN+α+μ+u1t−λ2βAtN−λ3α+u1t,λ˙2=−1+λ1βPtN−λ2βPtN−γ−μ−u2t−λ3γ+u2t,λ˙3=1+λ3μ,with the transversality conditions at time tf(50)λ1tf=0,λ2tf=1,λ3tf=−1.Furthermore, fort∈t0,tf, the optimal controls u1∗t and u2∗t are given by(51)u1∗t=min1,max0,1MPtλ1t−λ3t.(52)u2∗t=min1,max0,1NAtλ3t−λ2t.Proof.
The Hamiltonian in timet is given by(53)H=At−Rt+M2u12t+N2u22t+λ1Λ−βAtPtN−α+μPt−u1tPt+λ2βAtPtN−γ+μAt−u2tAt+λ3αPt+γAt−μRt+u1tPt+u2tAt.
Fort∈t0,tf, the adjoint equations and transversality conditions can be obtained by using Pontryagin’s maximum principle given in [22] such that(54)λ˙1=−∂H∂P,λ1tf=0,λ˙2=−∂H∂A,λ2tf=1,λ˙3=−∂H∂R,λ3tf=−1.
Fort∈t0,tf, the optimal controls u1∗t and u2∗t can be solved from the optimality condition(55)∂H∂u1=0,∂H∂u2=0,that is,(56)∂H∂u1=Mu1t+λ1t−Pt+λ3tPt=0,∂H∂u2=Nu2t+λ2t−At+λ3tAt=0.(57)u1t=1MPtλ1t−λ3t,u2t=1NAtλ3t−λ2t.
By the bounds inUad of the controls, it is easy to obtain u1∗t and u2∗t 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 stepi=t0 and i=tf. 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: Λ=3×106, α=0.293, β=0.700,γ=0.020, μ=0.054, and the initial values P0=10502796, A0=9041311, and R0=8020655.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 controlu1∗.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 from1.8765×107 to 2.0913×107 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 1.4632×107 to 1.3011×107 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 11% (Figure 5(a)). On the other hand, in (Figure 5(b)), we show that the registration rate on the lists is increased by 11.42%.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 controlu1∗ and u2∗.In this strategy, the two optimal controlsu1∗ and u2∗ are activated at the same time in order to improve the numerical results of case 1. The optimal control u2∗ represents the effort to convince the abstainers to change their opinion and reduce their negative effect on the potential electors.From Figure6(a), we can see that the number of registered individuals increases significantly from 1.8765×107 to 2.7204×107. Also, Figure 6(b) demonstrates that the number of abstainers decreased from 1.4632×107 to 5.8388×106 at the end of this political campaign.Therefore, we observe that the rate of registration abstinence on the electoral lists has decreased by105.25% (Figure 7(a)) and the registration rate on the lists is increased by 44.82% (Figure 7(b)).Finally, we conclude that the proposed strategy becomes more effective when we combined two optimal controlsu1∗ and u2∗.Figure 4
(a)
(b)Figure 5
(a)
(b)Figure 6
(a)
(b)Figure 7
(a)
(b)


## 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 abstainersAtt and maximize the number of registrants Rt during the time interval t0,tf and also to minimize the cost spent in an awareness program.In the model (1), we include two controls u1t and u2t for t∈t0,tf: the control u1 represents the awareness campaign effort and administrative and legal facilities effort to motivate the potential electors to register on the electoral lists. The control u2 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:(32)P⋅t=Λ−βAtPtN−α+μPt−u1tPt,A˙t=βAtPtN−γ+μAt−u2tAt,R⋅t=αPt+γAt−μRt+u1tPt+u2tAt,where P0≥0, R0≥0, A0t≥0, U0≥0, V0f≥0, and V0a≥0 are the given initial states.Then, the problem is to minimize the objective functional.(33)Ju1,u2,u3=Atf−Rtf+∫t0tfAs−Rs+M2u12s+N2u22sds,where the parameters M and N are the strictly positive cost coefficients; they are selected to weigh the relative importance of u1 and u2 at time t; tf is the final time.In other words, we seek the optimal controlsu1 and u2 such that(34)Ju1∗,u2∗=minu1,u2∈Uad3Ju1,u2,where Uad is the set of admissible controls defined by(35)Uad=uit:0≤ui≤1,fori=1,2,andt∈t0,tf.


## 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 u1∗,u2∗∈Uad2 such that(36)Ju1∗,u2∗=minu1,u2∈Uad2Ju1,u2.
If the following conditions are met:(1)
The set of controls and the corresponding state variables is nonempty.(2)
The control setUad 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 integrandLP,A,R,u1,u2 of the objective functional is convex on Uad and there exist constants c1,c2≻0 and β≻1 such that:(37)LP,A,R,u1,u2≥−c1+c2u12+u22β/2.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 P˙=FPt;P,A,R,A˙=FAt;P,A,R and R˙=FRt;P,A,R where FP,FA, and FR form the right-hand side of the system of equations (32). Let uit=ci for i=1,2 for some constants and since all parameters are constants and P,A, and R are continuous, then FP,FA, and FR are also continuous. Additionally, the partial derivatives ∂FP/∂P, ∂FP/∂A, ∂FP/∂R, ∂FA/∂A, ∂FA/∂A, ∂FA/∂R, and ∂FR/∂P, ∂FR/∂A, and ∂FR/∂R are all continuous. Therefore, there exists a unique solution (P,A,R) 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,Uad is closed.
Take any controlsu,v∈Uad and λ∈0,1. Then 0≤λu+1−λv
Additionally, we observe thatλu≤λ and 1−λv≤1−λ, then λu+1−λv≤λ+1−λ=1
Hence,(38)0≤λu+1−λv≤1,for allu,v∈Uadandλ∈0,1.
Therefore,Uad is convex and condition 2 is satisfied.Condition 3.
From the system of differential equations (32), we have(39)dNdt≤Λ−μN.
Then,(40)limsupt⟶∞Nt≤Λμ.
Therefore, all solutions of the model (32) are bounded. So, there exist positive constants B1,B2, and B3 such that ∀t∈t0,tf:(41)Pt≤B1,At≤B2,Rt≤B3.
We consider,(42)FP=P˙t≤Λ,FA=A˙t≤βPt−u2tAt,FR=R˙t≤αPt+γAt+u1tB1+u2tB2.
So, we can rewrite system (32) in matrix form as(43)Ft;P,A,R≤Λ¯+AXt−BUt,where(44)Ft;P,A,R=FPFAFRT,Λ¯=Λ00T,Xt=PART,Ut=u1u2T,A=000β00αγ0,B=P00APA.
It gives a linear function of control vector and state variable vector. Therefore, we can write(45)Ft;P,A,R≤Λ¯+AXt+BUt≤φ+ϕXt+Ut,where φ=Λ¯ and ϕ=maxA,B.
Hence, we see the right-hand side is bounded by a sum of state and control vectors. Therefore, condition3 is satisfied.Condition 4: The integrand in the objective functional (33) is convex on Uad.
It rests to show that there exist constantsc1,c2≻0 and β≻1 such that the integrand LP,A,R,u1,u2 of the objective functional satisfies(46)LP,A,R,u1,u2=At−Rt+M2u12+N2u22≥−c1+c2u12+u22β/2.
The state variables being bounded, letc1=2supt∈t0,tfA,R, c2=infM/2,N/2, and β=2, then it follows that(47)LP,A,R,u1,u2≥−c1+c2u12+u22β/2.


## 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 HamiltonianH in time t, defined by(48)Ht=At−Rt+M2u12t+N2u22t+∑i=13λifi,where fi is the right side of the system of differential equations (32) of the ith state variable.Theorem 7.
Given an optimal controlu∗=u1∗,u2∗∈Uad2 and solutions P∗, A∗, and R∗ of the corresponding state system (32), there exist adjoint functions λ1, λ2, and λ3 satisfying(49)λ˙1=λ1βAtN+α+μ+u1t−λ2βAtN−λ3α+u1t,λ˙2=−1+λ1βPtN−λ2βPtN−γ−μ−u2t−λ3γ+u2t,λ˙3=1+λ3μ,with the transversality conditions at time tf(50)λ1tf=0,λ2tf=1,λ3tf=−1.Furthermore, fort∈t0,tf, the optimal controls u1∗t and u2∗t are given by(51)u1∗t=min1,max0,1MPtλ1t−λ3t.(52)u2∗t=min1,max0,1NAtλ3t−λ2t.Proof.
The Hamiltonian in timet is given by(53)H=At−Rt+M2u12t+N2u22t+λ1Λ−βAtPtN−α+μPt−u1tPt+λ2βAtPtN−γ+μAt−u2tAt+λ3αPt+γAt−μRt+u1tPt+u2tAt.
Fort∈t0,tf, the adjoint equations and transversality conditions can be obtained by using Pontryagin’s maximum principle given in [22] such that(54)λ˙1=−∂H∂P,λ1tf=0,λ˙2=−∂H∂A,λ2tf=1,λ˙3=−∂H∂R,λ3tf=−1.
Fort∈t0,tf, the optimal controls u1∗t and u2∗t can be solved from the optimality condition(55)∂H∂u1=0,∂H∂u2=0,that is,(56)∂H∂u1=Mu1t+λ1t−Pt+λ3tPt=0,∂H∂u2=Nu2t+λ2t−At+λ3tAt=0.(57)u1t=1MPtλ1t−λ3t,u2t=1NAtλ3t−λ2t.
By the bounds inUad of the controls, it is easy to obtain u1∗t and u2∗t 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 stepi=t0 and i=tf. 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: Λ=3×106, α=0.293, β=0.700,γ=0.020, μ=0.054, and the initial values P0=10502796, A0=9041311, and R0=8020655.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 controlu1∗.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 from1.8765×107 to 2.0913×107 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 1.4632×107 to 1.3011×107 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 11% (Figure 5(a)). On the other hand, in (Figure 5(b)), we show that the registration rate on the lists is increased by 11.42%.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 controlu1∗ and u2∗.In this strategy, the two optimal controlsu1∗ and u2∗ are activated at the same time in order to improve the numerical results of case 1. The optimal control u2∗ represents the effort to convince the abstainers to change their opinion and reduce their negative effect on the potential electors.From Figure6(a), we can see that the number of registered individuals increases significantly from 1.8765×107 to 2.7204×107. Also, Figure 6(b) demonstrates that the number of abstainers decreased from 1.4632×107 to 5.8388×106 at the end of this political campaign.Therefore, we observe that the rate of registration abstinence on the electoral lists has decreased by105.25% (Figure 7(a)) and the registration rate on the lists is increased by 44.82% (Figure 7(b)).Finally, we conclude that the proposed strategy becomes more effective when we combined two optimal controlsu1∗ and u2∗.Figure 4
(a)
(b)Figure 5
(a)
(b)Figure 6
(a)
(b)Figure 7
(a)
(b)


## 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 numberℜ0. 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.



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*Source: 9738934-2020-01-03.xml*