1. Introduction

Participation in political life requires citizens’ attention, time, knowledge, money, and motivation. Citizens will participate if they receive benefits commensurate with the cost of participation. Perhaps, the debate on voting is the subject of the most studied political behavior. Individual economic, social, and psychological costs and voting benefits are well known [1]. Candidates and the media often use polls in election campaigns to determine which candidates are in the lead and who are likely to emerge victoriously. Opinion polls are surveys of intent for a sample of voters, while future results allow participants to negotiate and discuss opinions based on a particular outcome. Opinion polls revealed the expected voting quotas. Public opinion polls are widely used to identify people’s political positions, vote, and other behaviors by asking questions about their opinions, activities, and personal characteristics. Answers to these questions are then counted, statistically analyzed, and interpreted. Polls’ result also provides parties with the opportunity to refine their campaign strategies. In the long run, parties change their positions in public opinion [2].

Polls then play a key role in contemporary political campaigns. Party performance updates receive considerable media attention and often serve as a basis for political commentary in the weeks leading up to the election day. We know that knowing the positions of the electorate can affect voter behavior [3, 4]. However, it is unclear how voting shapes party leaders’ strategies during the election campaign. Public opinion is often criticized for influencing voter perceptions and focusing on popularity rather than politics. By strategically communicating the poll results, parties can shape voters’ vision of competition and encourage popular mobilization [5]. This explains why, at the beginning of the 20th century, more than thirty democratic countries banned the publication of opinion polls near elections [6, 7].

To fully understand the role of polls before the elections, it is necessary to take into account the offer of campaigns. This is particularly important in contexts of increased polarization, where holidays play an increasing role in determining the type of information that reaches voters [8]. At any time, party leaders may choose to speak of candidates from their party or other candidates for election, focusing on various political or gender issues. Voter signals in opinion polls can influence party leaders’ decisions to balance these different elements of campaign speech [9].

In the electoral process, where there are only two political parties, the chances of creating strong political polarization are increasing. For example, U. S. citizens face a polarizing scenario during a presidential election, where they must vote for a two-party decision: Republicans and Democrats. Previous studies have shown that U. S. elections are attracting the Internet, where both Twitter and blogs are showing high political polarization [10, 11].

Other political scenarios requiring a bilateral decision are second-round electoral processes. While citizens of the first round can vote from a wide range of different political parties, in the last round, they can vote only for the two final candidates. Voters may not be fully identified with either side, but they still have to take a side. Previous articles have shown that this second round increases political polarization in the country [12]. In [13], the authors analyzed the 2017 presidential election in Chile and measured the resulting political polarization. Of a minority of qualified users, they could estimate the opinion of the majority.

In this contribution, we examine how voters’ preferences and expectations compete with each other when dealing with voter support information to candidates. So, we start by developing a mathematical model to describe the evolution of opinions to predict the probability of results, and then we calculate and analyze the equilibrium points of the model by deriving the important stability thresholds for each equilibrium state.

Through the development of more efficient models, reliable statistical and mathematical methods are needed to improve the accuracy of modeling. One of the most recently popular ways is sensitivity analysis (SA) methods. Sensitivity analysis is used for a variety of reasons, such as developing decisions or recommendations, communicating, understanding or quantifying the system, and developing models. In the development of the model, it can be used to verify the validity or accuracy of the model, to simplify, calibrate, weak process, or lost data, and even to identify the important parameter for other studies [14].

The paper is organized as follows: Section 2 introduces our new model, giving some details about interactions between different compartments and parameters of the model. In Section 3, we derive basic reproduction numbers. Section 4 provides the results of the stability analysis of equilibria. In Section 5, sensitivity analysis is performed to identify the most important parameter in the proposed model, and Section 6 concludes the paper.

2. Presentation of the Model

There are many scenarios involving a binary decision. The poll model that we present here describes the opinions of agreement (or approval) and disagreement (or disapproval), regarding a candidate or idea, in polls preceding the elections. Note that the model can also describe situations in which there are more than two candidate parties because we can always reduce the situation to two decisions. For example, if there are four parts A, B, C, and D, we wish to study the political position of party A, and we can then examine the two subsets {A} and {B, C, D} of the investigation. Votes for A are considered agree, and votes for B, C, or D are considered disagree. More simply, we consider the opinion poll of voters on the performance of the party studied. As an example, we can cite the “approve or disapprove” poll done by Thomson Reuters on how Donald Trump fulfills his role as the president [15].

Without loss of generality, we devise a mathematical model describing the evolution of agree and disagree opinions during polls (surveys), and the types of surveys we consider here are surveys that can be answered in agreement, disagreement, or otherwise. Thus, the population targeted by the poll is regrouped into three groups: agree, disagree, and ignorant individuals.

Everyone has their reasons for agreement or disagreement. An ignorant person can be persuaded by someone who agrees at a rate β₁ or by someone who does not agree with the opinion at a rate β₂. A person agreeing with the opinion may be persuaded by someone who does not agree at a rate α₂, or a person who disagrees with the opinion may be persuaded by someone who agrees at a rate α₁. People can abstain from voting or lose interest without any direct contact with individuals from the opposite opinion group, then agree people become ignorants at the rate γ₁, and disagree people become ignorants at the rate γ₂. A flowchart describing different interactions between the compartments of the model is presented in Figure 1.

Therefore, all the solutions of systems (1)–(3) are nonnegative.

A summary of parameter description is given in Table 1.

3. Thresholds: Basic Reproductive Numbers

In epidemiology, the basic reproductive number R₀ (or epidemic threshold) is defined as the average number of secondary cases of an infection produced by a “typical” infected individual during his/her entire life as infectious when introduced in a population of susceptibles [17–21]. The threshold R₀ is mathematically characterized in terms of infection transmission as a “demographic process,” but offspring production is not seen as giving birth in a demographic sense, but it causes new infections through transmission [22–24]. Thus, the infection process can be considered as a “consecutive generation of infected individuals.” The following growing generations indicate a growing population (i.e., an epidemic), and the growth factor for each generation indicates the potential for growth. So, the mathematical characterization of R₀ is this growth factor [22]. Generally, if R₀ > 1, an epidemic occurs, whereas if R₀ < 1, there will probably be no epidemic.

Following this definition, we will define our thresholds as follows: R_D0 is the average number of new disagreements produced by an individual disagreeing introduced in a population of ignorant people during the period in which he or she was in this opinion. And, R_A0 is the average number of new agreements produced by an individual in agreement and that was introduced into a population of ignorant people during the period in which he or she was in that opinion.

For the analysis of epidemic models, the first step is to calculate the disease-free equilibrium (DFE). This equilibrium point is then used to calculate the basic reproductive number using the next-generation matrix method. The objective of this section is only the calculation of the thresholds and not the equilibrium states of the model. But, in this method, we have to determine the equilibrium states when A = 0 and when D = 0.

In this contribution, let R_X0 be the threshold of growing of the opinion X (either X = A “agree” or X = D “disagree”). Then, R_D0 is the threshold associated with the disagree-free equilibrium, while R_A0 is the one associated with the agree-free equilibrium.

We calculate the equilibria for the aforementioned model, and based on the next-generation matrix approach, we derive associated thresholds.

4. Stability Analysis

4.3. Examples

It can be seen from Table 3 that the equilibrium E₀ exists and it is stable in the four cases: example 1: R_D0 > 1 and R_A0 > 1, example 2: R_A0 < 1 and R_D0 < 1, example 3: R_D0 < 1 and R_A0 > 1, and example 4: R_D0 > 1 and R_A0 < 1.

Table 3. Steady states and parameters’ values used in example simulations, where I(0) = A(0) = D(0) = 100.

		β1	β2	α1	α2	γ1	γ2	RD0	RA0
1	E0 = (300,0,0)	0.0010	0.1010	0.1010	0.3010	0.5010	0.3010	1.9901	2.0730
2	E0 = (300,0,0)	0.0010	0.2010	0.2010	0.3010	0.1010	0.5010	0.5000	0.8543
3	E0 = (300,0,0)	0.0010	0.0010	0.2010	0.1010	0.1010	0.1010	0.5000	2.0001
4	E0 = (300,0,0)	0.0010	0.0010	0.1010	0.2010	0.1010	0.1010	2.0001	0.5000
5	E1 = (2.9703, 297.0297, 0)	0.1010	0.0010	0.2010	0.2010	0.0010	0.0010	0.6634	0.4925
6	E1 = (2.9703, 297.0297, 0)	0.1010	0.0010	0.0010	0.0010	0.0010	0.0010	0.5025	101
7	E2 = (2.9703, 0,297.0297)	0.0010	0.1010	0.2010	0.2010	0.1010	0.0010	0.4925	0.6634
8	E2 = (2.9703, 0,297.0297)	0.0010	0.1010	0.0010	0.0010	0.0010	0.0010	101	0.5025
9	$$	0.5010	0.1010	0.1010	0.0010	0.2010	0.0010	0.6688	0.5196
10	$$	0.1010	0.5010	0.0010	0.1010	0.0010	0.2010	0.5196	0.6688
11	E∗ = (133.3333, 10.5556, 156.1111)	0.1010	0.7010	0.3010	0.0010	0.2010	0.3010	467.7774	1.0672
12	E∗ = (214.2857, 80.4643, 5.2500)	0.3010	0.4010	0.0010	0.8010	0.2010	0.5010	1.0649	300.8004

Example 5 shows the existence and the stability of the steady state E₁ in the case of R_A0 < 1 and R_D0 < 1 and in the case of R_A0 > 1 and R_D0 < 1 in example 6. Examples 7 and 8 show the existence and the stability of the equilibrium E₂ in the cases of “R_A0 < 1 and R_D0 < 1” and “R_A0 < 1 and R_D0 > 1,” respectively.

Examples 9 and 10 show the possibility of the existence and stability of the two steady states E₁ and E₂ at the same time, that is, “β₁ > γ₁ and β₂ > γ₂ and R_A0 < 1 and R_D0 < 1.” These examples give an insight about the stability of each equilibrium for given parameters’ values. After some numerical calculations, we noted that, in this situation, the parameters γ₁ and γ₂ may switch from E₁ to E₂ and the inverse, while if γ₁ > γ₂, then E₁ will be more attractive, and if γ₂ > γ₁, then E₂ will be more attractive and that with the same initial conditions. We simulate the model with different initial conditions to illustrate the impact of initial conditions on the stability of E₁ and E₂ in this situation. Figure 2 depicts the stability of E₁ and E₂ at the same time, where we consider the same set of parameters from example 9 in Table 3. By changing the initial conditions, we can see that E₁ is stable when I(0) = 100, A(0) = 100, and D(0) = 80, while it can be seen that E₂ is also stable with this set of parameters but after choosing I(0) = 100, A(0) = 80, and D(0) = 100.

Example 11 shows the existence and the stability of the equilibrium E_∗, where “R_D0 > 1 and R_A0 > 1 and α(α − β₁ + β₂) > 0” and “β₁ < γ₁ and γ₂ < β₂ and α > 0,” while example 12 shows the existence and the stability of E_∗, where “R_D0 > 1 and R_A0 > 1 and α(α − β₁ + β₂) > 0” and “β₁ > γ₁ and γ₂ > β₂ and α < 0.”

Figure 3 depicts examples of the existence and stability of the equilibrium state E₀ for different parameters’ values and threshold R_D0 and R_A0 values simulated with initial conditions and parameters’ values from Table 3. Figure 4 depicts examples of the existence and stability of the equilibrium state E₁ for different parameters’ values and threshold R_D0 and R_A0 values simulated with initial conditions and parameters’ values from Table 3. It can be seen from Figure 4(b) that the function D decreases towards zero, but it will take a long time. In the Figure 4(c), we have considered 1,200 hours to show that the function D will go to zero but very slowly. Figure 5 depicts examples of the existence and stability of the equilibrium state E₂ for different parameters’ values and threshold R_D0 and R_A0 values simulated with initial conditions and parameters’ values from Table 3. We can see from Figure 5(b) that the function A will also take a very long time to go to zero. In Figure 5(c), we have considered 1,200 hours to show that function A will tend to zero but very slowly. Figure 6 depicts examples of the existence and stability of the equilibrium state E_∗ for different parameters’ values and threshold R_D0 and R_A0 values simulated with initial conditions and parameters’ values from examples 11 and 12 of Table 3.

5. Analysis of Thresholds

5.3. Statistical Analysis

Here, we use probability distribution functions of the six parameters given in Table 4 sampled by using the Latin hypercube sampling, see Figure 7. We compute the probabilities of equilibria existence and stability conditions. It can be seen from Table 5 that the probability of E₁ to exist is about 0.5960, while its probability of stability is about 0.53. The probability of E₁ to exist and to be stable is 0.4040. The probability of the existence of E₂ is about 0.6250, while the probability of its stability is about 0.5370. The probability of E₁ to exist and to be stable is 0.43. The equilibrium E_∗ has a probability of existence about 0.0240 and a probability of stability about 0.2190, while E_∗ exists, and it is stable with a probability of 0.001.

Table 4. Six input parameters’ probability distribution functions.

Parameters	Pdf
β1	Normal
β2	Normal
α1	Triangular
α2	Triangular
γ1	Triangular
γ2	Triangular

Table 5. Summary of probabilities of existence and stability conditions of equilibria.

Equilibrium	Probability of existence	Probability of stability	Probability of existence and stability
E1	0.5960	0.5300	0.4040
E2	0.6250	0.5370	0.4300
E∗	0.0240	0.2190	0.001

6. Sensitivity Analysis

Global sensitivity analysis (GSA) approach helps to identify the effectiveness of model parameters or inputs and thus provides essential information about the model performance. Out of the many methods of carrying out sensitivity analysis is the partial rank correlation coefficient (PRCC) method that has been used in this paper. The PRCC is a method based on sampling. One of the most efficient methods used for sampling is the Latin hypercube sampling (LHS), which is a type of Monte Carlo sampling [25] as it densely stratifies the input parameters. As the name suggests, the PRCC measures the strength between the inputs and outputs of the model using correlation through the sampling done by the LHS method [26–28].

Parameters β₁ and β₂ follow normal distribution with mean and standard deviation 0.5 and 0.01, respectively, while parameters α₁, α₂, γ₁, and γ₂ follow triangular distribution with minimum, maximum, and mode as 0.02, 0.8, and 0.51, respectively. A summary of probability distribution functions is given in Table 4. In Latin hypercube sampling approach, probability density function (given in Table 4) for each parameter is stratified into 100 equiprobable (1/100) serial intervals. Then, a single value is chosen randomly from each interval. This produces 100 sets of values of R_D0 and R_A0, from 100 sets of different parameter values mixed randomly, calculated by using equations (4) and (5), respectively.

Sensitivity analysis has been done concerning the basic reproduction numbers R_D0 and R_A0, and the main objective of this section is to identify which parameters are important in contributing variability to the outcome of basic reproduction numbers based on their estimation uncertainty.

To sort the model parameters according to the size of their effect on R_D0 and R_A0, a partial rank correlation coefficient is calculated between the values of each of the six parameters and the values of R_D0 and R_A0 in order to identify and measure the statistical influence of any one of the six input parameters on thresholds R_D0 and R_A0. The larger the partial rank correlation coefficient, the larger the influence is on the input parameter affecting the magnitude of R_D0 and R_A0.

As shown in Table 6, the transmission rate, β₁, and parameters γ₁ and γ₂ are highly correlated with the threshold R_D0 with corresponding values −0.1374 and 0.1613 and −0.2101, respectively. Moderate correlation exists between transmission rates α₁ and α₂ and R_D0 with corresponding values as −0.0633 and 0.0669, respectively. Weak correlations have been observed between the transmission rate β₂ and R_D0 with the corresponding value 0.0025.

As shown in the sensitivity index column of Table 6, the parameter γ₂ accounts for the maximum variability 0.8499 in the outcome of basic reproduction number R_D0. The parameter γ₁ is the next to account for the variability 0.7492 in the outcome of R_D0. Then, the transmission rate β₁ accounts for the variability 0.5596 in the outcome of R_D0 followed by the transmission rate α₂ that accounts for the variability 0.5269 in the outcome of R_D0. Transmission parameters α₁ and β₂ account for the least variability 0.1065 and 0.0192 in the outcome of basic reproduction number R_D0, respectively. Hence, parameters γ₁ and γ₂ and the transmission rate β₁ are the most influential parameters in determining R_D0.

It can be seen from Table 7 that the parameter γ₁ is highly correlated with the threshold R_A0 with the corresponding value −0.3343. Moderate correlation exists between transmission rates β₁, β₂, and α₂ and the parameter γ₂ and R_A0 with corresponding values as 0.2137, −0.1928, −0.2076, and 0.2523, respectively. Weak correlations have been observed between the transmission rate α₁ and R_A0 with the corresponding value 0.0857.

One can see in the sensitivity index column of Table 7 that the parameter γ₁ accounts for the maximum variability 0.8004 in the outcome of basic reproduction number R_A0. The parameter γ₂ is the next to account for the variability 0.7122 in the outcome of R_A0. The transmission rate β₁ accounts for the variability 0.5753 in the outcome of this threshold followed by the transmission rate α₂ that accounts for the variability 0.3872 in the outcome of R_A0. Transmission parameters β₂ and α₁ account for the least variability 0.2421 and 0.0159 in the outcome of basic reproduction number R_A0, respectively. Hence, parameters γ₁ and γ₂ and the transmission rate β₁ are also the most influential parameters in determining R_A0.

Scatter plots comparing the basic reproduction numbers R_D0 and R_A0 against each of the six parameters: β₁, β₂, α₁, α₂, γ₁, and γ₂ are shown in Figures 8 and 9, respectively, based on Latin hypercube sampling with a sample size of 100. These scatter plots clearly show the linear relationships between outcome of R_D0 and R_A0 and input parameters.

7. Conclusion

In this paper, an IAD model-type compartmental model has been considered to explore agree-disagree opinions during polls. The equations governing the system have been solved to compute equilibrium states, and the next-generation matrix method is used to derive basic reproduction numbers R_A0 and R_D0.

The model exhibits four feasible points of equilibrium, namely, the trivial equilibrium, agree-free equilibrium, disagree-free equilibrium, and the positive equilibrium. Sufficient equilibrium conditions of existence are given, and stability analysis is performed to show under which conditions equilibrium states are stable or unstable. The stability of these points of equilibrium is controlled by the threshold number R_A0 and R_D0. If the threshold, R_D0, is less than one, the disagree opinion dies out and the disagree-free equilibrium is stable. If R_D0 is greater than one, the disagree opinion persists and the disagree-free equilibrium is unstable. If the threshold, R_A0, is less than one, the agree opinion dies out and the agree-free equilibrium is stable. If R_A0 is greater than one, the agree opinion persists and the disagree-free equilibrium is unstable. We simulated some examples with different parameter values to show the existence and the stability of such equilibria.

The probabilities of the existence and probabilities of the stability of equilibria are computed based on the parameters’ distribution function sampled with the Latin hypercube sampling method. To identify the most influential parameter in the proposed model, global sensitivity analysis is carried out based on the partial rank correlation coefficient method and Latin hypercube sampling. This statistical study shows that the most influential parameters in the determination of thresholds of equilibria stability are β₁, the polarization parameter of ignorant people by people agreeing, the parameter, γ₁, of the loss of interest of the people agreeing, and finally γ₂, the parameter of loss of interest of disagreeing people.

Conflicts of Interest

The authors declare that they have no conflicts of interest.