SEDIS—A Rumor Propagation Model for Social Networks by Incorporating the Human Nature of Selection

Abstract

The explosive evolution of the internet has paved the path for the rise of social networks, which can help people connect remotely. Currently, social networks are commonly used for sharing thoughts, feelings, information, and personal life, which vary from individual to individual. The world has witnessed a tremendous increase in social media usage in the last decade, and more people are expected to spend their time online after the COVID-19 pandemic. This increases the rapid propagation of rumors and fake news within societies and communities. On one end, social networks act as an excellent platform for digital marketing and sharing information. However, on the other end, social network rumors and fake news create a significant impact on society, including riots. To study and analyze social network rumors, several mathematical rumor propagation epidemic models have been proposed. The majority are related to disease-spreading epidemic models and reject the human aspect of social selection. This paper introduces a new mathematical rumor propagation model for social networks by incorporating the human psychological aspect of selection as a separate state. Our mathematical analysis and computational simulation proved that the model exists within the system. It was also proven that the system is always non-negative and there always exists a solution in the system. Our implementation of an intervention mechanism within the discrete compartmental model simulation proved the necessity of an effective interference that can help to prevent the implications of uncontrolled rumor dissemination within social networks.

1. Introduction

With the onset of the 21st Century, social networks have gained momentum, allowing people to share information and news rapidly within a society. The content shared through these networks can be either factual or misinformation that can cause unexpected consequences around us. Though social networks were initially supposed to connect people across the world, the addition of new features in the later stage has allowed people to publicly share information, thoughts, personal life, news, and feelings with each other [1]. Even though rumors had been propagated in society before, more recently, fake news and false information have gathered their momentum at a rapid pace since social networks play an inevitable role in faster information diffusion across societies. Nowadays, individuals spend hours on numerous social networking websites engaging in various activities that encourage the quick spread of rumors among us.

Broadly, social networks are platforms for people to communicate with one another, whereas social media generally serves as platform for information dissemination. Fake news and cyberbullying have led to severe consequences in society and are frequently disseminated through social networks. Previous studies have proven that people are highly likely to accept fake news and rumors if they are circulated for a prolonged period without any interventions. The US Capitol riots that followed after the US Presidential election are an example of the implication of rumors being propagated through social networks. Moreover, it has been identified that informationally based determinants in social networks could influence the positive emotions of humans [2]. Reputation management plays a crucial role among online internet users, and misinformation is a major challenge in maintaining the reputation of social network users. Through interviews and surveys, it has been identified that online rumors, fake news, and cyberbullying could affect one’s reputation. In this study, several participants who were concerned about online reputation damage shared their experiences, which shed light on how these participants regarded managing their online reputation as both important and unpleasant, as well as disempowering. The majority of persons who experienced reputation loss via social networks were unable to identify workable repair methods and were, therefore, unable to find a solution to their issue. It was discovered through these interviews and surveys that a user’s reputation could impact their employment, academic possibilities, and social opportunities. Thus, it can be said that maintaining one’s online reputation is important because it is intimately related to one’s life and false rumors in online social networks need to be taken seriously [3]. Additionally, it has been asserted that preserving one’s online reputation is essential because it can significantly impact one’s personal life and profession. It has also been identified that an organization’s ignorance of dealing with social network risks concerning false rumors could adversely affect its reputation [4]. Response strategies have to be developed based on the different threats from social networks. An analysis of the hostile behavior of users explained that the aggressive behaviors of online users could further lead to cyberbullying. It also showcased that anonymous users tend to be more hostile than non-anonymous users [5]. Compared to people who do not share their identities on social media, those with a long history on a platform and active profiles tend to be less antagonistic.

Apart from reputation management, quantitative studies have found that cyberbullying can adversely affect the academic performance of students. It has been discovered that students who experience cyberbullying perform worse academically [6]. User influence plays a major role in the dissemination of information. Users are critical in rumor propagation, especially when it comes to community leaders within the network. It is stated that only a few models have been developed to measure user influence and the importance of social networks [7]. Social media posts by prominent individuals can have a direct impact on their social circle. If inaccurate information is spread about them at the same time, it could harm their reputation. Social media rumors about an individual are another cause of cyberbullying, especially among teenagers. A survey conducted among 310 students highlights that cyberbullying may contribute to low self-esteem among students [8]. A similar result has been observed in another study which states that victims of cyberbullying experience an increased level of loneliness. In short, it is noted that social network rumors can negatively affect individuals and societies [9]. It is noted that “online social media can operate as an “echo chamber” where personal ideas that affect individual medical decisions are predominantly repeated by others, especially during the pandemic time” [10]. An echo chamber refers to a group of people with similar interests. Studies have showcased that rumors are highly likely to disseminate within echo chambers.

Social media rumors can have both positive and detrimental effects on society. The rapid increase in social media usage has paved the way for the quick dissemination of rumors in society. It is crucial to create mathematical models for studying rumor propagation within a network across each individual and community. The basic epidemic models are the SI, SIR, and SIS models. The terminologies for these models are as follows:

1.

S: Susceptible node—refers to the nodes susceptible to getting infected.

2.

I: Infected nodes—nodes that are infected and are capable of spreading disease (rumors) to the susceptible ones.

3.

R: Recovered nodes—nodes recovered from infection by attaining immunity or recovered due to death.

The SI model states that all susceptible nodes within a network will get infected [11]. It acts as the basic model, which led to the development of other epidemic models. If β is the probability of getting infected, then the SI model can mathematically be defined as follows:

$$\frac{dI}{dt}=\beta \frac{SI}{n}$$

(1)

$$\frac{dS}{dt}=-\beta \frac{SI}{n}$$

(2)

where

• $S/n$ is the probability of meeting a susceptible one at a random unit of time.
• $SI/n$ is the average number of susceptible populations that infected nodes meet per unit time.
• $\beta \frac{SI}{n}$ is the average number of susceptible people infected from all infected per unit time.

The SI model mentioned in Figure 1, states that a person in the infected state is infected forever and no recovery from the infection is possible.

Later, Kermack and McKendrick proposed a new model by adding a “Recovery” state [12]. As seen in Figure 2, individuals in a recovered state cannot transmit disease to other susceptible ones. Let β be the infection rate and γ be the recovery rate. Let $s=S/n$ and $i=I/n$. Then the model can be explained as follows:

$$\frac{ds}{dt}=-\beta si$$

(3)

$$\frac{di}{dt}=\beta si-\gamma i$$

(4)

$$\frac{dr}{dt}=\gamma i$$

(5)

The likelihood of an infected node returning to the susceptible state is not assessed by the SI or the SIR model. When coming into a social network, people cannot stay infected or in a recovered/immunized stage forever. The SIS model, an expansion of the SI model, is shown in Figure 3. This model comes with a difference in that an infected node goes back to being a susceptible node instead of going to a recovered state. Considering $s=S/n,$ $i=I/n,$β as the infection rate, and γ as the recovery rate, the model can mathematically be explained as follows:

$$\frac{ds}{dt}=\gamma i-\beta si$$

(6)

$$\frac{di }{dt}=\beta si-\gamma i$$

(7)

Another extension of the former models is the SIRS model. This model is very similar to the SIS model with the insertion of a Recovered state after the Infected state.

By splitting the Infected state separately into positive and negatively infected states (P and N, respectively) based on the sentiment of rumors, the SPNR model was later proposed [13]. It estimates that a susceptible node might become either positively or negatively infected depending upon the sentiment of the rumor. The likelihood for a susceptible node to become either positively or negatively infected is 0.5:0.5 as a result of the model’s division of an infected state into positively and negatively infected states. This was followed by the RnSIR model, which adds a Restrained state, as seen in Figure 4 [14]. After testing the model through datasets, it was noted that a restrained population decreases with the progress of time. Later, the SDIR Model was proposed by incorporating a Doubter state [15]. Here, a doubter is a person who is doubtful whether a rumor is genuine or not but is not a spreader yet. As illustrated in the Figure 4, a doubter may be won over and may become a spreader for a certain probability. The chance of a stifler reverting to a doubter or spreader state, however, is not mentioned.

In social networks, it was found that opinion leaders play a major role in the diffusion of products [16]. Opinion leaders are those who explain the significance of media information to less sophisticated media consumers. Using the SIS and SIR models, it has been determined that, for opinion leaders in both models, the diffusion rate is comparatively high. Considering the public user posts and opinions in social networks, the S-SEIR model was proposed, where E stands for the Exposed node, referring to a population being exposed to a rumor but not propagating it [17]. Information propagation is created based on the textual mood of users on microblogging websites. The likelihood of a node under an exposed condition returning to a vulnerable state, however, is not addressed by this model.

After analyzing the E-Commerce domain, an epidemic model was proposed by classifying customers into cyber consumers, confused consumers, escapers, and recovered ones which is mentioned in Figure 5 [18]. People who purchase items online are categorized as cyber consumers. Those who have seen the false news but are not propagating it are categorized as confused consumers. Escapers, similar to an infected population, are the ones who get deceived by the fake news and start spreading the same news. People who have stopped spreading fake news are categorized as recovered consumers. However, this model fails to explain the possibility of a confused consumer being prevented from becoming an escaper. Later, a rumor propagation model with an optimal control approach for online social networks from the SIR model with manipulation by taking professional pages which spread fake news into account [19]. This model effectively conveys the significance of a control technique in the struggle against rumor dissemination. Effective control measures have been found to stop rumors from spreading quickly within networks.

During the COVID-19 pandemic, people were forced to work from home. Lack of social interaction might have an impact on people’s motivations. This problem is thought to have become more serious as a result of numerous social networking websites. However, when it comes to privacy and ethical concerns, these people-connecting technologies are quite likely to complicate matters, particularly when it comes to corporate surveillance and employee privacy [20]. Rumors are likely to spread more widely throughout society as a result of the surge in social network usage that has followed the outbreak. By understanding the changes in society during the pandemic period and by incorporating verified social media accounts, the SVIR (Susceptible–Verified–Infected–Recovered) model was formulated for OSN, by adding a Verified/Authenticated user state; in this state, people are generally not the spreaders of the rumor [21]. In contrast to earlier models, the recovered node here refers to those who do not believe the rumor. This concept operates under the premise that those responsible for disseminating false information have been booted out of the system. This model was followed by a new IDSRI rumor propagation model, which categorizes the total population in a network as ignorances, discussants, spreaders, and removers [22]. A discussant is a person who is assumed to know a rumor but is idle and does not spread it. However, discussants are likely to take part in the discussion related to the same topic. A remover is a person who knows the rumor but no longer spreads it. A new SEIS crisscross model was then proposed by hypothesizing that hostile atmospheres are likely to be created if different suggestions across the communities are raised by other people. If the Reproductive Number R₀ lies greater than one, the system is highly likely to be infected; and the system is free from infection if R₀ is less than one [23]. Several fact-checking websites, as well as online and television media, play a significant role in exposing false information that circulates on social media. The significance of these platforms was determined by a quantitative and qualitative analysis of data from five focus groups and an online survey with 350 participants. Although it was found that some participants do not trust fact-checking platforms and some do not know how to utilize fact-checking platforms, a sizable portion of the population nevertheless double-checks the material they come across on social media [24]. Another survey among Dutch youngsters to examine whether they could identify a fake website and news showed that people might believe fake news. However, it was also noted that, regardless of whether they believe the source, people might react if they learn that they have been exposed to erroneous information [25]. Digital education for young people and children is crucial because it can assist the next generation in spotting false stories on the internet. Future trends suggest that a larger proportion of people will use fact-checking websites to double-check news and information.

It can be observed that the popularity of fact-checking websites has increased after the COVID-19 pandemic, and people spend time on these platforms to analyze the authenticity of the information they encounter through social media. By understanding this change in societies after the COVID-19 pandemic, we formulate an epidemic model for social networks by incorporating the dimensions of social intelligence, which are not included in the classic epidemic models.

2. Problem Statement and Contributions

2.1. Problem Statement

Network epidemics play a significant role in manipulating society. From the educational sector to professional fields, the recent trends after the COVID-19 pandemic prove that the world is slowly shifting to a work-from-home strategy. This is expected to increase social media usage among all age categories. More recently, most people have begun to rely on social media platforms for sharing and gathering information based on their topics of interest. This also creates the probability of an increasing trend of fake news and false rumors through the networks. Rumor propagation models can help computer scientists make an early prediction of rumors, and a successful intervention can be made before the rumors have an impact on society. These models help to understand the spread of fake news and rumors and to plan early intervention mechanisms for the rumors.

Though the SEIR and SPNR models consider the user’s choice of information diffusion and rumor propagation, the states defined by these models are inappropriate considering the current trends in social networks. The Recovered state (R) is irrelevant in social networks when considering a network or community can be affected by multiple or consequent rumors. Moreover, we cannot say that a person has entered a recovered state if he/she does not spread the rumor. Several reasons prevent a particular user from propagating rumors within a network. If a user is not active in a network or has refrained from engaging in activities in social networks, it could affect rumor propagation through the web. A user can also stay silent for some time and share the story in the future. Apart from this, social media rumors are pretty different from pathogenic epidemic models, where there exists a choice among individuals whether a particular rumor should be propagated or not within a network. A user can also enter into a doubtful stage of whether to believe a specific rumor or not. Apart from the computational field, social media rumors are also a concern in the social science field. Hence, the psychological state of human nature of selection must be mentioned under Discrete Compartmental Modeling.

Most significantly, we must recognize that, in former times, people tended to share all relevant information without hesitation. Though only a few fact-checking websites focus on matters related to law, an increasing number and popularity of general fact-checking websites have changed the trend of decision-making [26]. This must be taken into account as many people now rely on these websites and media to check whether the information of interest, they receive through social media is genuine. It has also been highlighted that newspapers and television are now fighting harder than before to combat the spread of misinformation [27]. Many television news channels are currently broadcasting fake news propagated through social media. Several governments have now begun to take the issue of fake news seriously. Recently, the Government of India started plans to amend the IT Act to shut down apps and websites which publish fake news. It has become more common for people to be aware of, comprehend, and combat fake news in society. Our society has learned a lot about false information and its effects, and people have started to double-check whenever they come across a rumor in digital media. Although a significant shift in the trend has lately been observed, the majority of people continue to believe in fake news [28]. As rumors spread in different communities differently depending on the demographic and cultural environments, separate intervention mechanisms may have to be applied. Misinformation and hate speech are now creating a huge impact on our society. This has to be regarded seriously when compared to the traditional network epidemic models, as more people are now rethinking before forwarding a particular rumor of their interest. Hence, it is highly relevant to create a mathematical rumor propagation model by giving importance to the category of people who are “doubtful” about whether a piece of information is genuine or not.

2.2. Contributions

After implementing the model, the following contributions are achieved concerning the gaps in the literature and our understanding of the human aspect:

• Created a mathematical rumor propagation model by considering that the Recovered state is not practically possible in networks.
• Included a new state for social network rumor propagation by considering the human psychological aspect of social selection and current trends in society.

Moreover, it must be understood that not all susceptible nodes are likely to become infected within a network. The model must be formulated with the assumption that social network rumors contain an equal amount of factual and non-factual content.

Social awareness and situation-appropriate behavior have been viewed as indicators of social intelligence. As a result, unsuitable content is less likely to spread in societies where social and emotional intelligence is highly valued. Furthermore, based on a community’s interests and current circumstances, we may also assume that false information can be introduced into digital communities. The generated model can, hence, be applied to assess the social and emotional intelligence of communities that are frequently exposed to misinformation and share fake news.

3. The SEDIS Model

A user is likely to see multiple posts shared by their friends or acquaintances on social networks. However, these posts and rumors shared by one user will not be visible to all users in their feeds within a network or community. A user is likely to see a rumor only if a person in his contact or follower posts the same. However, this is not guaranteed at all times since some inactive users will not access social media at frequent regular intervals. Moreover, it is not guaranteed that a user will read the post. Thus, with reference to the S-SEIR model, we propose a new model—the SEDIS (Susceptible-Exposed-Doubter-Infected) with human selection choice of exposed and doubtful states. Here, an Exposed state refers to a population exposed to specific news or rumors. As mentioned earlier, we have to consider the fact that fact-checking websites and news media are gaining popularity in preventing the spread of misinformation within a network. The relevance of the Doubter state arises at this juncture. A Doubter refers to a person who is not yet infected but is quite confused about whether to believe a particular rumor or not. These individuals are highly likely to be infected if they encounter similar rumors repeatedly in the network and are also expected to return to a susceptible state if they start to believe that a rumor is fake after going through fact-checking sources. Depending on their level of psychology and their social and emotional intelligence, a person who has been exposed has a probability of either becoming Infected or becoming a Doubter. An individual can become infected or doubtful only if he/she enters the exposed state. However, not all exposed individuals will enter into a doubtful or infected state as they can give less priority to or reject the rumor they have encountered within the network. An exposed person is also less likely to spread rumors if he/ she is not active on social media. This new model does not take the “Recovered” state into account and incorporates the provision of a user to be susceptible again.

Rumor Propagation Model

Theoretically, the SEDIS model can be explained as follows:

• Susceptible users are the individuals on social media who have a chance to meet an Infected node—that is, the one who shares a rumor.
• Exposed ones are the individuals who have seen the rumors shared by the users from their contact/ following list.
• A susceptible node is likely to meet an infected node and may become exposed at probability α.
• Under certain probability, an exposed individual can either become a doubter or infected at probability β₁ and β₂, respectively.
• An exposed individual can reject a rumor and may go back to the susceptible state at probability µ₁.
• The doubter state refers to doubters who are doubtful whether a piece of information propagated towards them is genuine or not. These individuals can become infected at probability ϒ or return to the susceptible state at probability µ₂.
• There is no recovery stage. However, an infected individual may return to the susceptible stage at probabilityµ₃.

Figure 6 shows the SEDIS compartmental model. Individuals move from one compartment to another, as shown by the arrows in the figure. The parameter description of the model is illustrated in

Table 1. Parameter description of the Model.

Parameter	Physical Interpretation
S	Susceptible compartment
E	Exposed compartment
D	Doubter compartment
I	Infected compartment
α	Probability of transition from the susceptible to the exposed state
β1	Probability of transition from the exposed to the doubter state
β2	Probability of transition from the exposed to the infected state
ϒ	Probability of transition from the doubter to the infected state
µ1	Probability of transition from the exposed to the infected state
µ2	Probability of transition from the doubter to the susceptible state
µ3	Probability of transition from the infected to the susceptible state

.

4. Method

4.1. Discrete Compartment Modeling

Discrete Compartment Modeling is usually performed to showcase the epidemic growth of information. Let α be the probability of a susceptible node becoming exposed, and β₁ and β₂ be the probability of an exposed node entering into the Doubter state and Infected state, respectively. Let ϒ be the probability of a Doubter node becoming infected. Similarly, let µ1 be the probability for an exposed node to return to a susceptible state, µ₂ be the probability for a Doubter node to return to the Susceptible state, and µ₃ be the probability for an Infected node to return to the Susceptible state. Then, the model can mathematically be explained as follows for the rate of susceptible population:

$$\frac{dS}{dt}=\mu 3\frac{IS}{n}+\mu 2\frac{DS}{n}+\mu 1\frac{ES}{n}-\alpha \frac{SE}{n}$$

(8)

where

• $\frac{S}{n}$ is the probability of meeting a susceptible one at a random unit of time.
• ${\mu }_3\frac{IS}{n}$ is the average number of infected people becoming susceptible from all infected per unit time.
• ${\mu }_2\frac{DS}{n}$ is the average number of doubters becoming susceptible from all doubters per unit time.
• ${\mu }_1\frac{ES}{n}$ is the average number of exposed people returning to the susceptible state from all exposed people per unit time.
• $-\alpha \frac{SE}{n}$ is the average number of susceptible people becoming exposed from all susceptible per unit time.

Similarly, other states of the system can be explained as follows:

$$\frac{dE}{dt}=\alpha \frac{SE}{n}-{\mu }_1\frac{ES}{n}-{\beta }_1\frac{ED}{n}-{\beta }_2\frac{EI}{n}$$

(9)

$$\frac{dD}{dt}={\beta }_1\frac{ED}{n}-\gamma \frac{DI}{n}-{\mu }_2\frac{DS}{n}$$

(10)

$$\frac{dI}{dt}=\gamma \frac{DI}{n}+{\beta }_2\frac{ED}{n}-{\mu }_3\frac{IS}{n}$$

(11)

To simplify these equations, we consider $s=S/n, e=E/$n, $d=D/n$ and, $i=I/n$. The model can now be defined as follows:

$$\frac{ds}{dt}={\mu }_{3}i+{\mu }_{2}d+{\mu }_{1}e-\alpha s$$

(12)

This gives the susceptibility rate in the model. The state S transitions from Infected, Doubter, and Exposed nodes under the probability ${\mu }_3$, ${\mu }_2$ and ${\mu }_1$, respectively. Similarly, a node leaves from the Susceptible state to the Infected state under “α” probability. Similarly, we define other states as follows:

$$\frac{de}{dt}=\alpha s-{\mu }_{1}e-{\beta }_{1}e-{\beta }_{2}e$$

(13)

$$\frac{dd}{dt}={\beta }_{1}e-\gamma d-{\mu }_{2}d$$

(14)

$$\frac{di}{dt}=\gamma d+{\beta }_{2}e-{\mu }_{3}i$$

(15)

The basic reproduction number R₀ is estimated using the Next Generation Matrix [29]. We have $F$ which defines the rate of new infections and $V$ as the transfer of individuals out of minus into the next compartment.

$$F=\left[\begin{array}{ccc}\alpha S& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right] \mathrm{and} V=\left[\begin{array}{ccc}\left({\beta }_1+{\beta }_2+{\mu }_1\right)& 0& 0\\ -{\beta }_1& \left(\gamma +{\mu }_2\right)& 0\\ -{\beta }_2& -\gamma & {\mu }_3\end{array}\right]$$

$R_0$ is obtained from the dominant Eigen Value of $F{V}^{-1}$ as $\frac{\alpha }{{\beta }_1+{\beta }_2+{\mu }_1}$.

4.2. Analyzing the Model through R

R is an environment for statistical computing and graphics. Using the above-mentioned mathematical definitions, we formulated an algorithm and then implemented it using the R language. Compared to the previous SIS model, where the infected nodes gradually return to the susceptible state after a particular time, we witnessed a significant change in the number of susceptible and infected ones concerning the addition of two new states—Exposed and Doubter. The computational algorithm was implemented based on the mathematical definitions mentioned in Equations (12)–(15), as defined in the previous section. For evaluating the mathematical model, it is essential to understand how well the model can simulate graphically. The entire model was computationally implemented using Algorithm 1 using the R language.

Algorithm 1 Algorithm for the SEDIS Model
Require: Probability for susceptible to exposed state α
	Probability for the exposed state to the doubter state β1
	Probability for the exposed state to the infected state β2
	Probability for the doubter state to the infected state ϒ
	Probability for the exposed state to the susceptible state µ1
	Probability for the doubter state to the susceptible state µ2
	Probability for the infected state to susceptible state µ3
	State of nodes: State
	Node of the susceptible state = S
	Node of the exposed state = E
	Node of the doubter state = D
	Node of the infected state = I
Ensure: State of nodes after time interval t: state
1:	Generating scale-free network $N=V,E$ adjacent matrix A
2:	Initialization: original state: stat = [S,E,D,I]
3:	while interval < t do
4:	for i = 1: n do
5:	Switch state(st)
6:	case S: transfer to E with α
7:	case E: transfer to D with β1, I with β2 and S with µ1
8:	case D: transfer to I with ϒ and to S with µ2
9:	case I: transfer to S with µ3
10:	end Switch
11:	end for
12:	end while
13:	Return state

The time complexity of the algorithm is $O\left(t\right)$ for the worst-case scenario, where t is the required period for analysis. For the best case and average case, the time complexity remains the same as the algorithm is computed in t time.

5. Properties of the System

For time t, the densities of the susceptible, the exposed, the doubters, and the infected can be represented as $S\left(t\right), E\left(t\right), D\left(t\right)$, and $I\left(t\right)$, respectively. For each node, the specific evolution process can be explained as follows:

$$\frac{dS}{dt}\left(t\right)={\mu }_{3}I\left(t\right)+{\mu }_{2}D\left(t\right)+{\mu }_{1}E\left(t\right)-\alpha S\left(t\right)$$

(16)

$$\frac{dE}{dt}\left(t\right)=\alpha S\left(t\right)-{\mu }_{1}E\left(t\right)-{\beta }_{1}E\left(t\right)-{\beta }_{2}E\left(t\right)$$

(17)

$$\frac{dD}{dt}\left(t\right)={\beta }_{1}E\left(t\right)-\gamma D\left(t\right)-{\mu }_{2}D\left(t\right)$$

(18)

$$\frac{dI}{dt}\left(t\right)=\gamma D\left(t\right)+{\beta }_{2}E\left(t\right)-{\mu }_{3}I\left(t\right)$$

(19)

The number of nodes that each node directly contacts is almost the same and this number approximately obeys the Poisson distribution. Under this assumption, the following contents are discussed in the homogeneous network: $S\left(t\right), E\left(t\right), D\left(t\right)$, and $I\left(t\right)$ denote the densities of the population that are susceptible, exposed, doubters, and infected at time t, respectively. They satisfy the normalization condition:

$$S\left(t\right)+E\left(t\right)+D\left(t\right)+I\left(t\right)=1$$

(20)

Proposition 1.

There exists a solution$\left(S\left(t\right), E\left(t\right), D\left(t\right) I\left(t\right)\right)$to the system.

Proof: 

The space of the mentioned system can be defined as follows:

$$\Omega =\left\{\left(S\left(t\right),E\left(t\right),D\left(t\right),I\left(t\right)\right):S\left(t\right)\ge 0,E\left(t\right)\ge 0,D\left(t\right)\ge 0,I\left(t\right)\ge 0\right\}$$

Let

$$\mathbb{U}=\left\{\left(S\left(t\right),E\left(t\right),D\left(t\right),I\left(t\right)\right):S\left(t\right)\ge 0,E\left(t\right)\ge 0,D\left(t\right)\ge 0,I\left(t\right)\ge 0\right\}$$

then we can say that $\mathbb{U}\subseteq \Omega$ is an open set. The system can also be written as

$\frac{dx}{dt}=f$, where $x=\left(S\left(t\right),E\left(t\right),D\left(t\right),I\left(t\right)\right)$ and $f={\left(f_1,f_2,f_3,f_4\right)}^T$.

Based on this, the Jacobian Matrix $J$ can be obtained as follows:

$$J=\left[\begin{array}{cccc}-\alpha & {\mu }_1& {\mu }_2& {\mu }_3\\ \alpha & -{\mu }_1-{\beta }_1-{\beta }_2& 0& 0\\ 0& {\beta }_1& -\gamma -{\mu }_2& 0\\ 0& {\beta }_2& \gamma & -{\mu }_3\end{array}\right]$$

(21)

From the Matrix derived, it is pretty clear that all the elements of the Jacobian Matrix are continuous. Then $f:\mathbb{U}\to {ℝ}^4$ is a continuously differentiable map. Hence, we can say that there always exists a solution to the derived system. □

Proposition 2.

If$\left(S\left(0\right), E\left(0\right), D\left(0\right), I\left(0\right)\right)$is greater than 0, then the solution$\left(S\left(t\right), E\left(t\right), D\left(t\right), I\left(t\right)\right)$of the system is non-negative for time t > 0.

Proof: 

By contradiction, let us assume that there exists a time $t_1>0$ such that at least any of the $\left(S\left(t_1\right), E\left(t_1\right), D\left(t_1\right), I\left(t_1\right)\right)$ is negative.

We can assert that there exists $t_0\in \left(0,t_1\right)$ based on the continuity of solution such that any of the $\left(S\left(t_0\right), E\left(t_0\right), D\left(t_0\right), I\left(t_0\right)\right)=0$. Without losing generality, here we may assume that $t_0$ is the minimum time with such a property.

If $S\left(t_0\right)=0$, then $\left(E\left(t_0\right),D\left(t_0\right),I\left(t_0\right)\right)\ge 0$ shall be true, by implying the following truth.

$$\frac{ds}{d{t}_{t=t_0}}={\mu }_{3}I\left(t_0\right)+{\mu }_{2}I\left(t_o\right)+{\mu }_{1}E\left(t_0\right)>0$$

(22)

By the continuity of solution, $a>0$ exists such that $S\left(t\right)$ is strictly monotone, rising throughout the interval $\left(t_0-a, t_0+a\right)$.

Now, let us assume that ${\mathrm{t}}_2\in \left(t_0-a,t_0\right)$. We can then say that $I\left(t_2\right)<I\left(t_0\right)=0$ holds.

Since $t\left(0\right)>0$, there exists $t_3\in \left(0,t_2\right)$ such that $I\left(t_3\right)=0$ by Bolzano’s theorem which contradicts the assumption of $t_0$. If $E\left(t_0\right)=0$, then we can say that ${\frac{dE}{dt}}_{t=0}$ holds.

It can hence be verified that $E\left(t\right)=0$ for $t>0$ shall be the solution with the initial value $E\left(0\right)=0$. However, it has to be noted that $E\left(0\right)>0$ and contradicts the uniqueness of the solution. Similarly, it can be illustrated that $I\left(t_0\right)=0$ and $D\left(t_0\right)=0$ often lead to a contradiction. Thus, we can conclude that the solution $\left(S\left(t\right), E\left(t\right), D\left(t\right), I\left(t\right)\right)$ of the system is non-negative for time $t>0$. □

6. Analysis and Results

The model was computed and plotted graphically using the R package based on Algorithm 1. Assuming the probability for transition to all states to be 0.25, the following graphs are obtained.

Figure 7 shows a graphical plot obtained after assuming the probability of transition from each state to be 0.25 for 30 days. The infection probability for each transition is likely to vary depending upon the network, type of rumor, and psychology of individuals. However, we assume here a common possibility for all transitions. Initially, the susceptible population is assumed to comprise 49,999 nodes with one infected node within the network. The observation time is set for 30 days. In the graph, the exposed state is seen to increase exponentially as the infection spreads, while the susceptible population is seen to drop rapidly during the same period. This mainly happens since the susceptible population has moved to the exposed state after meeting the infected node. However, after a predetermined period, this rise is halted by an increase of infected and doubter nodes. We can see that, after a certain amount of time, the exposed count is surpassed by the infected nodes. All the states maintain a constant equilibrium after a specific time. The susceptible population percentage has fallen to approximately 50% while the equilibrium is maintained. Moreover, it is also noted that the doubter population is considerably low compared to all other states. This signifies that less population in the entire network enters into a doubtful state and tries to seek more information about the rumor to enter the infected state or go back to the susceptible state; however, from the graph, we can easily understand that this population comprises 5–10% of the entire population of the network and cannot be neglected.

Table 2. Temporal population for each state.

Day	S	E	D	I
0	49,999	1	0	0
1	40,163	7751	958	1128
2	34,197	10,093	2407	3303
3	30,578	10,398	3496	5527
4	28,383	10,057	4135	7425
5	27,052	9601	4438	8909
6	26,245	9208	4537	10,011
7	25,755	8913	4530	10,801
8	25,458	8709	4478	11,355
9	25,278	8572	4414	11,736
10	25,168	8483	4354	11,995
11	25,102	8427	4303	12,168
12	25,062	8391	4264	12,283
13	25,038	8369	4234	12,359
14	25,023	8355	4213	12,409
15	25,014	8347	4198	12,441
16	25,008	8342	4188	12,462
17	25,005	8338	4181	12,476
18	25,003	8336	4176	12,485
19	25,002	8335	4173	12,490
20	25,001	8334	4171	12,494
21	25,001	8334	4169	12,496
22	25,000	8334	4168	12,497
23	25,000	8334	4168	12,498
24	25,000	8333	4167	12,499
25	25,000	8333	4167	12,499
26	25,000	8333	4167	12,500
27	25,000	8333	4167	12,500
28	25,000	8333	4167	12,500
29	25,000	8333	4167	12,500
30	25,000	8333	4167	12,500

shows the population number at different time intervals for each state from the onset of the epidemic. The table was derived from the computational simulation based on the graph obtained in Figure 7. The total population is set at 50,000 with 49,999 nodes at the initial susceptible state and one node in the exposed state. It can be observed that the susceptible, exposed, doubter, and infected population values tend to become constant from day 10 and a constant value is maintained continuously for all states from day 25.

Figure 8 shows the obtained results after an effective and continuous intervention mechanism is applied to the model starting from the fourth day of an epidemic with an assumption of a decrease of 80% of infections. The figure indicates the need for an intervention mechanism at the beginning of a rumor spread within social networks. After the intervention mechanism is put into place inside the network, it is possible to see an increase in the susceptible population and a drop in the number of infected people.

Table 3. Temporal population during intervention.

Day	S	E	D	I
0	49,999	1	0	0
1	40,163	7751	958	1128
2	34,197	10,093	2407	3303
3	30,578	10,398	3496	5527
4	28,383	10,057	4135	7425
5	27,052	9601	4438	8909
6	31,477	5057	4040	9427
7	34,837	2977	3197	8989
8	37,390	2045	2413	8152
9	39,329	1644	1818	7210
10	40,802	1483	1406	6309
11	41,920	1430	1138	5512
12	42,770	1421	970	4838
13	43,415	1430	869	4285
14	43,906	1444	810	3840
15	44,278	1458	777	3487
16	44,561	1470	759	3209
17	44,776	1481	751	2993
18	44,939	1489	748	2825
19	45,063	1495	747	2695
20	45,157	1500	748	2596
21	45,229	1503	749	2519
22	45,283	1506	750	2461
23	45,324	1508	752	2416
24	45,356	1510	753	2382
25	45,379	1511	754	2356
26	45,397	1512	755	2336
27	45,411	1513	755	2321
28	45,422	1513	756	2309
29	45,430	1514	756	2300
30	45,436	1514	757	2294

shows the simulated results among 50,000 nodes for each state when a continuous intervention mechanism is applied based on Figure 8 with the above-mentioned assumption. From the fifth day forward, we can detect a significant drop in the number of exposed, infected, and doubters when compared to Table 2. It was observed that the susceptible population tends to grow from a fifth of the simulation. Furthermore, even 30 days after the start of the epidemic, a minor increase among the susceptible nodes and a slow reduction among the infected nodes are still discernible.

Implementing an intervention mechanism at the initial stage of an epidemic can help prevent a rumor from being disseminated to a large population within a network and can prevent any detrimental implications from the rumor. We believe that effective intervention mechanisms can be implemented either by the Government or service provider companies, which can effectively control the spread of fake news/rumors within a network.

7. Discussion

In social networks, not all people will be exposed to a rumor. Moreover, from the exposed state, a few people may get confused with the rumor and may move to the doubter state and the infected state. Hence, the exposed and doubter states are quite important when the dynamic social media usage patterns of users are considered. Even though the SIR and SEIR models have defined the Recovered state theoretically, it is not possible to compute, identify, and specify this category of people in real-world social networks [30]. Another flaw with these former models is that an infected population is highly likely to reach 0 after a particular time. This is not possible since a person who has once been infected is highly likely to be susceptible again to another or the same rumor of his/her interest, which still is another psychological factor. However, this proposition is considered in the SIRS and SIS models [31].

The infected and susceptible populations of both existing and proposed models during the starting and ending stage of analysis with the aforementioned conditions is shown in

Table 4. Population for each network model at the beginning and completion of the analysis.

Model	Time t (days)	Susceptible Population	Infected Population
SI Model	0	49,999	1
	30	28	49,972
SIS Model	0	49,999	1
	30	25,000	25,000
SEIS Model	0	49,999	1
	30	16,667	16,666
SEDIS Model	0	49,999	1
	30	25,000	12,500

. By the end of analysis, it can be noted that the susceptible population margins for the SI model are almost zero, with almost the whole population becoming infected. However, when it comes to the SIS model, we can see that the system has an equal number of infected and susceptible people. Additionally, the SEIS model shows an equal number of susceptible, exposed, and infected populations under the same transition rate. Many people are passive in social network discussions, and only a small portion of OSN users participate in group discussions, which supports our analysis and findings. This implies that even when a rumor or a piece of fake news is spread over a network, the majority of users are not exposed to it [32]. The high percentage of inactive social media users also explains why most OSN users will continue to be susceptible at the same time.

OSN users are more likely to take remedial action since they have more time, hence they are less likely to spread false information [33]. It is also anticipated that a large fraction of people who have accounts in OSNs occasionally log off, as already mentioned. This leads us to the conclusion that an infection cannot spread over an entire network at once. Considering these factors, the SEDIS model is defined specifically for online users who show behavioral outcomes when a rumor is being spread within a network. Therefore, the model can also be applied in the E-Commerce and digital marketing domain since rumors and fake reviews play a dominant role in digital and social media marketing. Moreover, using quantitative and qualitative methods, the model can be used to study the rumor dynamics of different communities for a specific period. This can help several communities in the future to prevent the spread of fake news and rumors through social networks.

8. Conclusions and Future Scope

Social network rumors are a concerning fact in modern society, where a good number of the population now depends on social networks for plenty of time daily. Though several rumor propagation mechanisms have been formulated, we have only a few models that consider the human aspect as a separate state for rumor propagation. Considering this fact, we proposed the SEDIS model of rumor propagation by adding the exposed and doubter states. Our implementation of a discrete compartment model based on the attained algorithm shows the importance of an intervention mechanism at the early stage of an epidemic to prevent the implications of rumor dissemination within a network. Our simulation analysis proved that continuous and effective intervention at the early stage of rumor propagation could prevent the epidemic spread within a network. Though the demand for a social network with an open space for free and uncontrolled opinions is high, service provider organizations need to implement intervention mechanisms whenever false news and information propagates within a network.

Though the SEDIS DCM has been implemented, it is essential to test the model with real-world data and compare it with other network epidemic models. It is also necessary to understand how well the model can simulate real-world data. In the present study, we have given an equal transition rate for each parameter within the model. There is a need to understand the population of each state at different times by altering the transition rate. Moreover, it must be understood that different social networks work in different ways. Platforms, such as Facebook, tend to show posts and ads related to users’ interest and the posts they referred to in the recent past. However, social networks, such as Twitter, work differently compared to this phenomenon. This itself tends to alter the exposed, doubter, and infection rates. Hence, it is essential to understand the population and rate of transition for each state at different probabilistic rates. Hence, in our future work, we also plan to compute different probabilistic values to analyze each state at different rates of transmission. Moreover, it is anticipated that communities with a high level of social intelligence are less likely to share misinformation. However, no studies have been conducted to assess the association between social intelligence and rumors in the digital world post-COVID-19 pandemic era. By applying the generated model, we believe that further explorations in identifying the relationship between social intelligence and misinformation in the digital world may yield fruitful results.