Modeling Rumor Spread and Influencer Impact on Social Networks

Social networks act as an indispensable component in the lives of individuals. However, misinformation and fake news are critical challenges in the digital world as people get persuaded towards false information. Though several fake news detection algorithms emerged, epidemic modeling is crucial in understanding the dissemination of fake news, which helps the policyholders to adopt control mechanisms to prevent the curb of infection within the networks. We propose a mathematical model of rumor spread by considering the human nature of selection and social influence within social networks by analyzing the stiffness of different global communities. The positivity of the model was mathematically proved, which proves the validity of the model within the real world. Our real-world data analysis showcases the possibility of a significant increase in fake news and misinformation within online digital networks during the COVID-19 pandemic. A comparative study using real-world data by extracting tweets shows that the proposed model outperforms the existing model. The significance of influencers in digital networks in disseminating rumors is discussed using the proposed model. The results can be used to analyze the impact of misinformation in different communities, which can help the policyholders implement necessary intervention mechanisms at the right time.


I. INTRODUCTION
The digital revolution after the 20th Century impacted the lives of every individual in the world.With the rapid increase in internet usage among all categories of people, social networks gained momentum in people's personal and professional lives.Even though social networks were initially designed to connect people worldwide [1], new features were added that enabled people to use social networks for various tasks, including news gathering and online shopping.However, on the other hand, rumors and misinformation pose a significant threat as they can influence communities during a crisis or pandemic [2], [3].Misinformation can even affect an individual or organization's reputation and alter the course of elections [4], [5], [6].People can easily be influenced by deliberate fake news campaigns, which could even change the governance of a nation.Influencers play a significant role in disseminating fake news within society [7], [8].They can The associate editor coordinating the review of this manuscript and approving it for publication was Giacomo Fiumara .
boost the propagation of misinformation or even curb the spread with necessary interventions.
Studies advocate that people with similar interests share information.These ''echo chambers'' can form a local community, facilitating the misinformation to spread faster in less time.However, on the other hand, an individual can be persuaded into terrorist or extremist organizations by external elements within the networks [9].People tend to get brainwashed and influenced into terrorist groups through persuasion, where an external influencer plays a crucial role in recruitment to extremist groups.Studies advocated increased suicidal activities in social networks with a strong association between cyberbullying and suicide among adolescents [10].
Fake news and false rumors have severe implications for society.The Capitol riots following the US Presidential election are the best example [11].Fake news may sometimes spark unforeseen violence, altering the direction of a society.In 2017, a mob in Myanmar assaulted and murdered Rohingya Muslims after being sparked by false news reports alleging that Rohingya Muslims were raping and murdering Buddhist women [12].Moreover, fake negative news has drastically affected the healthcare sector, especially during the COVID-19 pandemic.Several people were filled with false beliefs during the pandemic, which severely challenged healthcare workers [13].
Several mathematical epidemic models were put forward to understand the spread of misinformation within online social networks.The SI (Susceptible -Exposed) model, SIS (Susceptible -Infected -Susceptible) model, and SIR (Susceptible -Infected -Recovered) models are the most prominent among them [14].The SI model suggests that all the nodes within the network will become infected with no recovery after a stipulated amount of time.Compared to the SI model, the nodes return to the susceptible state in the SIS model, and a recovery mechanism is possible for the SIR model, where an infected node recovers from infection after a stipulated amount of time [14], [15], [16], [17].However, these models state that all susceptible nodes will get infected after a particular time, and the human nature of selection is not considered.
By considering the human aspect of selection, the SEDIS (Susceptible -Exposed -Doubter -Infected -Susceptible) model tends to be a more realistic model of rumor propagation than classic epidemic models [18].In traditional epidemic models, all users are thought to be equally likely to be exposed to a rumor.However, the SEDIS model allows for the potential that some users are more likely to be exposed to rumors than others based on their social network.The approach can be extended into digital networks as a pandemic model, in which disinformation is defined as a pandemic affecting digital network users [19].Various factors influence the propagation of disinformation, including the scale of the digital network, the density of the digital network, the amount of trust between users, and the availability of interventions.
Though several fake news detecting algorithms and methods have been developed [20], [21], the impact of misinformation on different demographic groups needs to be better understood.Understanding the demography and human developmental indices can help anticipate the propagation of fake news through stiffness analysis [22].Governments and policyholders could use this method to establish policies to prevent fake news from spreading through digital networks.Influencers significantly influence the spread of misinformation in digital networks [23].Many influential people are dedicated to disseminating truthful information.On the other hand, whether they are motivated by financial gain or to support a specific agenda, they are more likely to spread falsehoods.Furthermore, influencers, particularly those with a large following, can substantially impact disinformation dissemination [24], [25].
Several fact-checking websites have recently been shown to be a valuable medium for providing critical intervention to avoid the spread of misinformation through social media.The 2016 US Presidential campaign proves the same [26], [27].Fact-checking websites can also assist in slowing the spread of rumors by allowing users to confirm the veracity of material before sharing it.This is significant since rumors may spread swiftly on social media.Websites like Reuters Fact Check, PolitiFact, FactCheck.org,and Snope are significant examples of fact-checking websites that help ordinary people double-check the information they encounter.Factchecking websites can help limit the number of individuals exposed to rumors by offering a mechanism for people to verify information.This highlights the significance of an epidemic model in which human selection is essential.Although the SEDIS model emphasizes this feature, it fails to identify the propagation of a single viral content propagating over social networks.Moreover, the model considers all the misinformation in digital networks as a single entity, making tracking the spread tricky.As a result, it is critical to highlight a new epidemic model and comprehend the stiffness and role of influencers on various networks.
In this paper, we propose a new model derived from the SEDIS model considering a single viral misinformation propagating through digital networks.We study the stiffness of several nations using SIR and SEDIR/SEDIS models to analyze the spread of misinformation on the internet.The impact of influencers on the propagation of fake news in social networks was studied under several conditions.The results explain the need for an effective intervention against social media rumors in communities with higher internet users and lower levels of human development index.

II. SEDIR MODEL FOR SOCIAL NETWORKS
The suggested SEDIR (Susceptible -Exposed -Doubtful -Infected -Recovered) Model and SEDIS (Susceptible -Exposed -Doubtful -Infected -Susceptible) (see [19]) models that take into account the human nature of selection and social effect for a single viral misinformation can be considered for the study.A person is considered susceptible if they are neither infected nor immune to fake news.They become ''Exposed'' if they come into contact with misinformation.If they have doubts about the accuracy of the material, they can move to the ''Doubter'' state.The doubters do not spread misinformation but are highly susceptible to getting infected based on their community and the information they encounter.
For both SEDIS and SEDIR models, a person in the Doubter state has been exposed to disinformation but has not yet determined whether or not to believe it.This may happen for several reasons, including when the information is new to them, when it comes from a source they are unfamiliar with, or when it contradicts facts they already believe.Both the Doubter and Exposed stages are crucial in the SEDIS model because they may significantly influence misinformation transmission.For example, if a doubter chooses to believe and distribute disinformation, it might propagate misinformation to a larger audience.Similarly, if a person exposed to misinformation is exposed to it again, they are more likely to believe it and spread it to others.
The SEDIS model is quite similar to the proposed SEDIR model, with the difference being that the SEDIS model considers misinformation as a single disease within the networks.The model explains the pandemic nature of fake news and misinformation in digital and social networks.This means the model considers all the misinformation, rumors, and fake news as a single disease/infection and evaluates the same (see [19]).Compared to it, the SEDIR model considers a single viral misinformation propagating through the digital network.
Mathematically, the SEDIS model of network pandemic can be explained as follows; The reason why it is −αS and not −αSE for ( 1) and ( 7) is because the models assume that the infection is homogeneously distributed in the population.The algorithm of the proposed SEDIR model based on the equations ( 1) to ( 5) is as follows.For the worst-case situation, the algorithm's time complexity is O(t), where t is the time necessary for analysis.The time complexity stays the same for the best and average cases since the algorithm is computed in t time, which is dependent on the transmission rate.Figure 2 shows the compartmental representation for the SEDIS model.

B. ESTIMATING THE VALIDITY OF THE PROPOSED MODEL
The positivity of a solution or system is significant in the mathematical modeling of rumor propagation since it indicates that all of its components are larger than or equal to zero.This is important for validating the proposed model's validity and consistency with natural laws.It is critical to demonstrate that all system parameters are non-negative in the case of this model, which monitors the population for distinct classifications.Negative values would be meaningless in the context of population monitoring.A negative population size, for example, would be impossible, indicating that the system is invalid and does not obey natural laws.
Lemma 1: From equation ( 1); considering the first part of the model; Now considering equation (2); i.e.; taking the second part of the system; Similarly, considering the third part from (3); Solving the fourth part of the from (4); Finally; solving the final part of the proposed model; ≥ 0 Since all parts of the system provides non-negative solution, we can say that the solution of the system is positive for all t > 0. Since the system is positive at all conditions; we can say that the system is valid at real world.

C. COMPARISON WITH OTHER EXISTING MODELS
The statistical analysis and comparison with different epidemic models were done using the R language on an AMD Ryzen 5 CPU with 8GB RAM.Without interventions, we hypothesized a transmission rate 0.05 from the susceptible state and 0.03 from the recovered state.For the simulation, the population was 300,000 nodes for 100 days.By adding ten infected nodes, we initially considered a stable system free of rumors.The fundamental SI model and the traditional epidemic model with recovery mechanism (i.e., the SIR model) are compared for the study.
Figure 3 depicts the SI Model's outcome.We can extrapolate from the graph that the Susceptible or uninfected population within the network eventually declines to zero within 30 days.This is impossible since not all people on social networks will be active simultaneously in this short time.For larger network communities with thousands of people, even though the taste of all the participants can remain the same, it is less probable to make all people believe fake news forever.the susceptible population reaching zero after a specific time interval.People in social networks may not always act rationally.For example, even if the susceptible ones are aware that they are infected, they may continue to engage with others and take action to protect themselves from infection by checking fact-checking websites and other reliable sources about fake news.Furthermore, the model does not account for all the elements that can contribute to illness spread in social networks, such as individual behavior and asymptomatic carriers.Figure 5 shows the SEDIR model simulation under the same population and transmission rate.We could observe that the susceptible and infected population follows the Power law property [28] in the graph curve and does not reach zero even after 100 days.This signifies the presence of dormant users who rarely are not active on social networks along with the super-spreaders who deliberately share disinformation within the network and can also be the people who have an interest in sharing the fake news even after it loses relevance within the society.

III. STIFFNESS ANALYSIS TO PREDICT THE MISINFORMATION SPREAD
The mathematical approach of stiffness analysis is used to analyze the behavior of differential equations.Stiffness analysis can forecast how quickly a piece of fake news would move through a population in the context of fake news [29].The Jacobian matrix, in the case of fake news, is a function of the social network structure and the characteristics of the fake news itself.The Jacobian matrix's eigenvalues can be used to compute a stiffness index, which measures how stiff the differential equation is.A high stiffness index shows that the differential equation is extremely sensitive to changes in the initial conditions, implying that fake news will spread rapidly among the population.

A. STIFFNESS ANALYSIS USING SIR MODEL
Mathematically, the SIR (Susceptible -Infected -Recovered) model can be defined as; With ϕ as the recovery rate and ω as the infection rate; these parameters are related to two indices commonly used to assess worldwide societies' social, economic, and cultural performance.Generally, we can say that; ϕ = i 10 and ω = h 100 Here i refers to the internet penetration index of the nation and h refers to the HDI of each nation [30], [31], [32].Most often, the value of ϕ is less compared to that of ω since spreading misinformation is easier than reinforcing the truth.The Jacobian matrix of the linearised equation for the SIR model is; whose spectrum consists of one Eigenvalue equal to 0 and two real eigenvalues λ max and λ min .The ratio; gives the stiffness ratio, which is crucial for stiffness analysis [33].Table 1 shows the stiffness analysis of the selected list of nations based on the Human Development Indices, assuming that 10% of the population is initially infected.The results show that countries with higher HDI and internet penetration index are stiffer than nations with lower internet penetration index.This explains the need for an effective intervention mechanism for nations with high internet penetration index.

B. ANALYSIS USING SEDIR MODEL
Considering equations (1) to ( 5), the Jacobian Matrix for the SEDIR model is The basic Reproduction number R 0 for both SEDIS and SEDIR model is which is obtained from the dominant eigenvalue of FV −1  where F is the rate of new infections within the network and V is the transmission rate.Since the R 0 for both models is the same, they produce the same stiffness ratio upon calculation.Hence, we could use both the suggested SEDIR model and the SEDIS model for calculating stiffness by considering the human nature of selection and social intelligence.Table 2 gives the stiffness ratio of the aforementioned nations with similar ϕ and ω as that of 2022.Table 2 shows that the internet penetration ratio plays a crucial role, as nations with higher internet penetration are critical for transmitting fake news.Table 3 ranks countries based on stiffness values from the SIR and SEDIR models.We can observe that the ranking remains the same for all the selected nations under the stiffness ratio of both SIR and SEDIR models.Furthermore, countries such as Nigeria have a higher stiffness ratio when compared to other nations with a higher internet penetration ratio.This is primarily due to poor HDI, which also significantly impacts the rigidity of nations.Similarly, because of their greater HDI, countries like New Zealand have lower stiffness ratings than India.Fig. 6 compares the stiffness values of SIR and SEDIR models.We can observe a lower stiffness value for most nations under the SEDIR model compared to the SIR model.This is mainly because the addition of Doubter and Exposed states under the SEDIR model influences and reduces the stiffness rate.

IV. IMPACT OF COVID-19 PANDEMIC ON NETWORK STIFFNESS
COVID-19 has affected the global population in several ways, especially when people are forced to spend their time indoors due to government-imposed lockdowns.The educational scenario changed to online mode, and professionals were forced to work from home, which increased the use of digital networks worldwide.The changes in internet penetration rates before and after the pandemic validate this.
We randomly selected five nations to put into study, where the internet penetration rate, HDI, and stiffness ratio were observed.It was observed that the internet penetration rate increased during the pandemic period, and the stiffness ratio was observed for selected nations.However, countries with increased HDI and a stable or reduced internet penetration rate witnessed negative growth in stiffness ratio.Apart from the COVID-19 pandemic, several other factors influenced the stiffness of nations.Figure 7 shows the change in stiffness rate for selected countries, and Table 4 explains the scenario.Among the selected nations, India witnessed the most significant growth in stiffness rate and a considerable increase in internet penetration.At the same time, Canada saw a negative shift in stiffness rate and a steady rise in HDI without much change in internet penetration rate.Several factors, including socio-economic and political matters, internally impacted these nations apart from the COVID-19 pandemic, which are discussed below.
Ukraine, which was at war with Russia, suffered significant damage to life and property, resulting in negative growth in HDI [34].However, the number of internet users increased.The same can also be observed in Russia, which is facing sanctions due to the Ukraine war and suffered a decline in HDI in 2022 [35].
At the same time, India underwent the strictest COVID lockdown measures, leading to the markets' downfall during the pandemic period.As more people were forced to work from home, a significant increase in internet penetration rate was observed, from 0.35 to 0.58.Moreover, the unemployment rate reached a record high during the pandemic, significantly affecting the living standards of the people in India [36].
On the other hand, New Zealand had a stable government and stricter COVID measures during the lockdown period.According to the IMF, the economy of New Zealand rebounded strongly in 2021 due to the government's solid economic and health policies [37].Therefore, no significant decline in HDI was observed in New Zealand after the pandemic.Moreover, the increase in internet penetration rate boosted the stiffness rate of the nation.This was also visible in Canada, which had a negative growth in internet penetration rate with no significant change in HDI.
In short, we can conclude that an increase in internet penetration rate without a significant rise in HDI can boost the stiffness rate, which is crucial for spreading misinformation through social networks.

V. CASE STUDY: PLOTTING SIR AND SEDIR MODELS AGAINST ACTUAL DATA
On March 2, 2023, a fire broke out at the Brahmapuram waste treatment plant in Kochi, the largest city of Kerala state in India.It gradually spread to an area of 40 acres, creating panic all around the district [38].The Brahmapuram waste treatment plant is one of the largest waste treatment plants in Kerala, and it processes around 1,200 tons of waste daily.The fire caused widespread environmental damage and disruption to the local community.The toxic fumes engulfed several miles into the air, causing health problems for residents in the area.The incident highlighted the poor state of waste management in Kerala and led to several monthly discussions on social networks [39], [40].
The social media reaction to the Brahmapuram fire reflected the outpouring of rage and frustration felt by many people.The incident emphasized the significance of better trash management in Kochi, as well as the role of social media in increasing awareness of crucial issues.

VI. IMPACT OF INFLUENCERS WITHIN THE NETWORKS
Influencers play a crucial role in disseminating information and misinformation within social networks.Since they have high credibility, people tend to believe the information they share, often making them a target for spreading misinformation within social networks.There are several instances where an influencer spreads fake news, including paid promotions, and spreads information without checking facts and sources of information [41].This can negatively impact people's trust in digital networks and may cause social unrest if the misinformation is critical [42].Based on algorithm 1, we consider several scenarios of a social network community of 50 members using the SEDIR model where an influencer plays a crucial role in disseminating misinformation.The computational analysis was carried out using R, an open-source language enriched by a developer community contributing to its development.R's data processing and visualization capabilities make it particularly well-suited for analyzing epidemiological data, which can sometimes be complicated to manage.In our simulation, we considered influencers as the nodes with more significant connections.This is because they have many followers or subscribers on social media platforms, which allows them to reach a large audience with their content.The values to the parameters of the model are assigned randomly for all cases with α = 0.05, β 1 = 0.05, β 2 = 0.03, λ = 0.03, ϒ = 0.1, µ 1 = 0.01 and µ 2 = 0.03.

A. NETWORK WITH AN INFLUENCER
Influencers are critical in spreading fake news within communities as people have a higher chance to follow and share the content they post within the communities.This is a matter of concern in large communities where an influencer shares a viral post, which can severely affect society.Several possibilities exist where an influencer within a community can initially get infected.Influencers can also create and spread fake news on social networks.Studies advocate that habitual news sharers are responsible for disseminating 30-40% of fake news within the networks [43].Figure 9 shows the simulation of the SEDIR model where the influencer is initially infected, and Figure 10 shows the simulation using the SEDIR model, where the influencer is not initially infected.We can observe from both figures that the exposed population is higher for the network where the influencer acts as the initial spreader than the network where the influencer is not infected.To mathematically prove this, we can again use the SEDIR model equations mentioned in the previous sections.Lemma 2: The exposed rate is higher in a network where the influencer acts as the initial spreader than in networks where the influencer is not initially infected.
Consider the equations (1) to ( 5) to prove this.The only difference between the two cases is the initial conditions.
The rate of change in the number of exposed persons (de/dt) in each network can be used to compare the exposed rate in the two networks.By assuming that the population is homogenous, we can simplify the equations by assuming that S, E, D, I , and R are all percentages of the entire population.
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The equations can then be rewritten as follows: The term αs denotes the number of newly exposed individuals as a result of contact with infected individuals, whereas the term α(1 − i)i denotes the number of newly exposed individuals as a result of contact with either infected or susceptible individuals.For a strongly connected network, the value of i is higher on average, which means that the term α(1 − i)i is higher compared to a weakly connected network.This means that the exposed rate is higher in a network with an influencer getting infected initially compared to a network where the influencer is not initially infected.

B. STRONGLY AND WEAKLY CONNECTED NETWORKS
Strongly and weakly connected networks have significant importance in social networks.While strongly connected networks are vulnerable to misinformation spread since people are more likely to know each other and trust each other, people are more likely to be exposed to new information in weakly connected networks [44], [45].Figure 11 shows the simulation of the SEDIR Model in a strongly connected network of 50 members, and Figure 12 shows the simulation of the SEDIR model under a weakly connected network.We observe that strongly connected networks have a greater infection rate than all other networks in the simulation.However, because of a lack of influencers, a weakly connected network has a relatively low infection rate.There are numerous pathways between any two nodes in a densely connected network.This means that fake news can easily spread from one person to another, even if they are unrelated.Furthermore, strongly connected networks are frequently characterized by high degrees of homophily, implying that misinformation and rumors are more likely to be spread within these networks.Considering G as a strongly connected network and W as a weakly connected network, let f(G) be the probability that an will spread to all nodes in G and let f (W ) be the probability that an infection will spread to all nodes in W. Any node in a strongly connected network can reach any other node.This means that if an infection begins on one node in a strongly connected network, it can spread to all other nodes.
Hence, f(G) > f(W) can be proved.In a strongly connected network, there always exists a path from any node to any other node, and in a weakly connected graph, a node may not reach other nodes.Considering this fact, we can prove that f(G) > f(W).
Lemma 3: The infection in a strongly connected network is higher compared to weakly connected networks.
Let G be a strongly connected network, and W be a weakly connected network.Let p be the probability that an infection (misinformation) will spread to all the networks in a strongly connected network, and q be the rate of infection that will spread to all nodes in a weakly connected network.
We know that p > 0 and q may be equal to 0 at some cases (referring to the above arguments).
Therefore, we have; This demonstrates that the infection rate in a strongly connected network is higher than the infection rate in a weakly connected network.
Similarly, for a network with influencers (who are not strongly connected), there can be cases where an influencer is isolated from the rest of the network due to several scenarios (even if an influencer is blocked or banned from the community/network).In this case, the infection cannot spread to any other nodes, even if it starts at the influencer.
In such cases q = 0 where q is the probability that an infection will spread to all nodes in a network ''I'' with influencers.In such cases we have; which proves that the infection in a strongly connected network is higher than the infection in a network with influencers.

VII. DISCUSSION
Misinformation poses a severe challenge for governments and organizations to move forward.As individuals increasingly rely on social media for news and information during the lockdown period, the COVID-19 epidemic has resulted in an explosion of disinformation.Since there are now several sources of information and people are more prone to believe the networks around them, which made identifying and debunking disinformation much more difficult [46].Assuming that all the people within the network get infected, Figure 13 shows the simulation of the proposed SEDIR model under a high transmission rate for The result signifies that the susceptible population quickly reaches zero value, signifying that everyone will get infected in a network with higher infection levels.This can only happen if everyone within a dense network is active online.Figure 14 shows the simulation of the proposed model under a lower infection rate with a constant recovery rate.The result signifies that only a few people within the network are infected, even though several nodes are exposed to the rumor.
The results further clarify how rumors and misinformation could affect different populations based on the Internet penetration index and HDI.Rapid intervention mechanisms are necessary for networks with higher chances of rapid infection before the misinformation spreads among larger population groups.As a result, in locations with a higher internet penetration ratio, quick disinformation propagation may have a more significant potential.
Assuming the infected population is 10% and the susceptible population is 70% within the network, Table 5 compares stiffness using the SIR model for selected nations based on the values in 2018 (see [29]) and 2022.4, an increase in stiffness values can be observed for all three selected nations, which increased the internet penetration ratio.The results show that the stiffness value increases based on the increase in internet penetration rate for both the SIR and the SEDIS models.

Similar to Table
Table 6 shows the stiffness values of various nations against the educational and crime index using 2022 values.The data from the Human Development Data Center under UNDP (see [30]) and ''Numbeo'' (a Serbian crowd-sourced online database) [47] were considered for the study.
The results signify that both the education and crime index plays a minor role in the stiffness of each nation, even though the educational index is an essential metric in assessing the HDI of a nation.The significance of the internet penetration ratio is still portrayed here, even though the nations with high human development index have comparatively low stiffness ratios.
We can enumerate that nations with low HDI and higher internet penetration rates will be more vulnerable to the spread of misinformation than others.This means that irrespective of the educational and crime index, governments and policyholders should have to make faster intervention mechanisms in the nations with significantly high internet penetration rates.Social media sites could implement several intervention methods, such as labeling fake news, to prevent its future proliferation across larger populations.Applying required interventions to stop rumors at the proper moment, such as multi-feature rumor blocking [48], would offer productive results in limiting the spread of fake news among a wide audience on social networks.Several strategies may be used to limit rumor propagation, where influencers and communities play an essential role.When we wish to choose a seed set of users to stop the propagation of a rumor in a social network, techniques like the CC-DIM problem can be utilized to choose a seed set of users that is varied across several community structures, such as interests, demographics, or places [49].This will assist in preventing the misinformation from spreading throughout the social network.Identifying a broad group of susceptible and exposed individuals across multiple community structures for a viral disinformation campaign can be beneficial in stopping the fake news from spreading further.Alternatively, governments might request that social media platforms remove misinformation that has the potential to spread and cause societal violence.Moreover, effective measures should be adopted among nations to improve the HDI and provide awareness of rumors before they spread among a large population.

VIII. CONCLUSION AND FUTURE SCOPE
Social network rumors have a significant impact on society.From damaging trust for individuals and organizations to undermining the democratic process of the nation, social network rumors pose a serious issue that needs to be addressed.However, this can be addressed by creating awareness, checking the facts, and adopting policies to prevent misinformation among a wider audience.Recently, the importance of factchecking websites increased sharply among all categories of people.The significance of a Doubter state in mathematical epidemic modeling of rumors arises here.
In this work, we suggested a new mathematical epidemic model and analyzed the stiffness of various nations, which can significantly contribute to the fight against misinformation.By observing, we identified that the internet penetration rate plays a critical role in the stiffness of each nation.Apart from HDI and internet penetration rate, the content of rumors, the structure of social networking platforms, and the behavior of users play a crucial role that can contribute to the stiffness of a social network rumor.Our future goal is to address this part, which could help social networks to curb misinformation spread.
Though influencers play a vital role in the spread of misinformation, the density of the network and the connections among the community play another crucial role.Rumors tend to spread among the communities which are strongly connected compared to the others.On the other hand, identifying trustworthy influencers in a closely connected network can help to promote accurate information and dispel false rumors, which is another topic for future research.Our future goal is extensive research on stiffness analysis to analyze the spread of fake news in social media by understanding the influencers and their characteristics in different social networking platforms.Governments and policymakers can use the current study's findings to determine the best time to implement essential interventions before a rumor goes viral online.This can mitigate the negative consequences of misinformation and promote public awareness.
The ''Infected'' state refers to the people who propagate falsehoods.Individuals in the ''Restrained'' state for the SEDIR model have lost interest in information over time.Those who are no longer spreaders gradually become Restrained.Figure1depicts the discrete compartmental diagram of the SEDIR Model.

FIGURE 3 .
FIGURE 3. Simulation with the SI model.

Figure 4
Figure 4 depicts the SIR model for a similar population under the same transmission rate.We could also observe

FIGURE 4 .
FIGURE 4. Simulation with the SIR model.

FIGURE 6 .
FIGURE 6. Plotting the stiffness ratio of SIR and SEDIR model for selected nations.

FIGURE 7 .
FIGURE 7. Stiffness ratio of 2018 and 2022 for selected nations using SEDIR model.

Fig. 8
Fig.8shows the temporal plotting of actual data against SEDIR and SIR models (α = 0.0078, β = 0.087) among the dataset of 1600 individuals on Twitter.It can be observed that the SEDIR model fits well to the graph compared to the conventional SIR model.

FIGURE 8 .
FIGURE 8. Curve fitting of SEDIR and SIR models on Kochi Brahmapuram fire dataset.

FIGURE 10 .
FIGURE 10.SEDIR model with an influencer not infected initially.

FIGURE 11 .
FIGURE 11.SEDIR model under a strongly connected network.

FIGURE 12 .
FIGURE 12. SEDIR model under a weakly connected network.

TABLE 1 .
Stiffness analysis using SIR model.

TABLE 3 .
Ranking of nations based on the values from Table1 and 2.

TABLE 4 .
Change in internet penetration rate, HDI and stiffness rate before and after COVID-19 PANDEMIC.

TABLE 5 .
Comparison of stiffness values before and after Covid-19 pandemic using SIR model.

TABLE 6 .
Stiffness ratio against educational and crime index of 2022.