A public opinion propagation model for technological disasters

Technological disasters, defined as incidents involving failures in human-made systems¹, such as industrial, transportation, and structural breakdowns². Disasters result in significant economic losses and generate widespread public opinion online. If public opinion is not effectively managed, it can lead to secondary disasters and further losses. With rapid advancements in network technologies, the challenge of controlling public opinion has become more complex, as information can spread more quickly and extensively³. The easy accessibility of online platforms has allowed netizens to freely express their views. However, verifying the accuracy of publicly shared information is often difficult, and studies have shown that false information spreads faster and further than true information on social media⁴, complicating governance efforts. Online public opinion can negatively affect economic development⁵, and social stability⁶, drawing significant attention from researchers concerned with public opinion management during major emergencies.

Technological disasters, such as factory fires, traffic accidents, and terrorist attacks, are typically human-made⁷, meaning they can be prevented by addressing underlying risks. Disasters resulting from human activities, such as chemical or nuclear explosions, major structural collapses, and hazardous material spills, are often categorized as technological disasters⁸. In such events, public opinion has unique characteristics. The public’s attention often centers on the government’s response, especially regarding the investigation of the disaster’s causes and the attribution of accountability. When managed effectively, the government may receive praise; however, poor handling can lead to public dissatisfaction, which undermines governmental authority. Given the increasing frequency of technological disasters worldwide, it is essential to study the management of online public opinion in these contexts.

According to the 51st China Internet Development Statistics Report released on March 2, 2023, China had 1.067 billion internet users by the end of 2022, with an internet penetration rate of approximately 75.6%. With improvements in internet speed, portability, and accessibility, social networking platforms have become more influential in shaping public opinion during emergencies. Due to the intense public interest in technological disasters, information about such events can spread rapidly online. If these emergencies are mishandled or left unaddressed, public opinion can escalate, questioning the credibility of the responsible authorities. This can exacerbate public sentiment and even jeopardize public safety and government credibility⁹. Consequently, it is imperative for responsible authorities, including the government, to effectively guide and manage online public opinion in the wake of technological disasters.

There have been many significant achievements in the field of technological disaster research. However, few studies focus on public opinion related to technological disasters. Given the unique characteristics of technological disasters, specialized research is needed to understand and control the dynamics of public opinion. This study aims to examine the spread and evolution of public opinion related to technological disasters by analyzing the key factors that influence it. A public opinion dissemination model is proposed, incorporating factors such as disaster severity, government response, accountability, response speed, and the impact of both positive and negative media content. The model is used to analyze the primary factors that trigger online public opinion outbreaks, and to compare different government control strategies for public opinion. Finally, the model is applied to a case study, providing insights into governance strategies.

By analyzing the evolution of public opinion in the context of technological disasters, this study aims to help governments and authorities better manage public opinion and design effective interventions. The remainder of this paper is organized as follows: Section “Related work” reviews related studies, Section “Modeling” develops a public opinion dissemination model for technological disasters based on the SEIR model, Section “Findings” presents the simulation and analysis of the model, Section “Empirical analysis and results” discusses the case study and corresponding policies and recommendations, and Section “Conclusion” concludes the study.

The dissemination and control of online public opinion have been widely studied, leading to the development of two primary models for online public opinion propagation: differential dynamics models, and opinion dynamics models. In 1927, Kermack and McKendrick proposed the SIR model for epidemics¹⁰, which has since been adapted in various ways to model disease transmission. Given the similarities between information and disease spread, many researchers have applied disease transmission models to study the spread of information. For instance, Sudbury was one of the first to use the classic SIR model to study the spread of rumors¹¹. Xu et al.¹² proposed a dynamic propagation model based on the SIR framework, systematically analyzing recurring public opinion. Han et al.¹³ quantitatively studied the impact of user behavior on information dissemination using the SIR propagation model and empirical data. Zhao, et al.¹⁴ developed an variant of SIR public opinion propagation model that incorporated the role of opinion leaders and user interest in microblog content. Xu et al. used the SIR model to examine how public opinion on topics evolves over time on social media¹⁵. Zhang et al.¹⁶ introduced a new public opinion dissemination model based on the SIR model, considering a discussion mechanism.

Unlike disease spread, the dissemination of information depends on individuals’ belief in the information before they choose to share it. In disease transmission, susceptible individuals are exposed to the virus and have no choice but to become infected. In contrast, when public opinion spreads, people can decide whether or not to share the information after receiving it. Consequently, some individuals may hesitate to share information, especially if it pertains to unusual or sensitive topics, such as major disasters. In such cases, people often seek verification through multiple sources or use their own judgment to assess the information’s truthfulness. This hesitation, where people are aware of the information but reluctant to disseminate it, is similar to the exposed(latent) state in disease transmission models¹⁷. Therefore, the SEIR model is better suited to describe the propagation of public opinion, as it divides individuals into four distinct stages: susceptible, exposed, infected, and recovered^18,19. This model provides a more accurate and nuanced depiction of how public opinion spreads.

Several studies have contributed to public opinion dissemination using the SEIR model. Geng et al.²⁰ developed a SEIR model to simulate information transmission trends in the multilayer networks. Govindankutty et al.²¹ proposed an improved SEIR model considering the sentiment of solitary misinformation within networks and human cognitive characteristics. Geng et al.²² further extended the SEIR model by integrating the influence of network and government media and considering the interactions between different emotions. Yin et al.²³ introduced the E-SLFI model based on SEIR model, which describes the dynamics of sentiment propagation across ternary polarities. Opinion dynamics theory, first proposed by French Jr. in 1956 and later expanded by DeGroot in 1974, has also led to the development of numerous opinion dynamics models, including the Hegselmann-Krause model. Li et al.²⁴ combined the HK model with the SEIR disease transmission model to create the HK-SEIR public opinion evolution model.

In the field of public opinion model adaptation, Yuan et al.²⁵ simulated group attitude evolution by dynamically adjusting network structures based on the SIR model. Wu et al.²⁶ proposed a depolarization model that categorizes individuals based on their self-belief and suggests adaptive strategies for depolarization. Giráldez-Cru et al.²⁷ studied superstar rankings in the film industry, using real data to simulate mass communication processes. Guo et al.²⁸ developed an opinion dynamics model based on link prediction to improve the relationship and opinion update processes. Research on the adaptability of public opinion models mainly focuses on complex networks, opinion dynamics models, and machine learning techniques.

The study of public opinion on disasters has focused on the evolution, influencing factors, prediction, and management of disaster-related public opinion, as well as the responses of various stakeholders. Based on the data of Sina Weibo after the two earthquake cases, Wang et al.²⁹ analyzed Sina Weibo data from two earthquake cases, classifying public opinion topics and sentiment, and comparing public opinion before and after government interventions. Wu et al.³⁰ proposed a mixed-integer nonlinear programming model to mine tweet data on Weibo and respond to public demands during typhoon disasters. Wang et al analyzed the public’s response to the 7.20 Henan rainstorm in China using the LDA model and sentiment analysis, examining both the evolution of the disaster itself and public opinion^31,32. Qu et al.³³ also also studied the Henan rainstorm disaster and used social media data mining to identify key topics, proposing urban flood prevention measures. Dai et al.³⁴ analyzed the emotional and topical dynamics of the MU5735 air crash, creating a visual public opinion map to assist researchers and government administrators in understanding public opinion during disasters. Zhao et al.³⁵ developed an information diffusion model for subway emergencies based on the bounded confidence model. Muddiman et al.³⁶ highlighted the cognitive biases of online audiences during major emergencies, recommending enhanced public opinion management strategies. Zhang et al.³⁷ constructed sentiment classification models to analyze public opinion in response to food safety incidents. These studies primarily focus on analyzing specific disaster cases and proposing countermeasures for public opinion management. However, few models address the commonalities across different types of disasters.

Since the outbreak of COVID-19, numerous studies have examined public opinion related to the pandemic. Saricali et al. analyzed widespread discussions on social media about quarantine policies³⁸ and vaccines³⁹, while Jelodar et al.⁴⁰ used natural language processing techniques to categorize COVID-19-related internet reviews and propose practical strategies for managing adverse public opinion. Calandra et al.⁴¹ discussed the negative impact of COVID-19-related information on real life and the internet using artificial intelligence. Cheng et al.⁴² proposed an online public opinion stochastic resonance model to analyze resonance patterns in public health emergencies. Du et al.⁴³ studied the delayed and combinatory responses of online public opinion to the intensity of the pandemic. Xie et al.⁴⁴ developed a double-hazard opinion dynamics model for simultaneous natural disasters and public health events. Wang et al.⁴⁵ proposed a systematic approach to collect and analyze public opinion data from major social media platforms during the COVID-19 epidemic. Zeng et al.⁴⁶ introduced a multimodal fine-grained negative sentiment recognition model based on graph convolutional networks and ensemble learning to improve sentiment recognition during public health emergencies. Liu et al.⁴⁷ evaluated the internet public opinion risk level during local COVID-19 outbreaks. These studies provide valuable insights for better managing online public opinion during emergencies.

In summary, previous research on disaster-related public opinion has laid the foundation for our current study. However, much of the research has focused on specific case studies, there is extensive research on public opinion related to public health events and emergencies, there has been limited investigation into the propagation of public opinion during technological disasters. Technological disasters are frequent “hot topics” on the internet, and if not managed effectively, these discussions can threaten public safety and social order^9,34. Given that the public often attributes blame for life and property losses in technological disasters to political failure and government irresponsibility, it is crucial to develop timely and effective strategies for controlling information dissemination. Therefore, this study aims to explore the characteristics and evolution of online public opinion in the context of technological disasters. First, an SEIR-based model is developed that integrates disaster severity, government accountability and response, and the impact of positive and negative media content on public opinion. A case study is then presented to demonstrate the applicability of the proposed model. Finally, based on the identified characteristics of online public opinion, response strategies are proposed to manage and mitigate its impact.

The characteristics of public opinion on technological disasters and the factors influencing the spread of public opinion are first identified by distinguishing four state of individuals. Subsequently, the propagation thresholds, equilibrium points, and the stability of these equilibrium points are analyzed.

Public opinion influencing factors

A key aspect to consider is that technological disasters are human-made incidents, often resulting in direct casualties. Unlike other types of disasters, technological disasters are closely linked to questions of responsibility and post-disaster relief efforts. During such events, public emotions tend to run high. As these disasters are caused by human error, negligence, or corruption, while the public expresses sympathy for the victims, there is also significant resentment and antagonism toward those deemed responsible.

Technological disasters often attract substantial attention and generate significant online public discourse. Based on the characteristics of public opinion in the context of technological disasters, several influencing factors are analyzed. While many factors contribute to the formation of public opinion in the online media environment, four key factors are particularly influential in shaping public opinion regarding technological disasters.

• Technological disaster severity

  Technological disasters can have serious consequences, both directly and indirectly, such as casualties, property damage, and disruptions to daily life. Therefore, the severity of the disaster are positively correlated with the spread of public opinion. Immediately following a technological disaster, the public is eager to obtain any information related to the event. The greater the severity of the disaster, the more intense the public’s attention and the greater the spread of public opinion. In other words, the more severe the disaster, the higher the public interest and the more widespread the public discourse. Conversely, if the human and economic losses are relatively minor, there is less public discussion, and the conversation tends to subside quickly.

• Government response

  Following a technological disaster, the government’s response is crucial to post-disaster relief and accountability, both of which significantly affect the development and spread of public opinion.

  1. (1)

     Post-disaster relief effect. As the primary authority responsible for disaster management, the government’s handling of the emergency response is of significant public interest. This includes efforts such as treating the injured, restoring infrastructure, and providing comfort to the affected families. If the response is inadequate, the public’s dissatisfaction is likely to spread on online platforms, which can fuel further discussion and extend the duration of public debate.

  2. (2)

     Response speed. The responsible authorities and government must take swift and decisive action to address post-disaster issues and keep the public informed. When a disaster occurs, the public often speculates about its cause and assigns blame. Therefore, if the response is delayed or lacks urgency, it can fuel public speculation, which in turn can undermine government credibility.

• Degree of accountability

  Technological disasters are often caused by improper or illegal actions. Because these events can result in significant loss of life and property, the public seeks clarity about the cause of the disaster and accountability. If no individual or organization is held accountable, the responsible authorities must step in to provide timely information. The effectiveness of the authorities’ communication, the identification of those responsible, and the severity of any punishments play critical roles in shaping and amplifying public opinion.

• Positive and negative media content

  Following a technological disaster, media, especially, social media, plays a crucial role in shaping public opinion by providing real-time updates, expressing sympathy for the victims, and speculating on the causes and consequences. The tone or sentiment of media content-whether positive or negative-can significantly influence the spread of public opinion, alter people’s behaviors, perceptions, and judgments, and ultimately affect the public’s response to the disaster.

  1. (1)

     Positive media content. Positive media content could include supportive, encouraging, or optimistic narratives, and typically has a constructive or beneficial tone. It helps manage disaster-related information effectively. It includes the timely and positively release of first-hand information about the extent of the damage, ongoing rescue efforts, and the actions taken by responsible authorities. When such updates are communicated swiftly, it is more likely that media content will be seen as favorable, reinforcing public trust in the government’s ability to manage the disaster.

  2. (2)

     Negative media content. Negative media content typically carries a critical, pessimistic, or disapproving tone. It often involves reporting on failures, shortcomings, or negative aspects of the disaster response, such as poor government handling or mismanagement. For example, media content may focus on significant losses, including fatalities, injuries, and economic damage, or on investigations and blame directed at those responsible, questioning their accountability or negligence. Such negative media content can lead to a surge in public dissatisfaction, erode trust in government institutions, and fuel speculation about the government’s competence and ability to manage the crisis effectively.

The influencing factors for public opinion on technological disasters are shown in Fig. 1.

Influencing factors for public opinion on technological disasters.

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Model formulation

To analyze the evolution of public opinion on technological disasters, the public can be divided into four categories: the unaware S(t), the lurkers E(t), the spreaders I(t), and the stiflers R(t).

1. (1)

   The unaware S(t) are netizens who have no knowledge of the disaster but are easily influenced by disaster-related information, making them susceptible to public opinion.

2. (2)

   The lurkers E(t) are netizens who have received some information but have not yet decided whether to disseminate it, as they are still assessing its veracity.

3. (3)

   The spreaders I(t) are netizens who have received information and actively disseminate it.

4. (4)

   The stiflers R(t) are netizens who are either no longer concerned with the disaster or have received information but choose not to share it.

   To construct the SEIR propagation model, the following assumptions are made:

5. (1)

   Individuals who enter social networks are initially considered unaware.

6. (2)

   Due to the influence of new knowledge or changes in attitudes toward the disaster, individuals in any of the four categories-unaware, lurkers, spreaders, and stiflers leave the public opinion system with a probability c.

Information dissemination occurs through interactions with others, and at any given time, individuals exist in one of the four states: S(t), E(t), I(t), or R(t). It is also assumed that, at any moment, S(t), E(t), I(t), and R(t) are continuously differentiable functions. In the SEIR model, the relationships between the different netizen types are illustrated in Fig. 2. The definitions and meanings of the parameters in the system are provided in the Notations.

Based on the classic SEIR model, a public opinion-spreading model was developed that incorporates factors specific to technological disasters. The spreading rules are illustrated in Fig. 2 and are summarized as follows:

1. (1)

   If the damage is severe, it is expected that individuals in the unaware group will actively seek out contact with a spreader to obtain relevant information, with a transmission rate of \(b=b_{1}D\), where D represents the severity of the disaster. The rate of information dissemination is influenced by both positive and negative content. \(e^{ - {a_1}I}\) reflects a decrease in the transmission rate due to positive content, where \(a_{1}\) represents the impact of positive content on the dissemination rate. On the other hand, \(e^{ {a_2}I}\) indicates that the information dissemination rate increases due to the presence of negative content, where \(a_{2}\) represents the influence of negative content on the transmission rate. a represents the percentage of new individuals entering the system per unit of time, and c is the removal rate, i.e., the percentage of people leaving the system per unit of time. Therefore, the rate of change of the unaware population, expressed as a differential equation, is given by:

   $$\begin{aligned} \frac{dS(t)}{dt}=a - bSI{e^{-{a_1}I + {a_2}I}} - cS \end{aligned}$$

   (1)
2. (2)

   The unaware become lurkers with a probability of \(b{e^{ - {a_1}I + {a_2}I}},\) while lurkers transition to spreaders with a probability of d. Participants can exit the system with a probability of c. The probability d is influenced by the government’s response. A timely and effective “post-event” government response can positively regulate the conversion rate⁴⁸. If the government performs well in post-disaster relief and responds swiftly, it helps lurkers assess the veracity of the information, thereby reducing the likelihood of lurkers becoming spreaders. According to Eq. (1), the \(\frac{dE(t)}{dt}\) is proportional to the number of lurkers E(t), so:

   $$\begin{aligned} \frac{dE(t)}{dt}=bSI{e^{ - {a_1}I + {a_2}I}} - dE - cE \end{aligned}$$

   (2)
3. (3)

   Spreaders transition to stiflers with a probability of g, meaning that their thoughts and actions change, and they cease to spread information. This transition rate is influenced by the degree of accountability. In the context of a disaster, the concept of downward accountability by the government becomes particularly important⁴⁹. If the cause of the disaster is identified and those responsible are held appropriately accountable, the public’s attention to the issue diminishes. In other words, when the government actively pursues accountability, more spreaders are likely to become stiflers. Therefore, based on Eq. (2), the \(\frac{dI(t)}{dt}\) is proportional to the number of spreaders I(t), so:

   $$\begin{aligned} \frac{dI(t)}{dt}=dE - gI - cI \end{aligned}$$

   (3)
4. (4)

   The stiflers leave from the system at a constant rate c. From (3), there is

   $$\begin{aligned} \frac{dR(t)}{dt}=gI - cR \end{aligned}$$

   (4)

Structure of public opinion spreading process.

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By integrating Eqs. (1) through (4), the following global model for the public opinion spreading process in technological disasters is formulated:

The proposed model accounts for disaster severity, government response, accountability, and the influence of both positive and negative content. It is established based on the unique characteristics of technological disasters, providing a more accurate representation of the evolution of public opinion in response to such events. In the model, b is related to disaster severity; the more severe the disaster, the larger the value of b. d is influenced by government response, including the post-disaster relief effect and response speed. Notably, the better the government response, the smaller the value of d, meaning fewer latent individuals will transition into the spreader. g is determined by the level of accountability; as the level of accountability increases, more individuals in the spreading category will be converted into stiflers. \(a_{1}\) is related to positive media content, and \(a_{2}\) is related to negative media content; the more media content available for each type, the larger the corresponding value.

The above model takes into account the spread of public opinion during technological disasters. To investigate the mechanisms influencing the technological disaster factors, the properties of the model, including information propagation thresholds and the stability of equilibrium points, are analyzed.

Public opinion propagation equilibrium and propagation threshold

In this part, the model equilibrium point and propagation threshold of the model are calculated.

• Model equilibrium points

Before calculating the propagation threshold and propagation equilibrium points, the model solution boundedness was analyzed.

Lemma 1

\(A = \{ (S,E,I,R)|S,E,I,R \ge 0,S + E + I + R \le \frac{a}{c}\}\) is a positive invariant set of the system (5) and is also its attracting set.

Proof

\(N = S + E + I + R,\) deriving it to t(time), and it is obtained by the system (5)

So \(N = {e^{ - ct}}(N(0) - \frac{a}{c}) + \frac{a}{c},\) and for any \(N(0) \in A,\) there is

So A is the positive invariant set of the system (5).

If \(N(0) \in A,\) because A is a positive invariant set for the system of system (5), it is known that \(0 \le N \le a/c.\)

If \(N(0) \notin A,\) as indicated in the previous analysis, when \(t \rightarrow + \infty,\) there is \(0 \le N \le a/c.\)

Therefore, A is the attracting set for the system (5). In summary, A is the positive and invariant set(5) and attracting set for the system. \(\square\)

From Lemma 1, the equilibrium points of the model lie within the bounded region A. To determine the equilibrium points, set the equations in 5 equal to zero, yielding:

Two types of equilibrium points are identified: (1)Free-propagation equilibrium point: When both the lurkers E and spreaders I are zero, no information dissemination occurs. In this case, the equilibrium point is \({p_0} = (\frac{a}{c},0,0,0)\) representing the model’s free-propagation equilibrium. (2)Propagation equilibrium point: When \(I\ne 0,\) the solution to Eq. (8) becomes more complex. Therefore, the propagation equilibrium point is denoted as \({p^*}=({S^*},{E^*},{I^*}, {R^*}),\) where \(S = \frac{{(c + d)(g + c)}}{{bd}}{e^{({a_1} - {a_2})I^{*}}},\) \(E^{*}= \frac{(g+c)I^{*}}{d},\) \(R^{*}= \frac{gI^{*}}{c},\) \(I^{*}\) can determined by the follow equation:

Since the free-propagation equilibrium point represents an ideal state where public opinion information does not spread, the focus is placed on the propagation equilibrium point.

Theorem 1

When \(R_{0}> 1,\) there is a unique positive equilibrium point \({p^*}=({S^*}, {E^*}, {I^*}, {R^*})\) in the system (8).

Proof

From the second equation of the system (8), we get

This is obtained by substituting the above equation into the third equation for (8)

This is obtained by substituting the above equation into the first equation for (8),

This is obtained through calculation

The following discussion is organized into two scenarios:

1. (1)

   When \({a_1} - {a_2} > 0,\) \(F'(I) < 0,\) that is, F(I) is a subtraction function. When \({R_0} > 1,\) \(F(0) > 0,\) so (12) has a unique positive root \({p^*}.\)

2. (2)

   When \({a_1} - {a_2} < 0,\)

3. (a)

   If \(0 < {a_2} - {a_1} \le \frac{b}{c},\) then \(F'(I) < 0,\) when \({R_0} > 1,\) \(F(0) > 0,\) so (12) has a unique positive root \({p^*}.\)

4. (b)

   If \({a_2} - {a_1} > \frac{b}{c},\) the fixed point can be derived from \(F'(I) = 0,\) and it is

   $$\begin{aligned} {I_0} = \frac{1}{{{a_1} - {a_2}}}\ln \frac{b}{{c({a_2} - {a_1})}} \end{aligned}$$

   (14)

Because \(F''(I) < 0,\) \(F'(I)\) is a single subtraction function. (i) When \(0 < I \le {I_0},\) \(F'(I) > F'({I_0}) = 0,\) when \({R_0} > 1,\) \(F(0) > 0,\) so (12) has a unique positive root.(ii) When \(I > {I_0},\) \(F'(I) < F'({I_0}) = 0,\) F(I) is a single subtraction function, and when \(I \rightarrow + \infty,\) \(F(I) \rightarrow - \infty,\) so (12) has a unique positive root \({p^*}\) when \({R_0}>1.\)

Based on the above analysis, it is proven that when \({R_0} > 1,\) there is a unique positive equilibrium point \({p^*}({S^*},{E^*},{I^*})\) in the system (5). \(\square\)

• Propagation threshold

Based on the context of the model, the equilibrium points are considered to lie within the bounded region \(A = \{ (S,E,I,R)|S,E,I,R \ge 0,S + E + I + R \le 1\}.\) In infectious disease transmission studies, the basic reproduction number \({R_0}\) is a critical threshold for determining whether an infectious disease will spread. When \({R_0}>1,\) an outbreak is expected to occur within a certain range; when \({R_0} \le 1,\) the disease spread naturally dies out over time. Similarly, in the context of technological disasters, the propagation threshold \({R_0}\) for online public opinion spread is discussed below.

To calculate the basic reproduction number for the proposed model, the next-generation matrix method is employed⁵⁰. The decomposition of Eq. (5) into the components \({\mathscr {F}}_{\mathcalligra {i}}\) and \({\mathscr {V}}_{\mathcalligra {i}}\) is presented. The transmission model is defined with non-negative initial conditions and the following system of equations:

Let \(\mathscr {F}=(\mathscr {F}_{1}, \mathscr {F}_{2}, \mathscr {F}_{3}, \mathscr {F}_{4})^{'},\) \(\mathscr {V}=(\mathscr {V}_{1}, \mathscr {V}_{2}, \mathscr {V}_{3}, \mathscr {V}_{4})^{'},\) then we can get

Since individuals who have received information and either intend to spread it or are actively spreading it are categorized as lurkers and spreaders, \(\mathscr {F}_{2}, \mathscr {F}_{3},\) \(\mathscr {V}_{2},\) and \(\mathscr {V}_{3}\) are considered, and \(\mathscr {F}_{1}, \mathscr {F}_{4}, \mathscr {V}_{1}, and \mathscr {V}_{4}\) are not considered. Substituting \(p_{0}= (r/q, 0, 0, 0),\) a straightforward calculation gives;

At the same time, the inverse matrix of V can be calculated,

Then

The next-generation matrix method gives the basic reproduction number as a spectral radius of matrix \(FV^{-1}\)⁵⁰. The basic reproduction number of proposed model is given by trace of the matrix \(FV^{-1},\) which is marked as \(\rho (FV^{-1})\) ), and it is called the propagation threshold

From the formula (22), the propagation threshold is influenced by several factors, including the immigration rate a, the removal rate c, the parameter b related to the transmission rate and disaster severity, as well as the lurker conversion rate d and spreader conversion rate g. The rate d is affected by the government’s response, while g is influenced by level of accountability. However, neither the positive content coefficient \(a_{1}\) nor the negative content coefficient \(a_{2}\) have an impact on the propagation threshold. When \({R_0} \le 1,\) the system only has free-propagation equilibrium points, and public opinion in the system gradually fades out over time. On the other hand, when \({R_0} > 1,\) the model possesses a propagation equilibrium point, and without proper guidance and control, public opinion can explode within the system.

Equilibrium points and stability analysis

In the following, the model stability at the equilibrium points \({p_0}\) and \(p^{*}\) are discussed.

Theorem 2

Under the condition that \({a_2} < {a_1},\) when \({R_0} < 1,\) the equilibrium point \({p_0} = (\frac{a}{c},0,0,0)\) in the system (5) is globally asymptotically stable within A. When \({R_0} > 1,\) the system equilibrium point is unstable.

Proof

Given the liapunov function

where \(\theta = \frac{{a(c + d)(g + c)}}{{gdc}}(1 - {R_0}) - \varepsilon,\) \(\varepsilon\) is an arbitrarily small positive number. When \({R_0} < 1,\) L is a positive definite function.

L is derived along the system (5) to time t, it can be gotten

when \({a_2} < {a_1},\)

\(\theta = \frac{{a(c + d)(g + c)}}{{gdc}}(1 - {R_0}) - \varepsilon\) is substituted into the above formula, then we can get \(\begin{array}{ll} {a^2}bd - [ac(c + d)(g + c) - \theta gdc]{e^{({a_1} - {a_2})I}}& ~={a^2}bd - [ac(c + d)(g + c){R_0} + \varepsilon gd]{e^{({a_1} - {a_2})I}}\\ & ~\le {a^2}bd - ac(c + d)(g + c){R_0} = 0 \end{array}\)

So

Apparently, \(\frac{{dL}}{{dt}} = 0\) if and only if \((S,E,I,R) = (\frac{a}{c},0,0,0).\) The LaSalle invariant principle⁵¹ states that the boundary equilibrium point \({p_0}\) is globally attracting, therefore, as \(L\left( {\mathrm{t}} \right)\) is a positively definite infinity function, so \({p_0}\) is globally asymptotically stable.

The following proof demonstrates that when \({R_0} > 1,\) the boundary equilibrium point \({p_0}\) is unstable.

The Jacobian matrix for the system at the boundary equilibrium point is:

Its eigenvalues are

Apparently, \({\lambda _1},{\lambda _2},{\lambda _3}\) are less than 0. When \({R_0} > 1,\) there is \({R_0} - 1 > 0,\) namely

\(abd - ({c^3} + d{c^2} + g{c^2} + dgc) > 0,\) so \({\lambda _4} = \frac{{\sqrt{{d^2} - 2dg + \frac{{4abd}}{c} + {g^2}} - (2c + d + g)}}{2}\) is greater than 0. Using the method for determining instability⁵², because at least one of the eigenvalue is greater than 0, the free-propagation \({p_0}\) is unstable. \(\square\)

It is challenging to theoretically prove the stability of the propagation equilibrium point due to the complexity of the proposed model. Figure 3 illustrates the dissemination process of the system (5) under different initial conditions when \({R_0} > 1.\) The simulation analysis reveals that small changes in the initial values do not result in significant variations in the corresponding solutions, suggesting that the propagation equilibrium point is stable. This indicates that the technological disaster public opinion dissemination system is not highly sensitive to initial conditions, and the long-term trend of public opinion can be reliably predicted using the given system.

Dissemination process for four groups under the different initial conditions: (S(0),E(0),I(0),R(0))=(0.6 0 0.4 0), (0.62 0 0.38 0) and (0.64 0 0.36 0).

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Change curves for the four populations when \({R_0} < 1.\)

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Simulation analysis and control strategies

To validate the theoretical analysis and propose effective online public opinion control measures, Matlab was employed to simulate and analyze the proposed model. The simulation experiments aimed to verify the stability of the system and examine the sensitivities of the propagation threshold parameters in the model.

First, the global asymptotic stability of the free-propagation equilibrium point of the system (5) was simulated using the following parameters: \(a=0.6,\) \(b=0.1,\) \(c=0.3,\) \(d=0.1,\) \(g=0.5,\) \(a_{1} = 0.2,\) and \(a_{2} = 0.1.\) Based on the formula for\(p_{0}\) and \(R_{0},\) the equilibrium point is \({P_0}=(2,0,0,0),\) and the propagation threshold is \({R_0}= 0.08 < 1,\) which is less than 1, satisfying the condition in Theorem 3. Therefore, the equilibrium point \(p_{0}\) in the system (5) is globally asymptotically stable. Figure 4 illustrates that, given the initial values \(({S_0},{E_0},{I_0},{R_0}) = (0.6,0,0.4,0),\) the information eventually fades out and has no lasting impact. Despite the relatively large initial spreader value of 0.4, public opinion was not disseminated. In fact, in this scenario, regardless of the initial value size, the information did not spread.

When \({R_0} > 1,\) the global asymptotic stability of the equilibrium point in the system (5) was simulated and analyzed with the following parameters: \(a = 0.6,\) \(b = 0.7,\) \(c = 0.2,\) \(d = 0.2,\) \(g = 0.1,\) \({a_1} = 0.2,\) and \({a_2} = 0.1.\) Based on the formula for \(p_{0}\) and \({R_0},\) the propagation equilibrium point is \(P^{*} = (0.92, 1.04, 0.69, 0.52),\) and the propagation threshold is \({R_0} = 3.5 > 1.\) Figure 5 shows that, given the initial conditions \(({S_0}, {E_0}, {I_0}, {R_0}) = (0.6, 0, 0.4, 0),\) the relatively large initial proportion of spreaders and the high contact rate b lead to a greater likelihood of the unaware contacting spreaders. As a result, the number of lurkers increases rapidly. When the spreaders are converted into stiflers, the number of spreaders initially declines slightly. However, because \({R_0} > 1,\) the number of spreaders begins to rise again. At the same time, the number of stiflers increases toward the end of the transmission, and the system tends toward internal equilibrium. In this case, since spreaders remain in the system and continue to negatively impact society, it is essential for the government to take measures to control the spread of public opinion.

Change curves for four populations when \({R_0} > 1.\)

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Parameter b represents the severity of the technological disaster and the probability of contact between the unaware and the spreaders. Figure 6 illustrates the relationship between \({R_0}\) and changes in b. It can be observed that \({R_0}\) is directly proportional to b. Therefore, when the other parameters are fixed, an increase in b leads to a rise in \({R_0},\) indicating that \({R_0}\) is sensitive to variations in b. When b increases to approximately 0.2, \({R_0}\) reaches 1. If b continues to increase to 1, \({R_0}\) becomes significantly larger, approaching 5. However, when \({R_0} < 1,\) public opinion naturally fades over time. In contrast, when b increases further and \({R_0}\) exceeds 1, public opinion does not dissipate. Therefore, the government can regulate public opinion by controlling parameter b. Specifically, by reducing the value of b, the dissemination of public opinion can be controlled.

Parameter g represents the level of accountability for the technological disaster. Holding those responsible accountable and ensuring they receive appropriate punishment helps clarify the truth for the public and alleviates public anger. Governments need to take responsibility for the disaster, release timely updates on the progress of accountability, and ensure information transparency. Their ability to facilitate an orderly and effective rescue operation also helps reduce public anxiety about the consequences of the disaster. Therefore, the level of accountability influences the probability of a spreader becoming a stifler. If the level of accountability is high, the spreaders understand the truth, which calms the public and increases the likelihood of spreaders becoming stiflers. Figure 7 shows that when g changes in the vicinity of 0, the slope of the curve is steep, indicating that the threshold is highly sensitive to g in this region. In other words, a small increase in g significantly reduces the threshold. When g approaches 1, the threshold \(R_0\) becomes less than 1, meaning that public opinion will naturally dissipate. Therefore, to control public opinion, the government should aim to increase g by disclosing relevant information, maintaining transparency in disaster-related updates, and consistently publishing results related to accountability.

Parameter d is affected by government response, which includes the post-disaster relief effect and the speed of the response. In particular, a more effective government response results in a smaller value of d, leading to fewer latent individuals progressing to the spreader state. Therefore, to reflect the direct relationship between government response and the propagation threshold, we set the x-axis as 1-d and the y-axis as the propagation threshold. This setup allows us to analyze the relationship more easily. Figure 8 illustrates the relationship between government response and the public opinion propagation threshold. As shown in the figure, as \(1-d\) increases and the propagation threshold decreases, the government response has a clear impact on the threshold value, with the threshold steadily decreasing throughout the process. From the slope of the tangent line to the curve, we can see that the propagation threshold initially decreases slowly, but later decreases more rapidly. When \(1-d\) exceeds 0.8, the threshold decreases sharply. This indicates that once the government response reaches a certain level, the threshold significantly decreases, leading to the cessation of propagation. Therefore, if the government response improves through more timely and effective relief efforts, the threshold for public opinion propagation will decreases. This means that with better government intervention, the public is less likely to engage in widespread negative discourse, as the perceived effectiveness of the response helps to stabilize public sentiment.

Effects of parameter b on threshold \({R_0}.\)

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This section presents a case study to verify the applicability, feasibility, and effectiveness of the proposed dynamic model in analyzing the propagation of public opinion in technological disasters and developing appropriate control strategies.

Background to the 6.13 Shiyan gas explosion accident

At approximately 6:30 a.m. on June 13, 2021, a gas explosion occurred in Shiyan City, Hubei Province, China. The explosion resulted in 25 fatalities and 138 injuries, 37 of which were serious. The surrounding area was also severely damaged. Following the accident, the Shiyan local government implemented rescue measures. On June 14, an official investigation team was established, and more than 2,000 police officers were deployed to assist with the rescue efforts. The explosion also garnered significant media attention. At 9:35 a.m. on June 13, the first mainstream media outlet reported the disaster, and other official media outlets followed suit. On Sina microblog, both the media and concerned netizens called for relevant authorities to determine the cause of the explosion and expressed sympathy for the victims. According to the Qinglang Public Opinion Monitoring System, a prominent public opinion monitoring platform in China, discussions surrounding the event on microblog were viewed by hundreds of millions of people. The reason for selecting this case is that it represents a typical technological disaster, and the public opinion related to this event clearly influenced the government’s management of the disaster. Due to its severity, the incident attracted widespread attention, prompting substantial public discourse, which generated a significant amount of data on public opinion. This data can be used to derive the parameters needed for the model, based on factors such as the severity of the disaster, the government’s response, the accountability process, and the influence of positive and negative content.

Effect of parameter g on threshold \({R_0}\).

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• Disaster severity

Disaster severity is typically measured based on factors such as the number of fatalities or the economic losses incurred^53,54,55. According to the investigation report of the 6.13 Gas Explosion Accident released by the Hubei Provincial Emergency Management Department⁵⁶, the explosion resulted in 25 deaths and 138 injuries, 37 of which were serious. The direct economic loss was estimated at approximately CNY 53.9541 million. The investigation team classified the explosion as a major accident. According to the top ten technological disaster cases in 2021, as released by the Ministry of Emergency Management of China, the Shiyan gas explosion ranked first in terms of both fatalities and economic losses. To quantify disaster severity, a value of 1 indicates high severity, and 0 indicates low severity. Therefore, we assign a disaster severity coefficient \(D = 1\) for this incident.

• Government response

Post-disaster relief effect. At 7:40 a.m. on June 13, 2021, the government mobilized all available fire and rescue personnel to assist the injured, with local residents near the affected area also volunteering in the relief efforts. More than 2,000 rescue police officers were involved in the operations. By 11:00 a.m., the Shiyan local government had established an accident rescue command center and the “6.13 Accident Emergency News Center” to coordinate the release of information related to the disaster. To prevent secondary disasters, 913 residents living near the site of the explosion were urgently evacuated and resettled in nearby communities, while 853 relatives of the casualties were relocated to hotels. The government also focused on providing comfort to the families of the victims, closely monitored the investigation into the cause of the accident, and conducted safety appraisals of the surrounding buildings. Following the disaster, the government deployed 220 fire and rescue personnel, 1,200 police officers, 87 medical experts, 1,100 medical staff, and 58 psychological specialists⁵⁶. We evaluated the level of government response based on two dimensions: the scale and the structure of the disaster relief efforts. The government response was scored on a scale from 0 to 10, with 10 representing maximum satisfaction. Ten individuals knowledgeable about the disaster were randomly selected and asked to rate the government’s response in these two aspects. The average of these ratings was then calculated, and the final score was derived by dividing by two. The resulting average score for government response was 6.66. Since all coefficients fall within the range [0,1], the score was scaled down by a factor of 10, yielding a final Post-disaster relief effect level of \(d_{1}= 0.666\).

Effect of parameter d on threshold \({R_0}\).

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Response speed. The rescue teams successfully transported most of the injured to major hospitals in Shiyan within one hour of the accident. On the afternoon of June 13, 2021, the Information Office of the Shiyan Municipal People’s Government held its first press conference on the incident, and by the evening of June 14, 2021, the provincial government announced the establishment of a gas explosion accident investigation team. On July 16, 2021, interviews were conducted with relevant Shiyan departments to analyze the lessons learned from the gas explosion. Since the rescue site was in an urban area, we assume that the minimum time for the rescue teams to arrive was 0.5 hours, and the maximum time was 1.5 hours. The formula for calculating the rescue time satisfaction is (Actual rescue arrival time - Minimum time)/(Maximum time - Minimum time). In this case, the rescue arrival time was 1 hour, so the satisfaction with the rescue time is calculated as: response speed \(d_{2}=(1-0.5)/(1.5-0.5)=0.5\). Given that the weights of post-disaster relief effectiveness and response speed are both 0.5, the overall satisfaction with the response is: \(d=1-0.583=0.417.\)

• Degree of accountability

On June 18, 2021, the ’Shiyan Release’ news announced that, based on preliminary investigation results, the Shiyan public security authorities had detained eight suspects, including the general manager of the gas company. After 41 days, on July 24, 2021, the Hubei Provincial government determined that 34 public employees, including 11 provincial management cadres, were responsible for the gas explosion accident. All of them were dismissed. However, there was no mention of individuals being criminally punished or the severity of such punishment, and no official announcements were made regarding the progress of accountability during this period. We evaluate the degree of accountability based on three factors: the completion time of the accountability process, the scope and intensity of the punishment, and the reporting of accountability progress. The degree of accountability is rated on a scale of [0, 10], with 10 indicating maximum satisfaction. Ten individuals with knowledge of the disaster were randomly selected and asked to assess the accountability on the three aspects mentioned above. The average of their ratings was calculated and divided by three to determine the overall score for each individual. The final average score was 1.5. Since all coefficients were within the range [0, 1], the score was then scaled by a factor of 10. Therefore, the accountability coefficient, g, was determined to be 0.15.

• Positive and negative disaster content

In the Internet era, everyone has the ability to publish, spread, and even lead the dissemination of information. Therefore, the coefficients for positive and negative content from the media are represented by the proportion of positive and negative content in microblogs related to the disaster. These proportions were sourced from the public opinion analysis report by Qinglang Public Opinion Monitoring System on the 6.13 Gas Explosion⁵⁷. The report was published on the official WeChat account of “Qinglang Public Opinion” and was subsequently shared by the well-known news website NetEase News. Qinglang Public Opinion Monitoring System is a leading provider of online big data services in China, having issued numerous public opinion reports free of charge. With extensive experience in sentiment analysis, its results are considered highly reliable. According to this public opinion analysis report, public sentiment was categorized into positive, negative, and neutral. Public sentiment was predominantly negative, with approximately 78.29% of the comments reflecting negative sentiment. Netizens expressed sympathy, sadness, and other emotions for the victims, while also placing blame on the cause of the accident. Positive content, which accounted for 21.58%, mainly focused on affirmations of the government’s rescue efforts and response. Therefore, we set \(a_{1} = 0.216\) and \(a_{2} = 0.783.\)

The post-disaster relief effect and degree of accountability were determined using the expert survey method⁵⁸. One of the key advantages of the expert survey method is that the selected experts possess in-depth knowledge of the research field, which enhances the validity and reliability of the results. However, the method also has some limitations, such as the potential subjectivity of expert opinions, which may affect the accuracy of the outcomes. To minimize bias in the expert assessments, we selected experts with relevant knowledge and experience in disaster management, ensuring that they were sufficiently qualified to provide well-informed and appropriate evaluations⁵⁹. During the evaluation process, the evaluation criteria and objectives were clearly defined to help the experts understand the priorities and specific requirements. In this paper, based on the characteristics of the evaluation criteria, the survey subjects were classified into different levels, making it easier for the experts to comprehend the evaluation standards. Additionally, during the evaluation, experts were provided with adequate information and support, including relevant documents, data, and background information, to facilitate accurate assessments. Experts were given access to the detailed chronology of the incident, including the post-disaster relief and accountability processes that we compiled. Furthermore, throughout the evaluation, the experts conducted their assessments independently, ensuring the objectivity and impartiality of the evaluation process.

Data collection and simulation analysis

To investigate the propagation characteristics and patterns of public opinion during the “6.13 Shiyan Gas Explosion Accident,” a total of 18,632 Sina microblog posts were collected from June 13 to July 31, 2021, using the keyword “Shiyan Gas Explosion.” The dataset included information such as the user’s nickname, user ID, and the content of the microblogs. The number of posts per day was counted, as shown in Table 1. Time was used as the x-axis, while the total number of microblogs posted each day was represented on the y-axis. The trend in the number of daily microblogs following the Shiyan gas explosion is illustrated in Fig. 9.

The release of news about the gas explosion sparked significant public concern. As a result, on June 13, the widespread dissemination of the incident led to a sharp increase in the number of microblogs related to the event, reaching a peak of attention. As shown in Fig. 9, after June 13, the number of microblogs began to decline, reflecting a decrease in public attention. On June 18, when the ’Shiyan Release’ reported that the Shiyan City public security organization had identified eight individuals responsible for the incident, including the gas company general manager, and had placed one suspect in custody, the number of related microblogs began to rise again. However, after June 18, the number of related posts rapidly declined to zero. On July 23, 2021, Hubei Province announced that 11 provincial management cadres and 34 public officials responsible for the disaster had been dismissed. As a result, although there was a slight increase in the number of microblogs on June 23, public attention to the disaster quickly waned as the accountability process was resolved.

6.13 Shiyan gas explosion accident microblogs number of changes over time.

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To obtain the model case parameters, it was necessary to track the public opinion transmission process of the Shiyan gas explosion and gather detailed statistics on the public status changes within the system. Due to the complexity of the model and the inactivity of some system users, it was challenging to obtain the parameters directly. Therefore, the state of the microblog users was estimated using sampling statistics. To assess user continuity and activity, the microblog data were filtered and the database narrowed. Specifically, microblog users who had posted three or more times were selected. From this group, 315 users were randomly chosen, and the number of users and microblogs posted each day were recorded in Table 2. The proportion of spreaders was then calculated. These 315 users posted a total of 1,652 microblogs. As shown in Fig. 9, by around June 20, the online public opinion regarding the Shiyan gas explosion had nearly ceased to spread. Therefore, the number of users and microblogs from June 13 to June 20 was computed using the sample data, with the total number of microblogs being 1,515, as shown in Table 2. The parameters for the public opinion dissemination model based on this data are provided in Table 3.

Figure 9 shows that following the explosion on June 12, the number of spreaders peaked on June 13. After this, the number of spreaders gradually declined. Since the explosion occurred around 6:30 a.m., there were few active netizens initially; consequently, there were only 4 users before 10:00 a.m. on June 13. By the end of the day, the number of spreaders had increased to 236, resulting in \(b_{1} = 236 / 311 = 0.759.\) According to the definition of parameter b in the model, we calculate \(b = b_{1} D = 0.759 \times 1 = 0.759.\) Based on the above analysis, the following parameters were entered into the public opinion dissemination model: \(d = 0.417,\) \(g = 0.15,\) \(a_1 = 0.216,\) and \(a_2 = 0.783.\) Since many sample users continued to track the event and the number of people exiting the system was relatively small, the values for a and c were set to 0.01.

Figure 10 shows the actual and predicted spreader density changes for the Shiyan gas explosion in Hubei Province. The curve representing the actual number of spreaders indicates a slow decrease in the number of spreaders, reflecting the timely disaster relief efforts. Compared to Fig. 8, the trend of the spreaders curve is consistent with the one in Fig. 8, which suggests that the sample features correctly reflect the overall data. The peak of public opinion occurred on June 13, the day of the disaster, when the number of spreaders reached a maximum value of about 0.75. Afterward, the number of spreaders gradually decreased, indicating that the Shiyan gas explosion was initially a major public opinion issue. However, due to the rapid government response, the number of spreaders started to decline. On June 18, when the government announced the detention of eight individuals involved in the incident, there was another peak in the number of microblogs, occurring at \(t = 120.\) The predicted number of spreaders also follows a similar pattern: it peaks at the beginning of the disaster, then decreases and stabilizes around \(t = 170.\) This is consistent with the actual microblog data trends, confirming that the model aligns with real-world data. From the Fig. 10, it is evident that public opinion is on a downward trend, indicating that public concern can be mitigated as a result of effective post-disaster rescue operations, even though the accountability process was not handled as well. This outcome is consistent with the theoretical analysis. Regarding the propagation threshold \(R_0,\) government response is an important factor influencing the propagation threshold.

Densities of spreaders in the 6.13 Shiyan gas explosion accident.

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Comparative analysis

There are several public opinion propagation models based on dynamic systems. To compare the results of numerical simulations, the proposed public opinion propagation model for technological disasters was compared with the models presented by Zhang et al.⁶⁰ and Geng et al.²⁰. Zhang et al.⁶⁰ developed an extended SEIR model considering individual interactions to examine the propagation of network public opinion during the COVID-19 pandemic. Geng et al.²⁰ constructed an information dissemination model based on the SEIR framework, considering the structural characteristics of China’s three major social platforms. The reason for choosing these models is that they are variants of the SEIR model, and their parameters are relatively easy to obtain. Their parameters can be directly derived from the case study, with no additional parameters introduced. This helps minimize the influence of external factors on the comparative analysis. The proposed model, Zhang et al.’s model, and Geng et al.’s model are referred to as Model 1, Model 2, and Model 3, respectively. To compare the proposed model with Models 2 and 3 under the same parameters, simulations were conducted with initial values as shown in Table 3, and the parameter values aligned with those from the case study. Specifically, \(\alpha = 0.759,\) \(\lambda = 0.417,\) and \(\sigma = \omega = 0.15\) in Model 2; and \(\alpha = 0.759,\) \(\gamma = 0.15\) in Model 3. Since the parameter \(\xi\) does not exist in the proposed model, \(\xi\) represents the probability that a susceptible node becomes exposed. According to the assumptions of the proposed model, \(\xi = 1.\) The numerical results are shown in Fig. 10.

Figure 11 shows the proportion of spreaders in the three models. It is evident that the proportion of spreaders in Model 1 and Model 2 decreased more sharply compared to Model 3. In Model 3, the proportion of spreaders initially increased and then decreased. Upon closer inspection, it can be seen that in Model 2, the proportion of spreaders increased slightly at first, then decreased, whereas in Model 1, the proportion decreased continuously from the beginning to the end of the propagation process. The trends of the curves reveal that the curve in Model 3 behaves quite differently from the other two, especially at the beginning of the process. Additionally, the curves of Model 1 and Model 2 intersect and remain relatively close to each other throughout most of the process. Compared with Model 2, the proportion of spreaders in Model 1 decreases more rapidly in the same period. However, after the intersection point, the proportion of spreaders in Model 2 decreases faster than in Model 1. By the end of the propagation process, all three models show that the proportion of spreaders tends toward zero, but the curves in Model 2 and Model 3 approach zero more quickly.

The evolution of spreader in Model 1 (the proposed model) compared to the other two models.

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From Fig. 11, it is evident that the curve of Model 3 deviates significantly from the actual data and does not fit well. However, it is not immediately clear which of Model 1 or Model 2 provides a better fit. Therefore, we use the evaluation index SSE (Sum of Squared Errors) to assess the quality of the fit. SSE represents the sum of the squares of the deviations between the fitted data and the actual data, and the closer the SSE is to 0, the better the model fits the data. To eliminate the influence of singularities on the fitting degree, we excluded the data point at t = 120, then extracted the fitting values at t = 0, 24, 48, 72, 96, 144, and 168 using Matlab. We calculated the SSE for the three models and obtained the following values: 0.0168 for Model 1, 0.0208 for Model 2, and 0.0841 for Model 3. From these results, it is clear that the proposed model (Model 1) offers the best fit, as its SSE value is closest to 0, indicating a better match with the real data. In contrast, Model 3 has the worst fit. Although both Model 2 and Model 3 are public opinion propagation models, Model 2, which was designed for online public opinion during the COVID-19 pandemic and considers emergency features, produces more accurate results. However, model 3 is not designed specifically for emergencies or disasters, and therefore, it does not account for the unique characteristics of such events. As a result, the model’s fitness performance is relatively poor in reflecting the dynamics of public opinion in disaster scenarios.

In conclusion, among the three models, the proposed model is better equipped to describe the dissemination of public opinion in response to technological disasters. Unlike the other models, which were designed for general information diffusion or emergency situations like COVID-19, the proposed model specifically emphasizes the unique characteristics of public opinion during technological disasters. This focus allows it to provide a more accurate representation of how public opinion evolves in such events. The comparison between the models highlights the critical importance of considering the specific nature of an event when constructing a model. Tailoring a model to the characteristics of the event can significantly enhance its accuracy in capturing the real dynamics of public opinion propagation.

Policy suggestions

Based on the above analysis, some policy suggestions are given.

(1) Government response and accountability plays a crucial role in managing technological disasters. From the theoretical analysis, it can be seen that these factors are the key elements affecting the propagation threshold. Therefore, controlling these two parameters is beneficial for managing the spread of public opinion. The simulation results also indicate that a high level of government response can effectively reduce the likelihood of uninformed individuals becoming spreaders of misinformation. Furthermore, the case study analysis demonstrates that public opinion can be controlled when post-disaster rescue efforts are well-executed, even if accountability measures are delayed or insufficient. This highlights the critical importance of the government’s timely and effective response. To prevent the escalation of public opinion crises, The government should act swiftly, implementing immediate measures for rescue operations, resettlement, and active accountability enforcement. Such actions not only help to calm public discourse but also provide an opportunity to showcase the government’s emergency management capabilities.

(2) Based on the analysis, disaster severity is a key factor in the spread of public opinion. The simulation results show that the more severe the technological disaster, the greater the attention the government must pay to managing public opinion. Therefore, in cases of particularly severe damage, governments should take proactive steps in both rescue operations and disaster reporting. A lack of transparent and timely information can exacerbate public concern, potentially leading to secondary disasters driven by public opinion, which could make the situation more difficult to manage. Given the public’s sensitivity to government responses, all relevant authorities must act swiftly to ensure timely rescue operations and provide clear, transparent updates on the disaster’s progress, so that the public is well-informed and confident in the government’s handling of the situation.

Densities of spreaders when \(a_{2}-a_{1}=0.567,0.2,-0.1\).

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(3) Negative media content must be carefully monitored. From the theoretical analysis, it is evident that negative media content is not the primary factor affecting the propagation threshold in the public opinion of technological disasters. However, it is undeniable that negative media content can influence the spread of public opinion. To examine this, we analyze the impact of varying parameters of \(a_{2}-a_{1}\) on public opinion propagation, as shown in Fig. 12, which illustrates the densities of spreaders over time. As \(a_{2}-a_{1}\) changes from 0.567 to − 0.1, the figure shows that as positive media content intensity increases or negative media content intensity decreases, the density of public opinion spreaders decreases, and the peak of the spread becomes smaller. This suggests that public opinion spread is alleviated when \(a_{2}-a_{1}\) is reduced. Therefore, governments should take proactive measures to increase positive media content of the disaster while minimizing negative media content in order to better control the spread of online public opinion.

Public opinion surrounding technological disasters is shaped by unique influencing factors and characteristics. The process of public opinion propagation during such disasters is affected by factors including the severity of the disaster, government response, degree of accountability, and positive and negative media content. Given the prevalence of online discussion platforms, technological disasters often generate substantial public commentary. This study, therefore, analyzes the evolution and dissemination of public opinion in the context of technological disasters. A public opinion transmission model based on the SEIR framework was developed, incorporating the severity of the disaster, government response, accountability, and the influence of positive and negative media content. Using stability theory for ordinary differential equations, we analyzed the existence and stability of the zero-propagation and propagation equilibrium points. The next-generation matrix method was applied to determine the propagation threshold, and sensitivity analysis of key parameters was conducted via numerical simulations to identify the primary factors influencing public opinion. To demonstrate the practical application and effectiveness of the model, we analyzed microblogging data related to the gas explosion in Shiyan, Hubei Province, and performed comparative analysis with other models. Based on the findings, several governance strategies were proposed to effectively manage and control public opinion during technological disasters.

Some limitations in this study need to be addressed in subsequent researches.The current study collected data using the keyword “Shiyan Gas Explosion” on Sina microblogs. While this keyword captures approximately 90% of the microblogs related to the disaster, we acknowledge that using additional keywords could expand the coverage. In future data scraping tasks, we will aim to include more keywords to improve data accuracy. Furthermore, in this study, we assumed that the overall factors influencing public opinion remained constant. However, in reality, these parameters may vary over time. In future research, we plan to investigate how these factors change dynamically and incorporate time-varying parameters into the model, in order to more accurately reflect the evolution of public opinion during technological disasters.