Research on the Propagation Model of Unsafe Behaviors among Construction Workers Based on a Two-Layer NAN-SIRS Network

Abstract

Unsafe behaviors among construction workers are a leading cause of safety accidents in the construction industry, and studying the mechanism of unsafe behavior propagation among construction workers is essential for reducing the occurrence of safety accidents. Safety attitude plays a pivotal role in predicting workers’ behavioral intentions. We propose a propagation model of unsafe behaviors based on a two-layer complex network, in which the upper layer depicts the change in construction workers’ safety attitudes, and the lower layer represents the propagation of unsafe behaviors. In this model, we consider the impact of individual heterogeneity and herd mentality on the transmission rate, establishing a partial mapping relationship based on behavioral feedback. After that, by building a probability transition tree, we establish the risk state transition equation in detail using the microscopic Markov chain approach (MMCA) and analyze the established equations to deduce the propagation threshold of unsafe behaviors analytically. The results show that enhancing the influence of individual heterogeneity and behavioral feedback increases the threshold for the spread of unsafe behaviors, thereby reducing its scale, while herd mentality amplifies the spread. Furthermore, the coexistence of safety education and behavioral feedback may lead to one of the mechanisms fails. This research enhances understanding of the propagation mechanism of unsafe behaviors and provides a foundation for managers to implement effective measures to suppress the propagation of unsafe behaviors among construction workers.

1. Introduction

As a cornerstone of the national economy, the construction industry serves as a major employer and significantly contributes to overall development. However, ensuring safety within construction remains a pressing concern that impedes sustainable economic growth. Globally, the construction sector takes a staggering toll on human lives, with at least 108,000 fatalities reported annually, comprising roughly 30% of all fatal occupational accidents [1]. Table 1 provides an overview of construction safety across various countries, highlighting its persistent global challenge. Construction accidents not only lead to loss of life and substantial property damage but also have profound negative impacts on society [2]. The effective prevention of safety incidents within the construction industry is a matter of paramount importance for the government, society, and enterprises alike.

The direct causes of safety accidents can be classified into two main categories: unsafe human behaviors and hazardous conditions of objects [3]. Extensive research into the root causes of construction safety accidents reveals that approximately 80% of such incidents stem from unsafe behaviors exhibited by construction workers [4], underscoring the pivotal role of human behavior in safety accidents. Construction workers frequently engage in social interactions and form closely connected work groups during construction activities. When unsafe behaviors occur within the construction team, they can be replicated, evolved, and spread through interactions among construction workers, leading to the propagation of unsafe behaviors [5]. The continued propagation of unsafe behaviors among construction workers can result in a widespread group unsafe behaviors and ultimately lead to safety accidents. Therefore, understanding the transmission characteristics of unsafe behaviors among construction workers and effectively suppressing the evolution and spread of unsafe behaviors are of vital significance in reducing safety accidents in the construction industry.

Table 1. The status of construction safety in some countries.

Country	Description of the Status of Construction Safety
United States	The census data from the U.S. Bureau of Labor Statistics (BLS) showed that the number of fatalities among workers in the construction and mining industries in 2022 was 1056, accounting for 19.25% of all industries. The fatality rate (13 per 100,000 full-time equivalent workers) ranked fourth highest among all industries [6].
United Kingdom	The construction industry has the highest number of workers who die in fatal accidents annually, accounting for about one-third of all fatal accident deaths, and ranks second highest in non-fatal injury rates among all industries [7].
China	In 2020, there were a total of 689 production safety accidents in China’s housing and municipal engineering sector, resulting in 794 deaths. Compared to the same period in 2015, there was an average annual increase of 11.18% in accidents and 8.66% in deaths [8].
Singapore	In 2021, there were a total of 37 workplace fatal accidents, with a workplace fatality rate of 1.1 per 100,000 employees. The construction industry had the highest incidence of injuries and fatalities [9].
Australia	There were 31 fatalities recorded in 2023. This number of fatalities equated to three deaths per 100,000 workers, ranking second highest among all industries [10].
Korea	Of all industries, the construction industry has the highest proportion of fatalities, and its accident rate is significantly higher than that of other industries [11].

Table 1. The status of construction safety in some countries.

Country	Description of the Status of Construction Safety
United States	The census data from the U.S. Bureau of Labor Statistics (BLS) showed that the number of fatalities among workers in the construction and mining industries in 2022 was 1056, accounting for 19.25% of all industries. The fatality rate (13 per 100,000 full-time equivalent workers) ranked fourth highest among all industries [6].
United Kingdom	The construction industry has the highest number of workers who die in fatal accidents annually, accounting for about one-third of all fatal accident deaths, and ranks second highest in non-fatal injury rates among all industries [7].
China	In 2020, there were a total of 689 production safety accidents in China’s housing and municipal engineering sector, resulting in 794 deaths. Compared to the same period in 2015, there was an average annual increase of 11.18% in accidents and 8.66% in deaths [8].
Singapore	In 2021, there were a total of 37 workplace fatal accidents, with a workplace fatality rate of 1.1 per 100,000 employees. The construction industry had the highest incidence of injuries and fatalities [9].
Australia	There were 31 fatalities recorded in 2023. This number of fatalities equated to three deaths per 100,000 workers, ranking second highest among all industries [10].
Korea	Of all industries, the construction industry has the highest proportion of fatalities, and its accident rate is significantly higher than that of other industries [11].

Prior studies have delved into various factors influencing the unsafe behaviors of construction workers, including safety awareness [12], safety attitude [13], safety climate [14], regulatory systems [15], safety training [16], and work experience [17]. These factors operate across individual, organizational, and environmental levels, with individual attributes serving as primary drivers that impact the broader context [18]. At the individual level, the “unwillingness to perform safely” can be viewed as a cognitive decision-making process [19] and can be explained using theories from psychology. The Theory of Planned Behavior (TPB) delineates the determinants of unsafe behavior by elucidating the interplay between attitude, intention, and action [20], which underscores the pivotal role of behavioral intention in shaping conduct, with safety attitude emerging as a key predictor of workers’ intentions [21]. Thus, workers with positive safety attitudes are more inclined to work safely [22,23], and improving the safety attitudes of construction workers is a key factor in curbing unsafe behaviors [24]. Past research indicates a reciprocal relationship wherein safety attitudes not only influence safety behaviors positively but also undergo changes in response to behavioral outcomes, thereby forming a feedback loop [25]. Therefore, studying the interactive mechanisms between safety attitudes and the spread of unsafe behaviors can effectively prevent the occurrence of safety accidents.

Indeed, workers’ unsafe behaviors stem not solely from individual cognitive processes but also from their interactions within a broader social context [26]. Within the construction industry, workers often operate within interconnected groups formed through work dynamics, mentorship, and personal affiliations [17], forming intricate networks of interaction. Adjacent nodes within these networks (including co-workers, foremen, and managers) wield influence over workers’ attitudes and behaviors, shaping their safety practices [27,28]. Prior research has scrutinized the spread of behavior through network diffusion models, delineating the process into stages of imitation, recovery, and forgetting, often employing SIR models to dissect the mechanisms and features of behavior dissemination [29,30]. Some scholars have examined unsafe behavior propagation through the lens of small group dynamics, revealing ‘small-world’ characteristics in the networks facilitating such propagation [31]. Furthermore, by integrating insights from infectious disease models, researchers have highlighted the pivotal role of key individuals (such as team leaders, technical experts, and mentors) in driving the transmission of unsafe behaviors within groups [32]. Additionally, studies have underscored the phenomenon of individuals being more prone to safety rule violations upon witnessing heightened instances of unsafe behavior among their peers, a manifestation of conformity dynamics [33,34]. It is evident that the influence wielded by key figures within the workforce, coupled with herd mentality, significantly shapes the proliferation of unsafe behaviors.

Research into complex networks has attracted considerable attention from scholars. In real-world scenarios, complex systems often comprise multiple interconnected subsystems, prompting the development of the multilayer network concept by researchers. Numerous scholars have devised various mathematical models based on multilayer networks to elucidate the propagation of phenomena such as epidemics and risks. For instance, Xia et al. introduced a coupled diffusion model that accounts for the interplay between disease transmission and awareness dissemination [35]. Xu et al. investigated the impact of individual heterogeneity within a coupled disease transmission framework [36]. Furthermore, Huo et al. delved into the effects of risk preference and herding behavior on risk dissemination through the construction of a supply chain risk propagation model incorporating multiple network warning mechanisms [37]. Yao et al. established a localized mapping relationship within a two-layer complex network, building upon previous risk propagation models, and verified the significance of information disclosure awareness [38]. The applicability and practicality of multilayer network models have been further improved and developed.

In summary, the unsafe behaviors of construction workers have the characteristics of contagion, attracting increasing scholarly attention in order to understand their mechanisms from various perspectives, mainly employing the methods of constructing epidemic models or communication networks. However, previous studies have predominantly focused on singular propagation processes, overlooking the cognitive decision-making process underlying the genesis of unsafe behaviors, which is influenced by psychological factors among construction workers. It is crucial to recognize that shifts in safety attitudes can significantly impact the proliferation of unsafe behaviors. Concurrently, research on multi-layer coupled networks has advanced transmission models, acknowledging the intricate interplay between behaviors and awareness throughout the transmission process. While widely applied to studying the spread of risks and diseases, this approach has been underutilized in investigating the dissemination of unsafe behaviors within the construction industry. Hence, we employ multi-layer coupled networks to scrutinize the propagation dynamics of unsafe behaviors among construction workers. This research aims to investigate the interplay between safety attitudes and unsafe behaviors among construction workers, seeking to explore methods for reducing unsafe behaviors by enhancing safety attitudes, thereby effectively reducing the occurrence of safety accidents in the construction industry.

Therefore, this paper proposes a novel safety attitude–unsafe behavior (NAN-SIRS) spreading coupled model, which integrates herd mentality and individual heterogeneity. It formulates a local mapping relationship based on behavioral feedback to study the interaction between safety attitudes and unsafe behaviors among construction workers within a two-layer network framework. The threshold for the propagation of unsafe behaviors among construction workers is derived through the microscopic Markov chain analysis (MMCA). The primary contributions of this study are delineated as follows: (1) The model accommodates node heterogeneity and herd mentality, providing a more realistic description of the spread of construction workers’ unsafe behaviors. (2) The safety attitudes of construction workers are influenced by behavioral outcomes, albeit not all behaviors elicit feedback. Unlike prior studies employing two-layer networks with node-by-node mapping, this paper adopts a two-layer network approach with partial mapping, considering the influence of behavioral feedback on the propagation of unsafe behaviors. (3) By coupling the spread of unsafe behaviors within a two-layer network, this research facilitates a deeper understanding of the dynamic interplay between the dissemination of construction workers’ unsafe behaviors and shifts in safety attitudes.

2. Research Approach

This study established a two-layer network NAN-SIRS model to evaluate the interplay between safety attitudes and the dissemination of unsafe behaviors among construction workers. Figure 1 depicts the layout of the research methodology adopted in this paper.

The research methodology involves four principal stages. Before formally establishing the propagation model, we posited several fundamental assumptions. These assumptions delineated the possible states of nodes in both the safety attitude layer and the unsafe behavior propagation layer, as well as the scenarios of their transitions. The impact of individual heterogeneity, herd mentality, and the behavior feedback-based partial mapping relationships on the propagation of unsafe behaviors was described in detail. These basic assumptions facilitated the establishment of the propagation model. Subsequently, this paper established a two-layer NAN-SIRS model and furnished an elaborate description of their interactions. The upper layer network (Layer A) represented the variation in safety attitudes among construction workers, with nodes in two states: negative (N) and positive (A). The lower layer network (Layer B) depicted the propagation of unsafe behaviors, with nodes in three states: susceptible (S), infected (I), and recovered (R). Next, dynamic evolution equations were obtained using Granell et al.’s micro-Markov chain analysis (MMCA) method [39] to calculate the propagation threshold. Finally, numerical simulations were conducted to validate and analyze the model.

2.1. Applicability Analysis

In this paper, we apply the SIRS model, traditionally used in the field of disease propagation, to elucidate the dynamics of unsafe behavior dissemination among construction workers. It takes into account the forgetting rate of recovered individuals becoming susceptible again. The applicability of this model is mainly manifested in the following aspects.

• Similarity in the propagation environment: According to social network theory, construction workers, as a group, form a complex network among themselves due to relational ties. This network functions as the primary conduit for behavior propagation [40], wherein individual workers represent nodes within this network.
• Similarity in the propagation process: Communication and interaction among construction workers form the edges of this network. Unsafe behavior spreads through localized interactions among workers with proximate relationships. Additionally, key figures within the workforce (e.g., team leaders, technical experts, mentors) tend to possess more connections with other workers due to the nature of their roles.
• Similarity in propagation target characteristics: Each worker within the behavior propagation network operates independently, with infection and recovery rates influenced by factors such as safety knowledge, attitudes, and herd mentality, leading to varying rates of dissemination.
• Similarity in propagation direction: Similar to disease propagation, the edges within the behavior dissemination network are undirected, enabling propagation in any direction. Thus, the behavior dissemination network can be considered an undirected network.

2.2. Description of the Model

2.2.1. Basic Assumptions

For the sake of analysis, the following assumptions are made in this paper.

1.

Safety attitude layer (Layer A)

The upper layer represents the safety attitude (Layer A), delineating shifts in the safety mindset of construction workers. Attitude encompasses three dimensions: cognition, emotion, and behavioral tendency. Within the construction domain, safety attitude embodies workers’ perception of safety significance, emotional reactions towards safety protocols, and inclination to adhere to safety regulations [41]. As illustrated in Figure 2a, construction workers on this layer exhibit two states: N (negative) and A (positive). Among these, workers with a positive safety attitude demonstrate greater willingness to adhere to safety protocols, adopt appropriate work practices, and embrace safety-conscious behaviors. Enhancing the safety attitude of construction workers typically occurs through three avenues. Firstly, the influence of fellow team members: N-state workers are influenced by A-state counterparts, transitioning to the state A with a probability denoted by $\lambda$. Secondly, managerial safety education and training contribute to improving workers’ safety attitudes to a certain extent. When the proportion of A-state workers among all construction personnel surpasses a threshold represented by $\theta$, N-state workers undergo a transformation to the state A. Thirdly, experiences of accidents resulting from unsafe behaviors or managerial feedback on employee conduct impact workers’ safety attitudes. This reflects the partial mapping relationship between the two-layer networks, elaborated upon in subsequent sections. Furthermore, a forgetting rate exists, wherein over time, construction workers have a probability denoted by $\delta$ of transitioning from the state A to the state N.

2.

Unsafe behavior propagation layer (Layer B)

The lower layer constitutes the unsafe behavior propagation layer (Layer B), employing the SIRS model to elucidate the propagation of unsafe behavior among construction workers. Within this layer, construction workers are categorized into three states: S (susceptible) state, wherein individuals may imitate the unsafe behavior of others; I (infected) state, where individuals actively engage in unsafe practices and possess the capacity to propagate them; and R (recovered) state, where individuals temporarily rectify their unsafe conduct through corrective measures like punishment or education after engaging in unsafe behavior. As depicted in Figure 2b, when individuals in the S state interact with the individuals in the I state, there is a probability of $\beta$ that they will be infected, and the individual in the I state has a probability of $\mu$ of transitioning to the R state. Moreover, influenced by psychological factors such as luck and rebelliousness, there exists a certain probability of $\tau$ that individuals in the R state will transition back to the S state.

3.

Partial mapping relationships

There exists a partial mapping relationship between Layer A and Layer B, represented by $M=[{map}_1,{map}_2,\dots ,{map}_N]$. Here, M represents the set of partial mapping relationships between Layer A and Layer B.

$${map}_i=\left\{\begin{array}{c}1, a mapping relationship between Layer A and Layer B;\\ 0, no mapping relationship between Layer A and Layer B.\end{array}\right.$$

We argue that this partial mapping relationship delineates the variance in feedback received by construction workers regarding unsafe behavior. When construction workers engage in unsafe behavior, it may culminate in accidents, or managers may intervene to correct the behavior in order to avoid an accident. These intervention measures will change the workers’ attitudes and subsequently affect their subsequent safety behavior. Additionally, attitudes depend on how people perceive risks, and past accident experiences also have an impact on workers’ safety attitudes [15,25]. In conclusion, behavior feedback will impact the safety attitudes of construction workers. If there is no mapping relationship between the two layers (${map}_i=0$), construction workers will not receive behavior feedback after engaging in unsafe behavior, and their safety attitudes will not change. On the contrary, if there is a mapping relationship between the two layers (${map}_i=1$), construction workers’ unsafe behavior will lead to accidents or intervention from managers, and at this time, nodes of Layer A will adjust their safety attitude and transition from N to A with a certain probability. Therefore, the probability of the Layer B node receiving feedback on unsafe behavior and transitioning the Layer A node from N to A is defined as ${\chi }_i$:

$${\chi }_i=ma{p}_i\cdot {\zeta }_B.$$

(1)

Here, ${map}_i$ represents the mapping relationship between nodes in Layer A and Layer B. ${\zeta }_B$ represents the probability of construction workers in the unsafe behavior propagation layer (Layer B) receiving behavior feedback. Therefore, ${\chi }_i$ can be explained as the probability of construction workers receiving behavior feedback that results in a change in the safety state of Layer A (transitioning from N to A).

$${\chi }_i=\left\{\begin{array}{c}{\zeta }_B, {map}_i=1, there is a mapping relationship;\\ 0, {map}_i=0, there is no mapping relationship.\end{array}\right.$$

4.

Individual heterogeneity

In construction groups, key individuals like team leaders, technical experts, and mentors wield significant influence over workers due to their close connections and frequent interactions. Their work behaviors are closely observed and often emulated. Unlike disease transmission, where varying degrees of infection can affect susceptible individuals differently, in this context, the stronger an individual’s influence is, the more probable it is that their unsafe behaviors will propagate.

Assuming that the heterogeneity of individuals in Layer B depends on the degree of the individual’s nodes, the infection rate of susceptible individuals i can be defined as follows:

$${\beta }_{ij}=\frac{{{\omega }_{ij}}^{\alpha }}{{\max }_{l=1\dots N}{{\omega }_{lk}}^{\alpha }}\beta .$$

(2)

${\omega }_{ij}$ represents the influence of individual j on individual i in the unsafe behavior propagation layer, and $\alpha$ represents the extent to which individual influence affects the infection rates.

Where ${\omega }_{ij}$ is defined as

$${\omega }_{ij}=\frac{k_j^{\eta }}{{\displaystyle {\sum }_{s\in {\Gamma }_i}{k}_s^{\eta }}}.$$

(3)

$k_j$ is the node degree of individual j in Layer B, ${\Gamma }_i$ is the set of neighbors of individual i in Layer B, and $\eta$ controls the influence heterogeneity of individuals in Layer B. When $\eta =0$, ${\omega }_{ij}=1/k_i$, all neighbors of individual i have the same influence on it at this time. When $\eta$ is greater than/smaller than 0, the neighbors with greater/smaller node degrees have a greater impact on individual i. To align with the real-world scenario, we assume $\eta$ to be greater than 0.

5.

Herd mentality

Demonstrative imitation and contagious conformity are two ways in which unsafe behaviors spread [42]. Construction workers are susceptible to the influence of their peers while on duty. Instances of unsafe behavior within construction groups may prompt individuals to uncritically emulate their colleagues, driven by the sway of herd mentality, thereby resulting in erroneous decisions or actions. This phenomenon compels individual conduct to synchronize with group norms, culminating in enhanced group cohesion and the activation of conformity behaviors [34].

Therefore, we believe that the infection rate $M_{ij}$ is related to the “susceptibility” of the susceptible node i and the “transmission capacity” of the infected node j.

Assuming at time $t$ there are N infected node workers around construction worker i, with the conformity parameter set as $h$, the probability that construction worker i is infected by contacting the surrounding construction workers is $(1+h){\beta }_{ij}∆t$, and the more unsafe behaviors are encountered, the higher the likelihood of infection. When there are $g$ infected nodes around, the probability of infection is ${\beta }_{ij}{(1+h)}^g$, and the probability of not being infected is $1-{\beta }_{ij}{(1+h)}^g$.

Therefore, the probability of not being infected in the population of construction workers is as follows:

$${\phi }_i(g)={\displaystyle {\prod }_{n=1}^g\left[1-\beta {\left(1+h\right)}^n\Delta t\right]}.$$

(4)

By neglecting higher-order terms and applying Equation (2), we can obtain

$$M_{ij}=(1+h){\beta }_{ij}=(1+h)\frac{{{\omega }_{ij}}^{\alpha }}{{\max }_{l=1\dots N}{{\omega }_{lk}}^{\alpha }}\beta .$$

(5)

2.2.2. Construction of the Propagation Model

Drawing from the aforementioned assumptions, we formulate a two-layer network NAN-SIRS model, which describes the dynamic process of unsafe behavior propagation and changes in safety attitudes among the population of construction workers, as depicted in Figure 3. This two-layer network comprises identical nodes (individual construction workers) with distinct connections (interactions among workers). The upper layer represents the network of safety attitude changes, featuring a denser interconnection scheme. This layer encompasses not only the communication and collaboration occurring during construction activities but also various other interactions among construction workers, such as those with roommates, friends, and relatives, all of which can influence their safety attitudes.

Table 2. Parameters and corresponding descriptions in the current two-layered epidemic model.

Parameters	Descriptions
$a_{ij}$	Adjacency matrix in the safety attitude layer
$b_{ij}$	Adjacency matrix in the unsafe behavior propagation layer
${\beta }^N$	Infection rate for a susceptible individual in a negative state
${\beta }^A$	Infection rate for a susceptible individual in a positive state
$\gamma$	Parameters of the impact of Layer A on Layer B
$\mu$	Recovery rate for an infected individual
$\delta$	Forgotten rate from the positive state to a negative one
$\tau$	Transition rate of individuals from the restored state to a susceptible one
$\lambda$	Transition rate of individuals from a negative to a positive state influenced by their co-workers
$\theta$	Critical value for individuals to transition from a negative state to a positive state under the influence of safety training
$h$	The degree to which construction workers are influenced by herd mentality
$\chi$	Behavioral feedback influences the probability of safety attitude change
${\omega }_{ij}$	Level of influence of individual j on individual i
$\alpha$	The degree to which individual heterogeneity affects the infection rate

summarizes the key parameters of the model and their respective descriptions. The interactions between the two processes are as follows:

For individuals in Layer B who are infected, the probability of receiving behavioral feedback that causes the corresponding individual on Layer A to change from state N to state A is $\chi$. Within Layer A, individuals possessing a positive safety attitude demonstrate a higher propensity to adhere to safety protocols and adopt safe practices. Consequently, the susceptibility of individuals in Layer B to infection varies based on their status within Layer A. Assuming that the infection rates of susceptible individuals in the negative and positive states are ${\beta }^N$ and ${\beta }^A$, respectively, we can adjust the probability of individual infection using the parameter $\gamma$, where $0\le \gamma \le 1$. Then, ${\beta }^A=\gamma {\beta }^N$, where $\gamma$ represents the influence of the safety attitude change layer (Layer A) on the spread of unsafe behaviors layer (Layer B). When $\gamma =0$, the individuals in the positive state are completely immune to the infection, indicating their abstention from unsafe behaviors and lack of involvement in their propagation. When $\gamma =1$, safety attitude alterations do not impact the spread of unsafe behaviors, implying equivalent infection rates for individuals with positive and negative safety attitudes.

During the construction process, unsafe behaviors are typically classified into intentional and unintentional categories. Intentional unsafe behaviors entail workers knowingly deviating from established rules or procedures, fully aware that such actions contravene safety regulations or may precipitate accidents. Conversely, unintentional unsafe behaviors stem from workers’ insufficient capacity to execute tasks safely or from a lack of familiarity with proper procedures [43]. Individuals, regardless of possessing positive or negative safety attitudes, may engage in unintentional unsafe behaviors due to inadequacies in safety knowledge or skills, resulting in an inability to identify potential hazards or existing risks. On the other hand, workers with negative safety attitudes may also exhibit intentional unsafe behaviors driven by negative emotions, performance pressures, or conflicting values. Therefore, the two-layer network presents six potential node states: NS (negative and susceptible), NI (negative and infected), NR (negative and recovered), AS (positive and susceptible), AI (positive and infected), and AR (positive and recovered). At any time step $t$, each node i will become one of the above six seed states with a definite probability, denoted $p_i^{NS}\left(t\right), p_i^{NI}\left(t\right), p_i^{NR}\left(t\right), p_i^{AS}\left(t\right), p_i^{AI}\left(t\right), p_i^{AR}\left(t\right)$. The probability of all states satisfies the following equation:

$${p_i}^{NS}(t)+{p_i}^{NI}(t)+{p_i}^{NR}(t)+{p_i}^{AS}(t)+{p_i}^{AI}(t)+{p_i}^{AR}(t)\equiv 1.$$

(6)

2.3. Theoretical Analysis Using the MMCA

2.3.1. Dynamical Equations

Granell et al.’s [39] microscopic Markov chain approach (MMCA) has found extensive application in simulating interactions across multiple networks. By illustrating potential shifts in node states and their associated probabilities at each time step through a state transition probability tree, it becomes possible to derive the dynamic evolution equations characterizing the dynamics of the NAN-SIRS model.

Initially, assuming there are N nodes, $A={\left(a_{ij}\right)}_{N\times N}$ and $B={\left(b_{ij}\right)}_{N\times N}$ represent the adjacency matrices in the NAN and SIRS processes, respectively, where elements $a_{ij}=1$ and $b_{ij}=1$ if a link exists between node i and node j; otherwise $a_{ij}=0$ and $b_{ij}=0$. Assuming that the probability of a node being infected or influenced in its safety attitude by any neighbor is independent (assuming no dynamic correlation), in Layer A, the probability of worker i being in a negative state, without being influenced by positive-state neighbors and changing its safety attitude, is denoted $r_i\left(t\right)$, and the probability of not being influenced by safety education and transitioning to a positive state is denoted $g_i\left(t\right)$. In Layer B, the probability of not being infected by any infected neighbor under a positive (negative) safety attitude is denoted $q_i^A\left(t\right)$ ($q_i^N\left(t\right)$). The expressions are as follows:

$$\begin{array}{l}{r}_i(t)={\displaystyle \prod _j\left[1-a_{ji}{p_j}^A(t)\lambda \right]}\\ g_i(t)=H\left[\theta -\frac{{\displaystyle \sum _{j=1}^N{p_j}^A(t)}}{N}\right]\\ {q_i}^A(t)={\displaystyle \prod _j\left[1-b_{ji}{p_i}^I(t){M_{ji}}^A\right]}\\ {q_i}^N(t)={\displaystyle \prod _j\left[1-b_{ji}{p_i}^I(t){M_{ji}}^N\right]}\end{array},$$

(7)

where $H\left(x\right)$ represents the Heaviside step function, that is, either $H\left(x\right)=1$ if $x>0$ or $H\left(x\right)=0$ otherwise.

In order to describe the transitions of each state in the nodes more clearly, we constructed a probability transition tree, as shown in Figure 4. The root node of the probability tree represents the single state at the current time step, and the leaf nodes represent the possible states at the next time step, with the transition probabilities indicated on each branch of the tree. Leveraging the probability transition tree in Figure 4, the dynamic evolution equations for the six possible states can be obtained through the microscopic Markov chain approach (MMCA):

$$\begin{array}{ll}{p_i}^{NS}(t+1)=& {p_i}^{NS}(t)r_i(t)g_i(t){q_i}^N(t)+{p_i}^{NR}(t)r_i(t)g_i(t)\tau \\ & +{p_i}^{AS}(t)\delta g_i(t){q_i}^N(t)+{p_i}^{AR}(t)\delta g_i(t)\tau \\ {p_i}^{NI}(t+1)=& {p_i}^{NS}(t)r_i(t)g_i(t)\left[1-{q_i}^N(t)\right]\left(1-\chi \right)+{p_i}^{NI}(t)r_i(t)g_i(t)\left(1-\mu \right)\left(1-\chi \right)\\ & +{p_i}^{AI}(t)\delta g_i(t)\left(1-\mu \right)\left(1-\chi \right)+{p_i}^{AS}(t)\delta g_i(t)\left[1-{q_i}^N(t)\right]\left(1-\chi \right)\\ {p_i}^{NR}(t+1)=& {p_i}^{NI}(t)r_i(t)g_i(t)\mu +{p_i}^{NR}(t)r_i(t)g_i(t)\left(1-\tau \right)\\ & +{p_i}^{AI}(t)\delta g_i(t)\mu +{p_i}^{AR}(t)\delta g_i(t)\left(1-\tau \right)\\ {p_i}^{AS}(t+1)=& {p_i}^{NS}(t)\left\{r_i(t)\left[1-g_i(t)\right]{q_i}^A(t)+\left[1-r_i(t)\right]{q_i}^A(t)\right\}+{p_i}^{NR}(t)\left\{\left[1-r_i(t)\right]\tau +r_i(t)\left[1-g_i(t)\right]\tau \right\}\\ & +{p_i}^{AS}(t)\left\{\left(1-\delta \right){q_i}^A(t)+\delta \left[1-g_i(t)\right]{q_i}^A(t)\right\}+{p_i}^{AR}(t)\left[\delta \left[1-g_i(t)\right]\tau +\left(1-\delta \right)\tau \right]\\ {p_i}^{AI}(t+1)=& {p_i}^{NS}(t)\left\{r_i(t)g_i(t)\left[1-{q_i}^N(t)\right]\chi +r_i(t)\left[1-g_i(t)\right]\left[1-{q_i}^A(t)\right]+\left[1-r_i(t)\right]\left[1-{q_i}^A(t)\right]\right\}\\ & +{p_i}^{NI}(t)\left\{r_i(t)g_i(t)\left(1-\mu \right)\chi +r_i(t)\left[1-g_i(t)\right]\left(1-\mu \right)+\left(1-r_i(t)\right)\left(1-\mu \right)\right\}\\ & +{p_i}^{AS}(t)\left\{\left(1-\delta \right)\left[1-{q_i}^A(t)\right]+\delta g_i(t)\left[1-{q_i}^N(t)\right]\chi +\delta \left[1-g_i(t)\right]\left[1-{q_i}^A(t)\right]\right\}\\ & +{p_i}^{AI}(t)\left\{\left(1-\delta \right)\left(1-\mu \right)+\delta g_i(t)\left(1-\mu \right)\chi +\delta \left[1-g_i(t)\right]\left(1-\mu \right)\right\}\\ {p_i}^{AR}(t+1)=& {p_i}^{NI}(t)\left\{r_i(t)\left[1-g_i(t)\right]\mu +\left[1-r_i(t)\right]\mu \right\}+{p_i}^{NR}(t)\left\{r_i(t)\left[1-g_i(t)\right]\left(1-\tau \right)+\left[1-r_i(t)\right]\left(1-\tau \right)\right\}\\ & +{p_i}^{AI}(t)\left\{\left(1-\delta \right)\mu +\delta \left[1-g_i(t)\right]\mu \right\}+{p_i}^{AR}(t)\left\{\left(1-\delta \right)\left(1-\tau \right)+\delta \left[1-g_i(t)\right]\left(1-\tau \right)\right\}\end{array}$$

(8)

where $t$ represents the current time step, and $t+1$ represents the next time step. At each time step, the equation is satisfied, as shown in Formula (6). When the time step is large enough, the proportions of each state for each node reach a steady state. When $t\to \infty$, we have the following:

$$\begin{array}{l}{p_i}^{NS}{(t+1)}_{t\to \infty }={p_i}^{NS}{(t)}_{t\to \infty }={p_i}^{NS}\\ {p_i}^{NI}{(t+1)}_{t\to \infty }={p_i}^{NI}{(t)}_{t\to \infty }={p_i}^{NI}\\ {p_i}^{NR}{(t+1)}_{t\to \infty }={p_i}^{NR}{(t)}_{t\to \infty }={p_i}^{NR}\\ {p_i}^{AS}{(t+1)}_{t\to \infty }={p_i}^{AS}{(t)}_{t\to \infty }={p_i}^{AS}\\ {p_i}^{AI}{(t+1)}_{t\to \infty }={p_i}^{AI}{(t)}_{t\to \infty }={p_i}^{AI}\\ {p_i}^{AR}{(t+1)}_{t\to \infty }={p_i}^{AR}{(t)}_{t\to \infty }={p_i}^{AR}\end{array}.$$

(9)

2.3.2. Thresholds for the Propagation of Unsafe Behaviors

There exists a propagation threshold ${\beta }_C$ in the two-layer interaction process of the NAN-SIRS model, and this critical value determines whether unsafe behavior can spread or not. When the infection rate $\beta <{\beta }_C$, the initial unsafe behavior is quickly controlled and does not spread among the population of construction workers. When the propagation rate $\beta >{\beta }_C$, unsafe behavior will widely spread within the population of construction workers. According to the MMCA equations derived from the state transfer probability tree in the previous section and combined with Equation (7), the propagation threshold of unsafe behavior can be calculated.

Adding the second and fourth terms in Equation (8), we can obtain

$$\begin{array}{ll}{p_i}^I(t+1)=& {p_i}^{NS}(t)\left\{r_i(t)g_i(t)\left[1-{q_i}^N(t)\right]+r_i(t)\left[1-g_i(t)\right]\left[1-{q_i}^A(t)\right]+\left[1-r_i(t)\right]\left[1-{q_i}^A(t)\right]\right\}\\ & +{p_i}^{AS}(t)\left\{\delta g_i(t)\left[1-{q_i}^N(t)\right]+\left(1-\delta \right)\left[1-{q_i}^A(t)\right]+\delta \left[1-g_i(t)\right]\left[1-{q_i}^A(t)\right]\right\}+{p_i}^I(t)\left(1-\mu \right)\end{array}.$$

(10)

Near the propagation threshold, the probability of any node i in the unsafe behavior propagation network being infected tends to zero. Therefore, we can assume that $p_i^I=\epsilon \ll 1$. By combining Equations (2) and (5), we can further approximate the values of $q_i^A\left(t\right)$ and $q_i^N\left(t\right)$ in Equation (7) as follows:

$$\begin{array}{l}{q_i}^A(t)={\displaystyle \prod _j\left[1-b_{ji}{p_i}^I(t){M_{ij}}^A\right]}\approx 1-{M_{ij}}^A{\displaystyle \sum _j{b}_{ji}{\epsilon }_j}=1-(1+h){{\beta }_{ij}}^A{\displaystyle \sum _j{b}_{ji}{\epsilon }_j}\\ {q_i}^N(t)={\displaystyle \prod _j\left[1-b_{ji}{p_i}^I(t){M_{ij}}^N\right]}\approx 1-{M_{ij}}^N{\displaystyle \sum _j{b}_{ji}{\epsilon }_j}=1-(1+h){{\beta }_{ij}}^N{\displaystyle \sum _j{b}_{ji}{\epsilon }_j}\end{array}.$$

(11)

For simplicity, we make ${\alpha }^A=(1+h){{\beta }_{ij}}^A\sum _j{b}_{ji}{\epsilon }_j$ and ${\alpha }^N=(1+h){{\beta }_{ij}}^N\sum _j{b}_{ji}{\epsilon }_j$. In addition, the higher-order terms on ${\epsilon }_j$ can be neglected as they are sufficiently small near the propagation threshold. When $t\to \infty$, the system approaches a stable state. By substituting ${\alpha }^A$ and ${\alpha }^N$ into Equations (8) and (10) and assuming that all higher-order terms are ignored, we can obtain

$$\begin{array}{l}{p_i}^{NS}={p_i}^{NS}{r}_i{g}_i+{p_i}^{NR}{r}_i{g}_i\tau +{p_i}^{AS}\delta g_i+{p_i}^{AR}\delta g_i\tau \\ {p_i}^{AS}={p_i}^{NS}\left(1-r_i{g}_i\right)+{p_i}^{NR}\left(1-r_i{g}_i\right)\tau +{p_i}^{AS}\left(1-\delta g_i\right)+{p_i}^{AR}\left(1-\delta g_i\right)\tau \\ {p_i}^{NR}={p_i}^{NR}{r}_i{g}_i\left(1-\tau \right)+{p_i}^{AR}\delta g_i\left(1-\tau \right)\\ {p_i}^{AR}={p_i}^{NR}(1-r_i{g}_i)\left(1-\tau \right)+{p_i}^{AR}\left(1-\delta g_i\right)\left(1-\tau \right)\\ \mu {\epsilon }_i={p_i}^{NS}\left[r_i{g}_i{\alpha }^N+\left(1-r_i{g}_i\right){\alpha }^A\right]+{p_i}^{AS}\left[\delta g_i{\alpha }^N+\left(1-\delta g_i\right){\alpha }^A\right]\end{array}.$$

(12)

Furthermore, we can obtain

$$\mu {\epsilon }_i={\alpha }^N\left({p_i}^{NS}-{p_i}^{AR}\delta g_i\right)+{\alpha }^A\left[{p_i}^{AS}+{p_i}^{AR}\delta g_i-{p_i}^{NR}\left(1-r_i{g}_i\right)\right].$$

(13)

We define $p_i^A=p_i^{AS}+p_i^{AI}+p_i^{AR}$, $p_i^N=p_i^{NS}+p_i^{NI}+p_i^{NR}$. From Equation (6), we know that $p_i^A+p_i^N=1$. In the vicinity of the propagation threshold, $p_i^I=p_i^{NI}+p_i^{AI}={\epsilon }_i\ll 1$, and $p_i^{NR}\to 0$, $p_i^{AR}\to 0$. Therefore, we can obtain $p_i^{AS}\approx p_i^A$, $p_i^{NS}\approx {1-p}_i^A$, and based on ${\beta }_i^A\approx \gamma {\beta }_i^N$, we find that

$$\begin{array}{ll}\mu {\epsilon }_i& ={\alpha }^N{p_i}^{NS}+{\alpha }^A{p_i}^{AS}\\ & =\left[\left(1-{p_i}^A+\gamma {p_i}^A\right)\right]\left(1+h\right){{\beta }_{ij}}^N{\displaystyle \sum _j{b}_{ji}{\epsilon }_j}\\ & =\left(1+h\right)\beta \left\{\left[1-\left(1-\gamma \right){p_i}^A\right]{\left[\frac{{\omega }_{ij}}{{\max }_{l=1\dots N}{\omega }_{lk}}\right]}^{\alpha }\right\}\end{array}.$$

(14)

Furthermore,

$${\displaystyle \sum _j\left\{\left\{\left(1+h\right)\left[1-\left(1-\gamma \right){p_i}^A\right]{\left[\frac{{\omega }_{ij}}{{\max }_{l=1\dots N}{\omega }_{lk}}\right]}^{\alpha }\right\}{b}_{ji}-\frac{\mu }{\beta }{t}_{ji}\right\}}{\epsilon }_j=0,$$

(15)

where $t_{ji}$ is an element of the identity matrix, let the matrix $H=\left(h_{ji}\right)$, and the corresponding elements of the matrix are defined as follows:

$$h_{ji}=\left\{\left(1+h\right)\left[1-\left(1-\gamma \right){p_i}^A\right]{\left[\frac{{\omega }_{ij}}{{\max }_{l=1\dots N}{\omega }_{lk}}\right]}^{\alpha }\right\}{b}_{ji}.$$

(16)

The unsafe behavior propagation threshold ${\beta }_C$ should be the minimum value that satisfies Equation (16). Let $\wedge \mathrm{m}\mathrm{a}\mathrm{x}\left(H\right)$ be defined as the maximum eigenvalue of matrix $H$; then, the propagation threshold can be expressed as follows:

$${\beta }_c=\frac{\mu }{\wedge \max \left(H\right)}.$$

(17)

Therefore, the propagation threshold of unsafe behaviors among construction workers is related to their safety attitudes, especially the parameters $p_i^A$ of the safety state propagation layer. In addition, factors such as the herd mentality ($h$), the recovery rate ($\mu$), and the extent of individual heterogeneity ($\alpha$) all have important impacts on the propagation threshold of unsafe behavior in the model.

3. Numerical Simulation

In this section, we conduct a comprehensive analysis of our proposed model through numerical simulations, shedding light on the various factors influencing the mechanism of unsafe behavior propagation. The safety attitudes of construction workers are influenced by their colleagues, foremen, managers, and others, meaning that safety attitudes can change through the spread of social networks among construction workers. Recognizing the scale-free property as a fundamental trait of social networks [44], we regard the layer representing changes in construction workers’ safety attitudes (Layer A) as a scale-free network. Given the presence of pivotal figures such as team leaders, technical experts, and seasoned workers within the construction milieu, the connectivity (degree) of construction workers to neighboring nodes displays a notably uneven distribution, with these key individuals exhibiting greater connections (higher degrees) to other nodes. Therefore, we establish a two-layer BA scale-free network [45] for simulation, exhibiting a power-law distribution. Each layer of the network consists of $N=1000$ nodes, with the lower layer representing the network for propagating unsafe behavior and the upper layer signifying the stratum for evolving safety attitudes. In contrast to the lower layer network, the upper layer network has an additional 500 random links (which do not overlap with existing links). Both layers of the BA scale-free network start with three connected nodes, and the number of edges of the newly added nodes connecting to the existing nodes is set to 3. Additionally, the initial proportion of individuals in the I state is set to 5%. Given these initial conditions, the probabilities of individuals occupying different states at any given time can be derived from iterative computations based on Equation (7). The proportions of individuals holding positive safety attitudes and individuals propagating unsafe behaviors when the propagation of unsafe behavior in the two-layer network reaches a steady state are set to ${\rho }^A=\sum _{i=1}^N(p_i^{AS}+p_i^{AI}+p_i^{AR})/N$ and ${\rho }^I=\sum _{i=1}^N(p_i^{AI}+p_i^{NI})/N$, respectively. Similarly, the proportions of individuals in negative, susceptible, and infected states can also be determined. The results are averaged over 500 runs of each simulation.

Initially, we use Monte Carlo (MC) simulation to test the accuracy of microscopic Markov chain approach (MMCA). Monte Carlo (MC) simulation is a classic method of utilizing computers for stochastic simulation of dynamic processes. MMCA can handle propagation dynamics at the level of single nodes and characterize propagation rates. By comparing the results from iterative MMCA calculations with those from numerical Monte Carlo (MC) simulations, we can validate the accuracy of MMCA in solving the coupled dynamics proposed in this paper. Figure 5a illustrates the fluctuations in the proportions of individuals in N and A states obtained through MC and MMCA simulations, respectively. Figure 5b compares the proportions of individuals in S, I, and R states—denoted ${\rho }^S$, ${\rho }^I$, and ${\rho }^R$—as functions of the infection rate $\beta$ in the simulation results. From the figures, it can be observed that MMCA and MC show good consistency in their simulation results, indicating that MMCA is effective and can be used to simulate the spread of unsafe behaviors among construction workers. Therefore, MMCA is further utilized to examine the impacts of various parameters on the model.

3.1. The Propagation Process of Unsafe Behaviors

In the examination of how different parameters influence the dynamic evolution process, the effect of safety attitudes on the transmission rate of unsafe behavior is delineated across two levels: (a) a moderate impact of safety attitudes on the infection rate ($\gamma =0.8$), representing individuals with a positive safety attitude having an infection rate that is 0.8 times that of individuals with a negative safety attitude; and (b) a significant impact of safety attitudes on the infection rate ($\gamma =0.2$), representing individuals with a positive safety attitude having an infection rate that is 0.2 times that of individuals with a negative safety attitude.

Initially, we explore the effects of the parameter $\alpha$, which controls the degree of individual heterogeneity, and the parameter $h$, which represents the herd mentality, on the spread of unsafe behaviors. As shown in Figure 6, it depicts the trend of the proportion of infected individuals under different $\alpha$ values as the infection rate $\beta$ increases. It can be observed that the higher the degree of influence of individual heterogeneity, the lower the proportion of infected individuals, and the higher the transmission threshold. Furthermore, when $\alpha =0$, the network of unsafe behavior propagation becomes an unweighted network, and it can be learned that the unsafe behaviors in unweighted networks spread faster than the weighted networks under the same conditions. By comparing Figure 6a,b, it can be seen that the varying impact of safety attitudes on the infection rate of unsafe behaviors affects the proportion of infected individuals in the final network. This means that when the safety attitudes of construction workers have a more significant impact on the spread of unsafe behaviors (Figure 6b), the final scale of unsafe behavior infection is lower than that when the impact of safety attitudes is smaller (Figure 6a). This impact can be achieved through safety training, education, and the presentation of accident cases to bolster the awareness of construction workers and their ability to identify hazards, thereby mitigating the final scale of unsafe behavior propagation at the individual level.

Subsequently, Figure 7 illustrates the trend of the proportion of infected individuals with varying levels of herd mentality with the escalation of the unsafe behavior propagation rate $\beta$ upon the system reaching a stable state. It is evident from the figure that as the degree of herd mentality increases, the proportion of individuals infected with unsafe behaviors also rises, indicating that herd mentality exacerbates the spread of unsafe behaviors. Moreover, upon comparing Figure 7a,b, it is discernible that when the influence of safety attitudes on the infection rate of unsafe behaviors is significant ($\gamma =0.2$), the final proportion of infected individuals is smaller than when the influence of safety attitudes is less significant ($\gamma =0.8$). This suggests that the extent of influence of safety attitudes has a certain inhibitory effect on the spread of unsafe behaviors.

Upon comparing the curves in Figure 6 and Figure 7, it can be observed that when other parameters remain unchanged, the variations in the proportions of infected individuals caused by different levels of individual heterogeneity and different levels of herd mentality differ slightly, manifested by differences in the spacing between the curves. This observation initially suggests that individual heterogeneity exerts a greater influence on the eventual scale of unsafe behavior propagation. To provide a more intuitive assessment, we plotted the phase diagram shown in Figure 8a, which consists of a grid of 50 × 50 cells, with the color indicating the proportion of infected nodes in the population (${\rho }^I$) when the system reaches stability. The value of ${\rho }^I$ is the average of 100 runs of the modified Markov chain algorithm. Analysis of the color changes in the phase diagram reveals that the extent of unsafe behavior transmission escalates with increasing herd mentality ($h$) among construction workers, while it diminishes with heightened impact of individual heterogeneity ($\alpha$), aligning with the previous conclusion. Furthermore, owing to uniform alterations in the horizontal and vertical coordinates, the vertical axis transitions from light to dark (i.e., the proportion of infected nodes in the system decreases) at a faster rate than the horizontal axis. Therefore, the impact of individual heterogeneity on the scale of unsafe behavior transmission is greater than the influence of herd mentality.

Next, we analyze the curve of the proportion of infected individuals ${\rho }^I$ as the transmission rate $\beta$ varies, considering different values of the safety education threshold ($\theta$), behavior feedback effects ($\chi$), and recovery rate of infected individuals ($\mu$). By observing all the subplots in Figure 9, we can see that regardless of the variation in the values of $\theta$ and $\chi$, a higher $\mu$ not only increases the propagation threshold but also significantly reduces the scale of the final spread of unsafe behaviors. This phenomenon implies that smaller values of $\mu$ result in a greater weakening effect on the spread of unsafe behaviors by individuals with a high recovery rate. Comparing subplots (a) and (b) in Figure 9, it can be noted that when a lower safety education threshold is chosen ($\theta =0.1$), the impact of behavior feedback diminishes. We speculate that this could be attributed to the effective implementation of widespread safety education, where the impact of behavior feedback is replaced by safety education. The comparison between subplots (a) and (c) of Figure 9 shows that an increase in the safety education threshold ($\theta$) leads to an increase in the scale of final unsafe behaviors, highlighting the importance of effective safety education. By comparing subplots (c) and (d) in Figure 9, it is evident that behavior feedback increases the number of individuals with a positive safety attitude, thus better suppressing the spread of unsafe behaviors when the safety education threshold is higher ($\theta =0.9$). Additionally, comparing subplots (a) and (d), it can be seen that the coexistence of safety education training and behavior feedback will lead to the failure of one of the mechanisms. This is primarily due to the fact that a positive safety attitude can attenuate the spread of unsafe behaviors, regardless of whether it is instilled through safety education or behavior feedback.

3.2. Threshold for the Propagation of Unsafe Behaviors

In this section, we analyze the effects of herd mentality ($h$), individual heterogeneity ($\alpha$), behavioral feedback ($\chi$), recovery rate of infected individuals ($\mu$), and forgetting rate of safety attitude ($\tau$) on the threshold of unsafe behavior transmission (${\beta }_C$) in the NAN-SIRS model. To elucidate these effects, we turn to Figure 8b, which portrays the threshold of unsafe behavior transmission across different levels of herd mentality and individual heterogeneity. Notably, the figure highlights that diminishing workers’ herd mentality while amplifying the influence of individual heterogeneity serves to elevate the threshold of unsafe behavior propagation, consequently mitigating the overall extent of unsafe behavior propagation. Moreover, upon scrutinizing the phase diagram, it becomes evident that individual heterogeneity exerts a more pronounced influence on the transmission threshold compared to herd mentality.

Figure 10 shows the trend of the transmission threshold with respect to parameter $\lambda$ under different values of parameter $\chi$. The graph indicates that the transmission threshold (${\beta }_C$) gradually increases as the level of behavioral feedback ($\chi$) increases, i.e., when $\lambda$ remains constant, a higher degree of behavioral feedback received by individuals leads to a higher transmission threshold. This suggests that construction workers in the group are influenced by both the safety attitude transmission rate ($\lambda$) and the level of behavioral feedback ($\chi$). A higher transmission rate of positive safety attitudes and a stronger impact of behavioral feedback result in a smaller overall scale of unsafe behavior spreading.

In order to further explore the impact of propagation parameters on epidemic thresholds, the trend of transmission threshold changes under different rates of forgetting safe attitudes ($\delta$) and recovery rates of infected individuals ($\mu$) is plotted. As illustrated in Figure 11, the transmission threshold curve initially ascends and then stabilizes with increasing $\lambda$. Consequently, augmenting the transmission rate of safe attitudes proves particularly efficacious when $\lambda$ is relatively low. Moreover, a noteworthy observation emerges from the figure: the transmission threshold remains unchanged with rising $\lambda$ during the initial phase, indicating the existence of a threshold ${\lambda }_C$ within the safe attitude change layer. When $\lambda <{\lambda }_C$, alterations in safe attitudes exert no influence on the spread of unsafe behavior in Layer B, and when $\lambda >{\lambda }_C$, such alterations impact the spread of unsafe behavior. Furthermore, it is essential to highlight that the larger values of $\mu$ and smaller values of $\delta$ correlate with higher transmission thresholds.

4. Discussion

To further explore the interaction between changes in safety attitudes and the spread of unsafe behaviors among construction workers, this paper proposes a two-layer network NAN-SIRS model and conducts numerical simulations. In comparison with existing research [46,47], this paper not only analyzes the impact of safety attitudes of construction workers on unsafe behaviors but also delves into the mechanisms of their interaction. Furthermore, it considers the influence of factors such as conformity psychology and heterogeneity among construction workers on the spread of unsafe behaviors. The main findings of this paper are as follows:

• Enhancing the impact of individual heterogeneity among construction workers can effectively elevate the threshold for the spread of unsafe behaviors, thereby diminishing the ultimate scale of their propagation. The herd mentality of construction workers increases the scale of the spread of unsafe behaviors, but has a relatively minor impact on the threshold of propagation, aligning with extant research findings [34]. Within construction teams, pivotal figures such as team leaders and mentors assume a dual role in management and exemplification. Apart from facilitating the exchange of safety knowledge at the construction site, they must lead by example, demonstrating a greater commitment to safe practices. Furthermore, managers should not only monitor the behaviors of the workers but also those of these key figures. As the conduct of these influential individuals becomes more standardized, construction workers are likely to emulate safer behaviors.
• Lowering the thresholds for safety education and intensifying behavioral feedback levels both contribute to diminishing the propagation of unsafe behaviors, aligning with prior research highlighting the substantial impact of managerial feedback and safety training on workers’ unsafe practices [26]. However, we further reveal that the coexistence of both mechanisms can lead to the failure of one, as they both function by cultivating positive safety attitudes to mitigate the spread of unsafe behaviors. Therefore, creating a good safety atmosphere through effective safety education and training and encouraging workers to adopt a positive safety attitude can prevent potential problems. This not only reduces consumption and accidental losses caused by behavioral feedback but also effectively suppresses the spread of unsafe behaviors.
• The higher the rate of the propagation of positive safety attitudes and the stronger the impact of behavioral feedback, the higher the propagation threshold becomes. Managers ought to proactively gather information on unsafe behaviors among construction workers and implement timely and effective intervention measures to mitigate the spread of unsafe behavior by elevating the propagation threshold. This becomes especially crucial when traditional safety education and training methods fail to address the issue adequately.
• The recovery rate of infected individuals is directly proportional to the propagation threshold, while the forgetting rate of safety attitudes is inversely proportional to the propagation threshold. Managers must take proactive steps to enhance the recovery rate of affected individuals and diminish the rate of safety attitude decay through strategies such as instituting robust safety standards. This approach serves to elevate the propagation threshold, thereby diminishing the extent of unsafe behavior propagation.

By analyzing the interaction between safety attitudes and the propagation of unsafe behaviors among construction workers, this paper enriches the research findings in relevant fields. It provides new research perspectives for exploring the impact of psychological factors on the spread of unsafe behaviors among construction workers, which is of significant importance for advancing research in related areas. Additionally, this paper employs numerical simulations to identify key factors influencing the propagation of unsafe behaviors among construction workers. Our results thus offer theoretical support for project managers in enhancing safety management, suppressing the spread and evolution of unsafe behaviors among construction workers, and ultimately improving construction safety. Consequently, they will contribute to a reduction in safety incidents in the construction industry.

5. Conclusions

Given the frequent occurrences of safety incidents in the construction industry, with a focus on the spread of unsafe behaviors among construction workers and the aim of reducing safety incidents on construction sites, this paper constructs a two-layer network NAN-SIRS model. It separately describes the changes in safety attitudes among construction workers, the propagation of unsafe behaviors, and their interactions. The model takes into account the influence of herd mentality and node heterogeneity caused by key individuals on the spread of unsafe behaviors among construction workers and establishes a partial mapping relationship based on behavioral feedback. Utilizing a microscopic Markov chain approach (MMCA), the paper derives thresholds for the spread of unsafe behaviors among construction workers. Subsequently, through numerical simulations, the model’s feasibility is confirmed, and the impacts of parameter variations on the spread scale and threshold are thoroughly analyzed. The results indicate that enhancing the influence of behavioral feedback and individual heterogeneity will increase the threshold for the spread of unsafe behaviors, effectively reducing their propagation. The coexistence of safety education training and behavioral feedback will lead to the failure of one of these mechanisms.

Although our research findings offer some valuable insights into curbing the spread of unsafe behaviors among construction workers, there are still some limitations. Firstly, this study considers the heterogeneity of individual influence based on node degree, but individuals’ influence on others may also depend on factors such as their intimacy, trustworthiness, or relationship type. Future research will encompass a broader spectrum of heterogeneous influences. Secondly, although numerical simulations are conducted to explore key influencing factors, the integration of empirical case data will be pursued to validate the model’s efficacy comprehensively.