The impact of the self-recognition ability and physical quality on coupled negative information-behavior-epidemic dynamics in multiplex networks

Abstract

In recent years, as the COVID-19 global pandemic evolves, many unprecedented new patterns of epidemic transmission continue to emerge. Reducing the impact of negative information diffusion, calling for individuals to adopt immunization behaviors, and decreasing the infection risk are of great importance to maintain public health and safety. In this paper, we construct a coupled negative information-behavior-epidemic dynamics model by considering the influence of the individual's self-recognition ability and physical quality in multiplex networks. We introduce the Heaviside step function to explore the effect of decision-adoption process on the transmission for each layer, and assume the heterogeneity of the self-recognition ability and physical quality obey the Gaussian distribution. Then, we use the microscopic Markov chain approach (MMCA) to describe the dynamic process and derive the epidemic threshold. Our findings suggest that increasing the clarification strength of mass media and enhancing individuals' self-recognition ability can facilitate the control of the epidemic. And, increasing physical quality can delay the epidemic outbreak and leads to suppress the scale of epidemic transmission. Moreover, the heterogeneity of the individuals in the information diffusion layer leads to a two-stage phase transition, while it leads to a continuous phase transition in the epidemic layer. Our results can provide favorable references for managers in controlling negative information, urging immunization behaviors and suppressing epidemics.

1. Introduction

In the history of human development, infectious epidemics have posed important threats to the stability of the economic and social order, such as the smallpox virus, SARS, and the Black Death [1], [2], [3], [4]. Nowadays, during the global COVID-19 epidemic, we are committed to suppress the scale of epidemic transmission by reducing the impact of negative information diffusion and increasing the proportion of individuals who adopt immunization. How to better study the dynamics of epidemic transmission and understand the underlying mechanisms are important for controlling the epidemic infection [4], [5], [6], [7].

In the early stages of epidemic transmission research, Kermack et al. [8] obtained the dynamic critical values of epidemic transmission through a mathematical model, which also laid a foundation for later research on epidemic transmission dynamics. Afterward, some classical models such as SIS (Susceptible-Infected-Susceptible) [9] and SIR (Susceptible-Infected-Removed) [10] model have laid the foundation in the field of epidemic transmission. Moreover, with the continuous research on the dynamics of epidemic transmission, we found that information plays an important role in the epidemic transmission process. Epidemic-related information is constantly diffused through mass media and social networks, and people who are aware of the information tend to influence their perceptions and thus change their behavioral dynamics [11], [12]. For example, when individuals hear epidemic-related information, they usually take self-protective measures, such as wearing a mask and keeping a safe distance [13], [14]. Kabir et al. [15] developed SIR-UA dynamic system and explored the effect of awareness on epidemic model. In addition, with the advancement of research related to multiplex networks, it laid the foundation for studying epidemic transmission [16], [17]. Granell et al. [18] constructed a two-layer UAU-SIS model on multiplex networks to study the coupling effects of information diffusion and epidemic transmission. Kabir et al. [19] presented a two-layer SIR-UA epidemic model to analyze the effect of different heterogeneous networks in a population. Xia et al. [20] investigated the competing effects of positive and negative information on epidemics. Guo et al. [21] introduced the Heaviside step function to information diffusion by setting a local awareness rate to describe the herd effect of individuals in real life, and the results induced a two-stage effect of epidemic threshold and final epidemic size. In addition, during the COVID-19 epidemic, mass media played an important role in refuting negative information, correcting misinformation perceptions and guiding people to adopt correct behaviors [22], [23], [24]. Xia et al. [25] investigated the impact of mass media on epidemic transmission, and their results showed that mass media had an important impact on delaying epidemic outbreaks and reducing the final scale. Facts show that mass media can properly mitigate the negative impact of negative information by dispelling them in time [26], [27]. Yang et al. [28] focused on the design of routing strategies for traffic-driven epidemic spreading, proposed an adaptive routing strategy that incorporates topological distance with local epidemic information. These studies have suggested that it is important to study the coupling effect of information diffusion and the clarification strength of mass media on the dynamics of epidemic transmission.

There has been a long history of research on the effects of information on behavior. In fact, during epidemic transmission, information often influences individuals' decisions about behavioral adoption. When individuals aware epidemic-related information, they tend to make certain behavioral choices. Take vaccination behavior in COVID-19 as an example, when individuals hear positive information about vaccines for COVID-19, they will be more inclined to get vaccinated. Kuga et al. [29] considered two imperfect ways to protect against epidemic, namely imperfect vaccination and a defense against contagion such as wearing a mask, and revealed that a defense against contagion was marginally inferior to an imperfect vaccination as long as the same coefficient value was used. Conversely, when individuals hear negative information about vaccines, they are more hesitant to adopt vaccination behavior, which is often detrimental to epidemic prevention and control of COVID-19 [30], [31], [32]. In addition, other scholars have studied vaccination behavior in terms of benefit and cost [33], vaccine limitations [34], and social effect [35]. Gao et al. [36] constructed a model for city residents' green awareness-behavior spreading with policy regulation. The result has shown that promotion from government policy plays the most important role in city residents' green behavior spreading. Zhang et al. [37] investigated the effects of two types of incentives strategies on vaccination. Kabir et al. [38] considered a new mathematical framework for a vaccination game combined with SIR and UA situation and studied three different strategy updating rules concerning whether an individual committing or not vaccination. Further, Kabir et al. [39] established a two-layer SIR/V epidemic model that incorporates the effects of information buzz and information costs in the framework of a vaccination game. Li et al. [40] proposed an evolutionary vaccination game model in a multiplex network and explored how information diffusion affects vaccination and showed the role of information and behavior in controlling epidemics. In addition, the behavior of individuals will greatly influence the status of individuals in the physical contact network, thus affecting their infection probability, which in turn has an impact on the overall epidemic transmission process. Yang et al. [41] studied the interplay between individual behaviors and epidemic spreading in a dynamical network and investigated the effects of risk-averse migration on epidemic spreading. Ye et al. [42] proposed a heterogeneous epidemic-behavior-information transmission model, focusing on the impact of risk perception and behavioral changes on epidemic transmission. Yin et al. [43] constructed an information-behavior-epidemic three-layer network model that discussed the influence of negative vaccine-related information, vaccination behavior on epidemic transmission, and their results showed that this three-layer network structure could better describe the heterogeneity of real networks, and that factors related to vaccination behavior greatly influenced the epidemic transmission.

In addition to information diffusion and behavior adoption, the complex of epidemic transmission process in society is often greatly influenced by individual heterogeneity [44]. How to better describe the heterogeneity of individuals in real societies has been of great interest to scholars. Heterogeneity in individual psychological level, physical state, and social status may influence the dynamics of disease transmission [45]. Lu et al. [46] improved the SIR-A model to consider the community size and used continuous awareness values of individuals. Peng et al. [47] proposed a behavior spreading model to describe the heterogeneity of individuals in adopting the behavior, in which the adoption threshold follows the Gaussian distribution. Sun et al. [48] distinguished susceptibility and infectivity between aware and unaware individuals, relaxed the degree of immunization, and took into account three types of generation mechanisms of individual awareness. Tian et al. [49] considered edge weight distribution and adoption heterogeneity on the networks separately, where the adoption thresholds of nodes obey truncated Gaussian distribution. Cui et al. [50] constructed a multi-layer weighted social network to catch individual intimacy heterogeneity, and then build an adoption threshold model with a tent-like probability function to explore the IFPT characteristics. Zhang et al. [51] introduced local and global epidemic information into individual precaution degrees to reflect individual heterogeneity. In previous studies, most scholars have considered the heterogeneity of infectivity and susceptibility of individuals in the physical contact network, but few scholars have discussed the impact of individual physical quality heterogeneity on epidemic transmission. In fact, there is a cumulative effect on the individual's immune system in the process of being infected and only when the accumulation reaches a certain level does the infection manifest itself [52], [53], [54], [55]. Moreover, for the immunization behavior, many scholars have studies some counterintuitive phenomena. Zhang et al. studies the doubled-edged sword effect of the peer pressure [56] and rational behavior [57]. Pires et al. [58] besides we observe the emergence of an intermediate optimal segregation level where the community structure enhances negative opinions over vaccination but counterintuitively hinders the global disease spreading. Zhang et al. [59] found a counter-intuitive phenomenon analogous to the well-known Braess's Paradox, namely a better condition may lead to worse performance. From this point of view, it is worth discussing how to describe the intricate psychological and physiological phenomena of individuals in real transmission.

Although many scholars have discussed the influence of information on the epidemic in depth, the co-evolution among information, behavior, and epidemic has rarely been studied, and the intricate mechanisms can be better described by constructing a coupled model in multiplex network. Many scholars have hypothesized that when individuals become aware of information, they will immediately adopt protective behaviors to protect themselves from infection. However, in reality, awareness of information and adoption of behavior are often not synchronized. In the case of COVID-19, for example, when individuals become aware of epidemic-related information, they may choose not to get vaccinated because they do not trust the information or do not care about the risk of the epidemic. In addition, when individuals hear negative information related to immunization behavior, such as the side effects of vaccines, they may also choose to adopt the immunization behavior and get vaccinated in order to reduce their own probability of infection. Therefore, it is necessary to study the process of information diffusion separately from the process of behavior adoption. In addition, the individual state change is subject to a decision and adoption process. The individual first decides whether to make a decision, which is a 0 and 1 process. Therefore, it is more reasonable to use the Heaviside step function to describe the decision process. When the individual decides to make a decision with 1, they will choose to adopt with a certain probability. In real life, the self-recognition and physical qualities of individuals are different. Taking COVID-19 as an example, individuals with poor self-recognition abilities always can't recognize unconfirmed information and be misled by negative information, especially the immunization-related negative information, and their probability of receiving the immunization behavior might decrease. Moreover, the infection process is affected by both the physical quality of the individual and the environment in which the individual lives. On the one hand, individuals with low physical quality are more likely to get infected. That is to say, when individuals' physical quality is high, their infection risk can be greatly reduced. On the other hand, when the number of infected neighbors around a susceptible individual is large, the risk of the susceptible individual becoming infected is relatively high. It's shown that patients with COVID-19 exhibit lymphopenia and high cytokine levels and its infection is closely related to immune system function [60]. And the individuals will show the corresponding transmission character when the infection reaches a certain level. Therefore, it is also necessary to study the impact of the environment and the heterogeneity of individual physical abilities.

Motivated by the above analysis, we propose the coupled UAU-DKD-SIS model with considering mass media to explore the co-evolution among the negative information diffusion, immunization behavior adoption and epidemic transmission. We focus on the influence of individuals' self-recognition ability and physical quality on the transmission dynamics in the multiplex network. Among them, we introduce the Heaviside step function for each layer to portray the decision-adoption process and the Gaussian distribution to explore the effect of individual heterogeneity on the transmission. By utilizing the microscopic Markov chain approach (MMCA), we derive the expression of the epidemic threshold. The correctness of the dynamical equations and the epidemic threshold is verified by extensive simulations.

The framework of this paper is as follows: In Section 2, we describe our three-layer UAU-DKD-SIS model with considering mass media with related assumptions. In Section 3, we apply a microscopic Markov chain approach and derive the threshold of epidemic transmission. In Section 4, we analyze the model by using extensive numerical simulation. Finally, in Section 5, we make a summary of this work.

2. The three-layer coupled model

The impact of information on epidemic transmission is essentially subject to individuals' immunization behavior. During the COVID-19 epidemic, timely clarifying the unconfirmed information by mass media is an important means of suppressing negative information diffusion. By releasing official information promptly, mass media can clarify the unconfirmed information to reduce the possibility of individuals believing negative information. In the case of vaccination, for example, negative information tends to make people hesitate to adopt the COVID-19 immunization behavior, which influences immunization. In addition, different individuals have different self-recognition qualities. Individuals with low self-recognition are easily affected by negative information. Also, individuals are influenced by the herd effect when making a decision, when most of the neighbors around an individual believe the negative information or adopt the immunization behavior, the individual may also follow the trend and thus believe the negative information or adopt the immunization behavior. Moreover, in the epidemic transmission process, individuals with different infection probabilities due to their different states and physical qualities, which also influence epidemic transmission.

To better describe this phenomenon, as is shown in Fig. 1 , we consider a three-layer network coupled model including of negative information transmission, immunization behavior adoption and epidemic transmission, in which we introduce a Heaviside step function for each layer of the transmission process to describe the decision-adoption process, and focus on the influence of individual's self-recognition ability and physical quality on the transmission dynamics. The model assumptions in this paper are as follows.

Assumption 1

Negative information diffusion process.

Fig. 1.

Structure of the three-layer UAU-DKD-SIS propagation network framework with considering mass media.

In the negative information diffusion layer, we mainly consider the negative information related to immunization behavior (risk of adopting immune behavior, side effects of immunization, etc.). In this layer, the nodes include two states, U (unaware) and A (aware). Individuals in state U are unaware of negative information, and individuals in state A are aware of negative information and spread it. In real life, individuals who are aware of negative information often influenced by two factors. The first factor is the individual's self-recognition ability [61], [62], which means one's ability to distinguish true information from unconfirmed information. When the individual's self-recognition ability is high, he or she will be able to distinguish the unconfirmed information with high probability and will be more difficult to believe negative information, that is, they have a higher threshold. When the individual's self-recognition ability is low, he or she will be easily misled by negative information. To better describe the heterogeneity of individuals' self-recognition ability, we set that the individuals' self-recognition ability ${\omega }_i\left(0\le {\omega }_i\le 1\right)$, and assume it follows the Gaussian distribution $G\left({\omega }_i\right)~N\left({\mu }_1,{\sigma }_1^2\right)$, where ${\mu }_1$ can be regarded as the average self-recognition ability of the whole population. The second factor is the influence of the herd effect [63], [64], [65]. The herd effect is that each individual's decision-adoption process is under pressure from the group to develop changes in their decision and speech in a direction consistent with the group. When there are more negative information spreaders in the neighborhood, individuals may be influenced by their neighbors and thus believe negative information with a higher probability. Here, we consider the process by which individuals believe negative information is a decision-adoption process, and therefore we introduce a Heaviside step function to describe the decision process, i.e., if $x>0$, $F\left(x\right)=1$, else $F\left(x\right)=0$. Taking the above factors into account, we assume that nodes in state U will aware negative information with probability ${\lambda }_i\left(t\right)$ that,

$${\lambda }_i\left(t\right)=\mathit{\lambda F}\left(\frac{\sum _{j=1}{P}_j^A\left(t\right)}{k_i}-{\omega }_i\right)$$ (1)

where, $\lambda$ is the initial information diffusion rate., $\frac{\sum _{j=1}{P}_j^A\left(t\right)}{k_i}$ denotes the proportion of individuals who know the negative information among the neighbors $j$ of node $i$ at moment $t$. $P_j^A\left(t\right)$ is the probability that individual $j$ is in state $A$ at moment $t$. ${\omega }_i$ represents the individual's self-recognition ability, and a larger ${\omega }_i$ indicates a higher self-recognition ability. $0<{\omega }_i<1$, on the one hand, the larger ${\omega }_i$ is, the higher the ability to identify unconfirmed information, and the lower the probability of individuals to believe in negative information is. On the other hand, the more neighbors around an individual believe in negative information, the greater the probability that he or she believes in negative information. When ${\omega }_i=1$, the individual has the highest self-recognition ability and is completely unaffected by the surrounding neighborhood and will not believe negative information. When ${\omega }_i=0$, individuals have the lowest self-recognition ability, and they are more vulnerable to neighbors' influence.

While nodes in state $A$ will forget the negative information with probability $\delta$.

Assumption 2

Mass media negative information clarification process.

In real life, clarifying unconfirmed information timely is important to optimize the information diffusion environment by using mass media [66], [67]. Here, we assume that the officially authorized mass media will release some official information to clarify the unconfirmed information, and use measures to control the negative information spreaders. For instance, during COVID-19, there was a lot of negative information about vaccines on social networks, and timely refuting of these negative information by the media could reduce the impact of negative information, and make the public realize the importance of vaccines [68]. Here, the mass media mainly plays the role of eliminating the spreaders of negative information. We assume that an individual in state A is eliminated with probability m by the mass media, and then change to state U.

Based on the above discussion, we can obtain the propagation process of the negative information diffusion in Fig. 2 .

Assumption 3

Immunization behavior adoption process.

Fig. 2.

Propagation process of the negative information diffusion.

In the immunization behavior adoption layer, we mainly consider the adoption of immunization behavior, such as vaccination. In this layer, nodes include two states, K (not adopted) and D (adopted). In real life, when we are faced with whether to adopt immunization behavior, we are often influenced by two factors. The first is the influence of our mental subjective factor, and the second receives the objective influence of the herd effect from our neighborhood. Taking vaccination behavior as an example, if the individual's acceptance of vaccination behavior is low, they will usually subjectively hesitate to adopt vaccination behavior, which means a higher adoption threshold and a lower possibility of adoption. And when the individual's acceptance of immunization behavior is high, the threshold of its adoption will also increase. Besides, when individuals learn negative information related to vaccination, they may reduce their willingness to be vaccinated. Thus, the behavior adoption process can be described as a decision-adoption process. We assume ${\theta }_i$ to represent the i individual's adoption threshold. The larger the ${\theta }_i$, the more difficult it is for individual to adopt immunization behavior. We argue that individuals have lower ${\theta }_i$ when they are aware of negative information, so we assume that,

$${\theta }_i=\left\{\begin{array}{cc}\theta & \mathit{thestateof}A\\ \mathit{\xi \theta }& \mathit{thestateof}U\end{array}\right)$$ (2)

where $\theta$ denotes the individuals' adoption threshold which is not influenced by other factors. The larger the $\theta$, the more difficult it is for the individual to adopt the immunization behavior. $\xi$ is used to portray the effect that individuals who are unaware of the negative information have a lower adoption threshold. A smaller $\xi$ means of being unaware of the negative information can benefit more the immunization behavior adoption process, $\xi \in \left[01\right]$.

As for the effect of the herd effect, when an individual is surrounded by the percentage of neighbors who adopt immunization behaviors, he or she will tend to follow the trend and adopt immunization behaviors and vice versa. Thus, we also introduce the Heaviside step function to portray the immunization behavior adoption process.

In summary, we hypothesize that nodes in state D will adopt the immunization behavior with the probability that,

$${\alpha }_i\left(t\right)=\mathit{\alpha F}\left(\frac{\sum _{j=1}{P}_j^K\left(t\right)}{k_i}-{\theta }_i\right)$$ (3)

where, $\frac{\sum _{j=1}{P}_j^K\left(t\right)}{k_i}$ means the proportion of individuals who adopt the immunization behavior among the neighbors j of node i at moment t. $\alpha$ is the basic behavior adoption rate that is not influenced by other factors. $P_j^K\left(t\right)$ is the probability that individual j is in state A at moment t. Individuals will adopt immune behaviors with probability $\alpha$ only when $\frac{\sum _{j=1}{P}_j^K\left(t\right)}{k_i}>{\theta }_i$.

While nodes in state K will face immunization failure with probability $\eta$.

Based on the above discussion, we can obtain the propagation process of the immunization behavior adoption in Fig. 3 .

Assumption 4

Epidemic transmission process.

Fig. 3.

Propagation process of the immunization behavior adoption.

In the epidemic transmission layer, nodes include two states S (susceptible) and I (infected). We hypothesize that an individual is infected concerning not only the individual's physical quality but also the environment in which the individual lives [69], [70]. For the actual situation, individuals with good physical quality tend to have higher immunity to epidemic, while individuals with poor physical quality tend to be more susceptible to epidemic. To better describe the heterogeneity of individuals' physical quality, we set that the individuals' physical quality $h_i\left(0\le h_i\le 1\right)$, and assume it follows the Gaussian distribution $Q\left(h_i\right)~N\left({\mu }_2,{\sigma }_2^2\right)$, where ${\mu }_2$ can be regarded as the average physical quality of the whole population. In addition, we believe that the adoption of immunization behaviors can improve individuals' physical quality. When individuals adopt immunization behaviors, their physical quality is often stronger than those who do not adopt. We assume that,

$$h=\left\{\begin{array}{cc}\gamma h_i& \mathit{thestateof}D\\ h_i& \mathit{thestateof}K\end{array}\right)$$ (4)

where $\gamma$ portrays the reduction degree of physical quality for individuals who do not adopt immunization behaviors, and $0\le \gamma \le 1$. Furthermore, the infection rates for individuals in different states are discussed in detail in Assumption 6.

While a node in state I will be cured and return to state S with probability $\mu$.

Assumption 5

The node state.

Based on Assumption 1, Assumption 2, Assumption 3, Assumption 4, we can conclude that there are eight node states coupled in the three-layer network, namely, UDS, UDI, UKS, UKI, ADS, ADI, AKS, and AKI. Here, we assume that once individuals become infected, they will know the dangers of negative information and then adopt immunization behaviors. The individuals with four states of UDI, ADI, and AKI will be transformed into UKI individuals with probability 1, i.e.

$$\mathit{UDI},\mathit{ADI},\mathit{AKI}\stackrel{1}{\to }\mathit{UKI}$$

As a result, we can obtain five valid node states: UDS, UKS, ADS, AKS and UKI.

Assumption 6

Coupling effect of the negative information diffusion layer and immunization behavior adoption layer on the epidemic transmission layer.

We hypothesize that the infection probability in a physical contact network is influenced by a number of components. One component is the individual's state in the information diffusion process and the immunization behavior adoption process. In real life, when individuals believe negative information or do not adopt immunization behaviors, it tends to make individuals less vigilant in contact with others (e.g., not adopting effective self-protection measures), and this raises the infection probability in the epidemic transmission layer. Another component is the individual's physical quality. It is generally assumed that individuals with higher physical quality tend to have higher resistance to infection, thus reducing their risk of infection. Based on the above hypothesis, we believe that the probability of infection is reduced ${\phi }_1$ when individuals do not believe the negative information, and the probability of infection is reduced ${\phi }_2$ when individuals adopt immunization behavior. From this, we can obtain the basic infection rate ${\beta }^X$ as follows:

$${\beta }^X=\left\{\begin{array}{cc}\beta & \mathit{thestateof}\mathit{ADS}\\ {\phi }_1\beta & \mathit{thestateof}\mathit{UDS}\\ {\phi }_2\beta & \mathit{thestateof}\mathit{AKS}\\ {\phi }_1{\phi }_2\beta & \mathit{thestateof}\mathit{UKS}\end{array}\right)$$ (5)

where $X\in \left\{\mathit{UKS},\mathit{AKS},\mathit{UDS},\mathit{ADS}\right\}$, ${\phi }_1$ and ${\phi }_2$ is used to characterize the attenuation factor of infection rate in individuals with information and behavior respectively, ${\phi }_1,{\phi }_2\in \left[01\right]$.

To simplify the model, we consider the impact of information has the same extent as behavior, namely ${\phi }_1={\phi }_2=\phi$. The simplified parameters are as follows.

$${\beta }^X=\left\{\begin{array}{cc}\beta & \mathit{thestateof}\mathit{ADS}\\ \mathit{\phi \beta }& \mathit{thestateof}\mathit{AKS},\mathit{UDS}\\ {\phi }^2\beta & \mathit{thestateof}\mathit{UKS}\end{array}\right)$$ (6)

where $\phi$ is used to characterize the attenuation factor of infection rate in individuals with different status, $\phi \in \left[01\right]$.

In fact, as we described above, the infection process is closely related to the environment in which the individual is exposed and the individual's physical quality. On the one hand, the larger the $h_i$ is, the higher the individual's physical quality is, and the smaller the probability of infection is. On the other hand, the more infected neighbors around an individual with susceptible state, the greater the risk of his environment and the greater the infection probability is. Therefore, we also introduce Heaviside step function to characterize the epidemic transmission process. In addition, we believe that when individuals do not adopt immunization behavior, their physical quality reduces the probability of $\gamma$ compared to individuals who adopt immunization behavior.

In summary, the probability of infection for a susceptible individual is not only influenced by their own state, but also by the surrounding environment. The infection probability for individuals in different states is as follows.

$${\beta }_i^{\mathit{ADS}}\left(t\right)={\beta }^{\mathit{ADS}}F\left(\frac{\sum _{j=1}{P}_j^I\left(t\right)}{k_i}-\gamma h_i\right)$$ (7)

$${\beta }_i^{\mathit{AKS}}\left(t\right)={\beta }^{\mathit{AKS}}F\left(\frac{\sum _{j=1}{P}_j^I\left(t\right)}{k_i}-h_i\right)$$ (8)

$${\beta }_i^{\mathit{UDS}}\left(t\right)={\beta }^{\mathit{UDS}}F\left(\frac{\sum _{j=1}{P}_j^I\left(t\right)}{k_i}-\gamma h_i\right)$$ (9)

$${\beta }_i^{\mathit{UKS}}\left(t\right)={\beta }^{\mathit{UKS}}F\left(\frac{\sum _{j=1}{P}_j^I\left(t\right)}{k_i}-h_i\right)$$ (10)

where, $\frac{\sum _{j=1}{P}_j^I\left(t\right)}{k_i}$ denotes the proportion of infected individuals among the neighbors j of node i at moment t. $h_i$ denotes the individuals' physical quality, $\gamma$ portrays the reduction degree of physical quality for individuals who do not adopt immunization behaviors. $P_j^I\left(t\right)$ is the probability that individual j is in state I at moment t. Individuals will adopt be infected with probability ${\beta }^X$, $X\in \left\{\mathit{UKS},\mathit{AKS},\mathit{UDS},\mathit{ADS}\right\}$ only when $\frac{\sum _{j=1}{P}_j^I\left(t\right)}{k_i}$ reaches the physical quality h.

Based on the above discussion, we can obtain the propagation process of the epidemic transmission in Fig. 4 .

Fig. 4.

Propagation process of the epidemic transmission.

In summary, the propagation process of each layer is shown in Fig. 5 . The definitions of parameters are shown in Table.1 .

Fig. 5.

Schematic diagram of propagation process and individual state at each layer.

Table. 1.

Definition of parameters.

Parameter	Definition
$N$	Number of nodes in the network
$m$	The clarification strength of mass media
$\lambda$	The initial negative information diffusion rate
${\omega }_i$	The self-recognition ability of individual i
$\delta$	The negative information forgetting rate
$\theta$	The adoption threshold
$\xi$	The reduction factor of adoption threshold
$\alpha$	The basic behavior adoption rate
$\eta$	The immunization failure rate
$h_i$	The physical quality of individual i
$\gamma$	The reduction factor of physical quality
$\mu$	The epidemic recovery rate
$\beta$	The epidemic transmission rate
${\phi }_1$	The reduction factor of epidemic transmission rate for U individuals
${\phi }_2$	The reduction factor of epidemic transmission rate for K individuals
${\mu }_1$	The expectation of Gaussian distribution for self-recognition ability
${\mu }_2$	The expectation of Gaussian distribution for physical quality
${\sigma }_1$	The standard deviation of Gaussian distribution for self-recognition ability
${\sigma }_2$	The standard deviation of Gaussian distribution for physical quality ability

3. The theoretical analysis based on MMCA

According to our three-layer UAU-DKD-SIS model with considering mass media, based on assumption 6, the N individuals can be in five states: UDS, UKS, ADS, AKS, and UKI. The probabilities that individual i being in these five states at time step t can be represented as $P_i^{\mathit{UDS}}\left(t\right)$, $P_i^{\mathit{UKS}}\left(t\right)$, $P_i^{\mathit{ADS}}\left(t\right)$, $P_i^{\mathit{AKS}}\left(t\right)$, and $P_i^{\mathit{UKI}}\left(t\right)$, respectively, satisfying the normalization condition $P_i^{\mathit{UDS}}\left(t\right)+P_i^{\mathit{UKS}}\left(t\right)+P_i^{\mathit{ADS}}\left(t\right)+P_i^{\mathit{AKS}}\left(t\right)+P_i^{\mathit{UKI}}\left(t\right)=1$.

Let $A={\left(a_{\mathit{ij}}\right)}_{N\times N}$, $B={\left(b_{\mathit{ij}}\right)}_{N\times N}$ and $C={\left(c_{\mathit{ij}}\right)}_{N\times N}$ be the adjacency matrices of the negative information diffusion network, the immunization behavior adoption network and the epidemic transmission network, respectively. Then, we amuse $r_i\left(t\right)$ to represent the probability of unaware individual $i$ not being informed by any neighbor at time $t$ in the negative information diffusion layer, ${\alpha }_i^U\left(t\right)$ to represent the probability of $\mathit{UDS}$ individual $i$ adopt immunization behavior at time $t$, ${\alpha }_i^A\left(t\right)$ to represent the probability of $\mathit{ADS}$ individual $i$ adopt immunization behavior at time $t$, $q_i^{\mathit{UDS}}\left(t\right)$ to represent the probability of $\mathit{UDS}$ individual not being infected by any neighbor at time $t$, $q_i^{\mathit{UKS}}\left(t\right)$ to represent the probability of $\mathit{UKS}$ individual not being infected by any neighbor at time $t$, $q_i^{\mathit{ADS}}\left(t\right)$ to represent the probability of $\mathit{ADS}$ individual not being infected by any neighbor at time $t$, and $q_i^{\mathit{AKS}}\left(t\right)$ to represent the probability of $\mathit{AKS}$ individual not being infected by any neighbor at time $t$, respectively. Assuming the absence of dynamical correlations in the three-layer networks, they can be calculated as follows:

$$r_i\left(t\right)=\mathit{\lambda F}\left(\frac{\sum _{j=1}{P}_j^A\left(t\right)}{k_i}-{\omega }_i\right)$$ (11)

$${\alpha }_i^U\left(t\right)=\mathit{\alpha F}\left(\frac{\sum _{j=1}{P}_j^K\left(t\right)}{k_i}-\mathit{\xi \theta }\right)$$ (12)

$${\alpha }_i^A\left(t\right)=\mathit{\alpha F}\left(\frac{\sum _{j=1}{P}_j^K\left(t\right)}{k_i}-\theta \right)$$ (13)

$$q_i^{\mathit{ADS}}\left(t\right)=\prod _j\left[1-c_{\mathit{ji}}{P}_j^I\left(t\right){\beta }_i^{\mathit{ADS}}\left(t\right)\right]$$ (14)

$$q_i^{\mathit{AKS}}\left(t\right)=\prod _j\left[1-c_{\mathit{ji}}{P}_j^I\left(t\right){\beta }_i^{\mathit{AKS}}\left(t\right)\right]$$ (15)

$$q_i^{\mathit{UDS}}\left(t\right)=\prod _j\left[1-c_{\mathit{ji}}{P}_j^I\left(t\right){\beta }_i^{\mathit{UDS}}\left(t\right)\right]$$ (16)

$$q_i^{\mathit{UKS}}\left(t\right)=\prod _j\left[1-c_{\mathit{ji}}{P}_j^I\left(t\right){\beta }_i^{\mathit{UKS}}\left(t\right)\right]$$ (17)

where $P_j^A\left(t\right)=P_j^{\mathit{ADS}}\left(t\right)+P_j^{\mathit{AKS}}\left(t\right)$, $P_j^K\left(t\right)=P_j^{\mathit{UKS}}\left(t\right)+P_j^{\mathit{AKS}}\left(t\right)$, $P_j^I\left(t\right)=P_j^{\mathit{UKI}}\left(t\right)$.

Then, we can construct the transition probability trees of each state in our UAU-DKD-SIS model with considering mass media, as shown in Fig. 6 . Based on the transition probability trees, we can develop the transition equations for five possible states using the MMCA [71], [72] in Eqs. (14), (15), (16), (17).

$$\begin{array}{c}{P}_i^{\mathit{UKI}}\left(t+1\right)=P_i^{\mathit{UDS}}\left(t\right)\left\{r_i\left(t\right)+\left[1-r_i\left(t\right)\right]m\right\}\left\{{\alpha }_i^U\left(t\right)\left[1-q_i^{\mathit{UKS}}\left(t\right)\right]+\left[1-{\alpha }_i^U\left(t\right)\right]\left[1-q_i^{\mathit{UDS}}\left(t\right)\right]\right\}\\ +P_i^{\mathit{UDS}}\left(t\right)\left\{1-r_i\left(t\right)\right\}\left(1-m\right)\left\{{\alpha }_i^A\left(t\right)\left[1-q_i^{\mathit{AKS}}\left(t\right)\right]+\left[1-{\alpha }_i^A\left(t\right)\right]\left[1-q_i^{\mathit{ADS}}\left(t\right)\right]\right\}\\ +P_i^{\mathit{UKS}}\left(t\right)\left\{r_i\left(t\right)+\left[1-r_i\left(t\right)\right]m\right\}\left\{\left(1-\eta \right)\left[1-q_i^{\mathit{UKS}}\left(t\right)\right]+\eta \left[1-q_i^{\mathit{UDS}}\left(t\right)\right]\right\}\\ +P_i^{\mathit{UKS}}\left(t\right)\left\{1-r_i\left(t\right)\right\}\left(1-m\right)\left\{\left(1-\eta \right)\left[1-q_i^{\mathit{AKS}}\left(t\right)\right]+\eta \left[1-q_i^{\mathit{ADS}}\left(t\right)\right]\right\}\\ +P_i^{\mathit{ADS}}\left(t\right)\left[\delta +\left(1-\delta \right)m\right]\left\{{\alpha }_i^U\left(t\right)\left[1-q_i^{\mathit{UKS}}\left(t\right)\right]+\left[1-{\alpha }_i^U\left(t\right)\right]\left[1-q_i^{\mathit{UDS}}\left(t\right)\right]\right\}\\ +P_i^{\mathit{ADS}}\left(t\right)\left(1-\delta \right)\left(1-m\right)\left\{{\alpha }_i^A\left(t\right)\left[1-q_i^{\mathit{AKS}}\left(t\right)\right]+\left[1-{\alpha }_i^A\left(t\right)\right]\left[1-q_i^{\mathit{ADS}}\left(t\right)\right]\right\}\\ +P_i^{\mathit{AKS}}\left(t\right)\left[\delta +\left(1-\delta \right)m\right]\left\{\left(1-\eta \right)\left[1-q_i^{\mathit{UKS}}\left(t\right)\right]+\eta \left[1-q_i^{\mathit{UDS}}\left(t\right)\right]\right\}\\ +P_i^{\mathit{AKS}}\left(t\right)\left(1-\delta \right)\left(1-m\right)\left\{\left(1-\eta \right)\left[1-q_i^{\mathit{AKS}}\left(t\right)\right]+\eta \left[1-q_i^{\mathit{ADS}}\left(t\right)\right]\right\}\\ +P_i^{\mathit{UKI}}\left(t\right)\left(1-\mu \right)\end{array}$$ (18)

$$\begin{array}{c}{P}_i^{\mathit{UDS}}\left(t+1\right)=P_i^{\mathit{UDS}}\left(t\right)\left\{r_i\left(t\right)+\left[1-r_i\left(t\right)\right]m\right\}\left[1-{\alpha }_i^U\left(t\right)\right]q_i^{\mathit{UDS}}\left(t\right)\\ +P_i^{\mathit{UKS}}\left(t\right)\left\{r_i\left(t\right)+\left[1-r_i\left(t\right)\right]m\right\}\eta q_i^{\mathit{UDS}}\left(t\right)\\ +P_i^{\mathit{ADS}}\left(t\right)\left[\delta +\left(1-\delta \right)m\right]\left[1-{\alpha }_i^U\left(t\right)\right]q_i^{\mathit{UDS}}\left(t\right)\\ +P_i^{\mathit{AKS}}\left(t\right)\left[\delta +\left(1-\delta \right)m\right]\eta q_i^{\mathit{UDS}}\left(t\right)\end{array}$$ (19)

$$\begin{array}{c}{P}_i^{\mathit{UKS}}\left(t+1\right)=P_i^{\mathit{UDS}}\left(t\right)\left\{r_i\left(t\right)+\left[1-r_i\left(t\right)\right]m\right\}{\alpha }_i^U\left(t\right)q_i^{\mathit{UKS}}\left(t\right)\\ +P_i^{\mathit{UKS}}\left(t\right)\left\{r_i\left(t\right)+\left[1-r_i\left(t\right)\right]m\right\}\left(1-\eta \right)q_i^{\mathit{UDS}}\left(t\right)\\ +P_i^{\mathit{ADS}}\left(t\right)\left[\delta +\left(1-\delta \right)m\right]{\alpha }_i^U\left(t\right)q_i^{\mathit{UKS}}\left(t\right)\\ +P_i^{\mathit{AKS}}\left(t\right)\left[\delta +\left(1-\delta \right)m\right]\left(1-\eta \right)q_i^{\mathit{UDS}}\left(t\right)\\ +P_i^{\mathit{UKI}}\left(t\right)\mu \end{array}$$ (20)

$$\begin{array}{c}{P}_i^{\mathit{ADS}}\left(t+1\right)=P_i^{\mathit{UDS}}\left(t\right)\left[1-r_i\left(t\right)\right]\left(1-m\right)\left[1-{\alpha }_i^A\left(t\right)\right]q_i^{\mathit{ADS}}\left(t\right)\\ +P_i^{\mathit{UKS}}\left(t\right)\left[1-r_i\left(t\right)\right]\left(1-m\right)\eta q_i^{\mathit{ADS}}\left(t\right)\\ +P_i^{\mathit{ADS}}\left(t\right)\left(1-\delta \right)\left(1-m\right)\left[1-{\alpha }_i^A\left(t\right)\right]q_i^{\mathit{ADS}}\left(t\right)\\ +P_i^{\mathit{AKS}}\left(t\right)\left(1-\delta \right)\left(1-m\right)\eta q_i^{\mathit{ADS}}\left(t\right)\end{array}$$ (21)

$$\begin{array}{c}{P}_i^{\mathit{AKS}}\left(t+1\right)=P_i^{\mathit{UDS}}\left(t\right)\left[1-r_i\left(t\right)\right]\left(1-m\right){\alpha }_i^A\left(t\right)q_i^{\mathit{AKS}}\left(t\right)\\ +P_i^{\mathit{UKS}}\left(t\right)\left[1-r_i\left(t\right)\right]\left(1-m\right)\left(1-\eta \right)q_i^{\mathit{AKS}}\left(t\right)\\ +P_i^{\mathit{ADS}}\left(t\right)\left(1-\delta \right)\left(1-m\right){\alpha }_i^A\left(t\right)q_i^{\mathit{AKS}}\left(t\right)\\ +P_i^{\mathit{AKS}}\left(t\right)\left(1-\delta \right)\left(1-m\right)\left(1-\eta \right)q_i^{\mathit{AKS}}\left(t\right)\end{array}$$ (22)

Theorem 1

Given that the topology of three-layer multiplex networks, the epidemic dynamics on the top of them is described in Eqs.(14), (15), (16), (17). Then, the threshold determining whether the epidemics can break out is represented as

$${\beta }_c^U=\frac{\mu }{{\mathrm{\Lambda }}_{\mathit{max}}\left(H\right)}$$ (23)

where${\mathrm{\Lambda }}_{\mathit{max}}\left(H\right)$is the maximum eigenvalues of the coupled matrix$H$related to the information diffusion, behavior adoption and the contact network.

Proof

. When the system tends to steady state, we can obtain that $P_i\left(t+1\right)=P_i\left(t\right)=P_i$ holds for any nodes $i$ and all possible states. Then, we can further compute the epidemic threshold. The number of infected of individuals in the system approaches 0 near the epidemic threshold, and we can assume $P_i^I=P_i^{\mathit{UKI}}={\epsilon }_i<<1$. Accordingly, Eqs. (14), (15), (16), (17) are approximated as

$$q_i^{\mathit{UDS}}\left(t\right)=\prod _j\left[1-c_{\mathit{ji}}{P}_j^I\left(t\right){\beta }_i^{\mathit{UDS}}\left(t\right)\right]\approx 1-{\beta }_i^{\mathit{UDS}}\sum _j{c}_{\mathit{ji}}{\epsilon }_{\mathit{ji}}$$ (24)

$$q_i^{\mathit{UKS}}\left(t\right)=\prod _j\left[1-c_{\mathit{ji}}{P}_j^I\left(t\right){\beta }_i^{\mathit{UKS}}\left(t\right)\right]\approx 1-{\beta }_i^{\mathit{UKS}}\sum _j{c}_{\mathit{ji}}{\epsilon }_{\mathit{ji}}$$ (25)

$$q_i^{\mathit{ADS}}\left(t\right)=\prod _j\left[1-c_{\mathit{ji}}{P}_j^I\left(t\right){\beta }_i^{\mathit{ADS}}\left(t\right)\right]\approx 1-{\beta }_i^{\mathit{ADS}}\sum _j{c}_{\mathit{ji}}{\epsilon }_{\mathit{ji}}$$ (26)

$$q_i^{\mathit{AKS}}\left(t\right)=\prod _j\left[1-c_{\mathit{ji}}{P}_j^I\left(t\right){\beta }_i^{\mathit{AKS}}\left(t\right)\right]\approx 1-{\beta }_i^{\mathit{AKS}}\sum _j{c}_{\mathit{ji}}{\epsilon }_{\mathit{ji}}$$ (27)

Substitute Eqs. (24), (25), (26), (27) into Eqs. (18), (19), (20), (21), (22), we can obtain the following equations

$$\begin{array}{c}{P}_i^{\mathit{UDS}}=P_i^{\mathit{UDS}}\left(t\right)\left[r_i+\left(1-r_i\right)m\right]\left(1-{\alpha }_i^U\right)+P_i^{\mathit{UKS}}\left(t\right)\left[r_i+\left(1-r_i\right)m\right]\eta \\ +P_i^{\mathit{ADS}}\left(t\right)\left[\delta +\left(1-\delta \right)m\right]\left(1-{\alpha }_i^U\right)+P_i^{\mathit{AKS}}\left(t\right)\left[\delta +\left(1-\delta \right)m\right]\eta \end{array}$$ (28)

$$\begin{array}{c}{P}_i^{\mathit{UKS}}=P_i^{\mathit{UDS}}\left(t\right)\left[r_i+\left(1-r_i\right)m\right]{\alpha }_i^U+P_i^{\mathit{UKS}}\left(t\right)\left[r_i+\left(1-r_i\right)m\right]\left(1-\eta \right)\\ +P_i^{\mathit{ADS}}\left(t\right)\left[\delta +\left(1-\delta \right)m\right]{\alpha }_i^U+P_i^{\mathit{AKS}}\left(t\right)\left[\delta +\left(1-\delta \right)m\right]\left(1-\eta \right)\end{array}$$ (29)

$$\begin{array}{c}{P}_i^{\mathit{ADS}}=P_i^{\mathit{UDS}}\left(t\right)\left(1-r_i\right)\left(1-m\right)\left(1-{\alpha }_i^A\right)+P_i^{\mathit{UKS}}\left(t\right)\left(1-r_i\right)\left(1-m\right)\eta \\ +P_i^{\mathit{ADS}}\left(t\right)\left(1-\delta \right)\left(1-m\right)\left(1-{\alpha }_i^A\right)+P_i^{\mathit{AKS}}\left(t\right)\left(1-\delta \right)\left(1-m\right)\eta \end{array}$$ (30)

$$\begin{array}{c}{P}_i^{\mathit{AKS}}=P_i^{\mathit{UDS}}\left(t\right)\left(1-r_i\right)\left(1-m\right){\alpha }_i^A+P_i^{\mathit{UKS}}\left(t\right)\left(1-r_i\right)\left(1-m\right)\left(1-\eta \right)\\ +P_i^{\mathit{ADS}}\left(t\right)\left(1-\delta \right)\left(1-m\right){\alpha }_i^A+P_i^{\mathit{AKS}}\left(t\right)\left(1-\delta \right)\left(1-m\right)\left(1-\eta \right)\end{array}$$ (31)

$$\begin{array}{c}{\mathit{\mu \epsilon }}_i=P_i^{\mathit{UDS}}\left[r_i+\left(1-r_i\right)m\right]\left[{\alpha }_i^U\left({\beta }_i^{\mathit{UKS}}\sum _j{c}_{\mathit{ji}}{\epsilon }_{\mathit{ji}}\right)+\left(1-{\alpha }_i^U\right)\left({\beta }_i^{\mathit{UDS}}\sum _j{c}_{\mathit{ji}}{\epsilon }_{\mathit{ji}}\right)\right]\\ +P_i^{\mathit{UDS}}\left(1-r_i\right)\left(1-m\right)\left[{\alpha }_i^U\left({\beta }_i^{\mathit{UKS}}\sum _j{c}_{\mathit{ji}}{\epsilon }_{\mathit{ji}}\right)+\left(1-{\alpha }_i^U\right)\left({\beta }_i^{\mathit{UDS}}\sum _j{c}_{\mathit{ji}}{\epsilon }_{\mathit{ji}}\right)\right]\\ +P_i^{\mathit{UKS}}\left[r_i+\left(1-r_i\right)m\right]\left[\left(1-\eta \right)\left({\beta }_i^{\mathit{UKS}}\sum _j{c}_{\mathit{ji}}{\epsilon }_{\mathit{ji}}\right)+\eta \left({\beta }_i^{\mathit{UDS}}\sum _j{c}_{\mathit{ji}}{\epsilon }_{\mathit{ji}}\right)\right]\\ +P_i^{\mathit{UKS}}\left(1-r_i\right)\left(1-m\right)\left[\left(1-\eta \right)\left({\beta }_i^{\mathit{AKS}}\sum _j{c}_{\mathit{ji}}{\epsilon }_{\mathit{ji}}\right)+\eta \left({\beta }_i^{\mathit{ADS}}\sum _j{c}_{\mathit{ji}}{\epsilon }_{\mathit{ji}}\right)\right]\\ +P_i^{\mathit{ADS}}\left[\delta +\left(1-\delta \right)m\right]\left[{\alpha }_i^U\left({\beta }_i^{\mathit{UKS}}\sum _j{c}_{\mathit{ji}}{\epsilon }_{\mathit{ji}}\right)+\left(1-{\alpha }_i^U\right)\left({\beta }_i^{\mathit{UDS}}\sum _j{c}_{\mathit{ji}}{\epsilon }_{\mathit{ji}}\right)\right]\\ +P_i^{\mathit{ADS}}\left(1-\delta \right)\left(1-m\right)\left[{\alpha }_i^A\left({\beta }_i^{\mathit{AKS}}\sum _j{c}_{\mathit{ji}}{\epsilon }_{\mathit{ji}}\right)+\left(1-{\alpha }_i^A\right)\left({\beta }_i^{\mathit{ADS}}\sum _j{c}_{\mathit{ji}}{\epsilon }_{\mathit{ji}}\right)\right]\\ +P_i^{\mathit{AKS}}\left[\delta +\left(1-\delta \right)m\right]\left[\left(1-\eta \right)\left({\beta }_i^{\mathit{UKS}}\sum _j{c}_{\mathit{ji}}{\epsilon }_{\mathit{ji}}\right)+\eta \left({\beta }_i^{\mathit{UDS}}\sum _j{c}_{\mathit{ji}}{\epsilon }_{\mathit{ji}}\right)\right]\\ +P_i^{\mathit{AKS}}\left(1-\delta \right)\left(1-m\right)\left[\left(1-\eta \right)\left({\beta }_i^{\mathit{AKS}}\sum _j{c}_{\mathit{ji}}{\epsilon }_{\mathit{ji}}\right)+\eta \left({\beta }_i^{\mathit{ADS}}\sum _j{c}_{\mathit{ji}}{\epsilon }_{\mathit{ji}}\right)\right]\end{array}$$ (32)

Then, based on Eqs. (28), (29), (30), (31), Eq. (32) can be reduced to

$${\mathit{\mu \epsilon }}_i=\left[\left(P_i^{\mathit{UKS}}\phi +P_i^{\mathit{AKS}}\right){\beta }_i^{\mathit{AKS}}+\left(P_i^{\mathit{UDS}}\phi +P_i^{\mathit{ADS}}\right){\beta }_i^{\mathit{ADS}}\right]\sum _j{c}_{\mathit{ji}}{\epsilon }_j$$ (33)

can be further simplified to

$${\mathit{\mu \epsilon }}_i=\left[\left(P_i^{\mathit{UKS}}\phi +P_i^{\mathit{AKS}}\right){\mathit{\phi \sigma }}_1+\left(P_i^{\mathit{UDS}}\phi +P_i^{\mathit{ADS}}\right){\sigma }_2\right]\beta \sum _j{c}_{\mathit{ji}}{\epsilon }_j$$ (34)

where ${\sigma }_1=F\left(\frac{\sum _{j=1}{P}_j^I\left(t\right)}{k_i}-h_i\right)$, ${\sigma }_2=F\left(\frac{\sum _{j=1}{P}_j^I\left(t\right)}{k_i}-\gamma h_i\right)$.

Next, we can rewrite Eq. (34) as the following form

$$\sum _j\left\{\left[\left(P_i^{\mathit{UKS}}\phi +P_i^{\mathit{AKS}}\right){\mathit{\phi \sigma }}_1+\left(P_i^{\mathit{UDS}}\phi +P_i^{\mathit{ADS}}\right){\sigma }_2\right]c_{\mathit{ji}}-\frac{\mu }{\beta }{\zeta }_{\mathit{ji}}\right\}{\epsilon }_j=0$$ (35)

where ${\zeta }_{\mathit{ji}}$ is the element of the identity matrix.

Let $h_{\mathit{ji}}=\left[\left(P_i^{\mathit{UKS}}\phi +P_i^{\mathit{AKS}}\right){\mathit{\phi \sigma }}_1+\left(P_i^{\mathit{UDS}}\phi +P_i^{\mathit{ADS}}\right){\sigma }_2\right]c_{\mathit{ji}}$ be the element of matrix $H$. According to Eq. (32), the epidemic threshold of the proposed model can be described as follows

$${\beta }_c={\beta }_c^U=\frac{\mu }{{\mathrm{\Lambda }}_{\mathit{max}}\left(H\right)}$$ (36)

where ${\mathrm{\Lambda }}_{\mathit{max}}\left(H\right)$ denotes the maximum eigenvalue of the matrix $H$. It can be clearly seen form Eq. (36) that ${\beta }_c$ explicitly depends on the information diffusion process, the immunization behavior adoption process, the recovery probability $\mu$, the reduction degree of physical quality $\gamma$, the individuals' physical quality $h_i$, and the attenuation factor of infection rate $\phi$.

Fig. 6.

The probability transmission trees for 5 possible states.

4. The numerical simulations

In this section, we perform Monte Carlo (MC) simulations to verify the applicability of our proposed model. We investigate the effect of different parameters on the epidemic outbreak threshold ${\beta }_c$ and the proportion of infected individuals ${\rho }^I$. The networks adopted in this paper is the Barabási-Albert (BA) scale free network. For the three-layer BA scale free networks, the average network degree for each layer is $<k>=12$. In all the experiments, the network size is fixed to be $N=3000$ nodes. Each point in the figures is obtained by 50 simulations. For simplicity, we amuse that there is no correlation among the three-layer networks. In addition, the initial numbers of nodes A, K, and I were assumed to be 0.1, respectively.

First, we investigate the clarification strength of mass media m on the epidemic outbreak threshold ${\beta }_c$ versus the proportion of infected individuals ${\rho }^I$. in Fig. 7 . We can see that the ${\rho }^I$ decreases and ${\beta }_c$ increases with the increase of m. This evolutionary trend coincides with the reality that individuals will be more likely not to believe negative information when the government concerned increases the strength of clarifying the unconfirmed information. As individual realize from government actions that immunization behaviors are not harmful, they will also be more willing to adopt immunization behaviors. Therefore, timely clarifying the unconfirmed information by mass media and the implementation of measures to reduce the diffusion of negative information are important for epidemic control. Moreover, by comparing Fig. 7(a), (b) and (c), we find that ${\rho }^I$ becomes larger as the negative information diffusion rate increases and the ${\beta }_c$ shows a reduction. In other words, it is important to increase scientific publicity and reduce the probability of believing negative information for epidemic control.

Fig. 7.

${\rho }^I$ as a function of the infection probability $\beta$ and the clarification strength of mass media m. Three phase diagrams are obtained by MMCA for each point in the grid 40*40. $\lambda$ is set as follows: (a)$\lambda =0.5$, (b) $\lambda =0.8$, (c) $\lambda =1$, from left to right. The remaining parameters are set to be $\delta =0.1$, $\eta =0.4$, $\mu =0.7$, $\phi =0.5$, $\theta =0.5$, $\xi =0.1$, $\gamma =0.2$, $\alpha =0.5$, ${\mu }_1=0.1$, ${\mu }_2=0.1$, ${\sigma }_1={\sigma }_2=0$.

In addition, Fig. 8 shows more clearly the effect of $m$ on the epidemic outbreak threshold ${\beta }_c$. The clarification strength of media on the unconfirmed information has a certain impact on the proportion of infected individuals. The epidemic outbreak threshold ${\beta }_c$ increases as $m$ increases, also indicating that mass media clarify the unconfirmed information and eliminate negative information in timely plays an important role in delaying the outbreak of the epidemic.

Fig. 8.

${\rho }^I$ as a function of the infection probability $\beta$ under different clarification strength of mass media $m$_. The remaining parameters are the same as Fig. 7(b).

In Fig. 9 , we carry out the effect of reduction factor $\xi$ on the epidemic outbreak threshold ${\beta }_c$ and the proportion of infected individuals ${\rho }^I$. We can observe a seemingly counterintuitive phenomenon that ${\rho }^I$ has a sudden decrease with $\xi$ increase. In other words, there is a two-stage effect of reduction factor $\xi$ on the epidemic outbreak threshold ${\beta }_c$.This two-stage phenomenon is similar to the results obtained in Ref [21] by introducing the Heaviside step function and analyzing the critical point on one-dimensional lattice model. And for the model we proposed, this same conclusion is applicable for the immunization behavior adoption process. The reason for this non-monotonicties two-stage effect is when the Heaviside step function is introduced to describe the adoption process, the steady-state equation for individuals without adoption is two-stage near the epidemic outbreak threshold. Specifically, we assume that the probability for one node not adopting the immunization behavior is $P_D$ and the probability for one node adopting the immunization behavior is $P_K$. Let $P^{\left(j\right)},j=0,1,2$ be the probability that one node has $j$ neighbors who adopt the immunization behavior, $P^{\left(0\right)}=P_D^2$, $P^{\left(1\right)}=P_D{P}_K$ and $P^{\left(2\right)}=P_K^2$. Therefore,

$$\left\{\begin{array}{cc}{P}_i^D=P_i^K\eta +P_i^D{P}^{\left(0\right)}=P_i^K\eta +P_i^D{P}_D^2\approx P_i^K\eta & 0<{\theta }_i<0.5\\ P_i^D=P_i^K\eta +P_i^D\left(1-P^{\left(2\right)}\right)\approx P_i^K\eta +P_i^D& {\theta }_i\ge 0.5\end{array}\right)$$ (37)

Fig. 9.

${\rho }^I$ as a function of the infection probability $\beta$ and the reduction factor $\xi$. Three phase diagrams are obtained by MMCA for each point in the grid 40*40. ${\mu }_1,{\mu }_2$ are set as follows: (a) ${\mu }_1=0.1,{\mu }_2=0.1$, (b)${\mu }_1=0.1,{\mu }_2=0.15$, (c) ${\mu }_1=0.2,{\mu }_2=0.1$, from left to right. The remaining parameters are set to be $\delta =0.1$, $\eta =0.1$, $\mu =0.5$, $\phi =0.5$, $m=0.2$, $\gamma =0.5$, $\lambda =0.5$, $\alpha =0.5$, ${\sigma }_1={\sigma }_2=0$.

The Eq. (37) means that around the epidemic threshold ${\beta }_c$, $P^D=\frac{1}{N}{\sum }_{i=1}^N{P}_i^D\approx 1$ when ${\theta }_i\in \left[0.51\right]$, whereas $P^D\approx P^K\eta \approx 1/\left(1+\frac{1}{\eta }\right)$ when ${\theta }_i\in \left[0.51\right]$. Therefore, as a result of different fractions of unadopted individuals around ${\theta }_i=0.5$, there exist an abrupt transmission for the epidemic threshold ${\beta }_c$. In Fig. 9, as $\xi$ increases, the individual's immunization behavior adoption threshold ${\theta }_i$ increases, and when ${\theta }_i$ reaches to the critical point, there lead to the two-stage effects on the epidemic threshold ${\beta }_c$. Besides, comparing Fig. 9(a), (b) and (c), we investigate the effect of the average self-recognition ability ${\mu }_1$ and physical quality ${\mu }_2$ of the whole population. As we can see from the figure, raising both ${\mu }_1$ and ${\mu }_2$ can significantly increase the epidemic outbreak threshold and inhibit the epidemic transmission. In addition, we can also find that raising ${\mu }_1$ and ${\mu }_2$ can improve where the segmentation range of the stage effect, which is about 0.475, 0.55, and 0.65 for Fig. 9(a), (b) and (c), respectively. As $\xi$ decreases, the threshold for individuals who do not aware negative information to adopt immunization behavior decreases, and individuals will be more likely to adopt the immunization behavior and thus reduce the risk of infection. In addition, raising ${\mu }_1$ and ${\mu }_2$ can reduce the probability of individuals believing negative information and the probability of infection, respectively. In reality, when individuals have a reduction in their adoption threshold, they are more likely to be influenced by their neighbors and choose to adopt immunization behaviors. In addition, when the average self-recognition ability and physical quality of the entire population increases, people are less likely to believe negative information and are more likely to resist infection by epidemic. Therefore, it is important for the prevention and control of epidemic that the authorities increase scientific and health promotion efforts to improve the self-recognition ability and physical quality of the whole population.

In Fig. 10 , we can similarly observe the two-stage effect of $\xi$ on the epidemic outbreak threshold ${\beta }_c$. Decreasing $\xi$ have important implications for epidemic suppression. Individuals who believe negative information tends to be less likely to adopt the immunization behavior. In turn, individuals who don't adopt immunization behaviors tend to be associated with a higher risk of infection. In addition, we can observe that since the parameters act mainly on the Heaviside step function, due to the characteristics of the function, there is little effect on the proportion of infected individuals ${\rho }^I$ especially for a big infection rate $\beta$. That is, when the rate of epidemic transmission is large, controlling only the diffusion of negative information has little impact on controlling the scale of the epidemic. $\xi$ can be seen as the adverse effect of negative information on the adoption of immunization behavior. For smaller $\beta$ of epidemic transmission, decreasing $\xi$ has a greater effect on ${\beta }_c$ and ${\rho }^I$. Therefore, for managers, increasing scientific information dissemination and dispelling negative information will be of great importance to increase the proportion of individuals adopting the immunization behaviors, and indirectly increase the epidemic outbreak threshold.

Fig. 10.

${\rho }^I$ as a function of the infection probability $\beta$ under different reduction factor $\xi$. The remaining parameters are the same as Fig. 9(a).

We go on to discuss in Fig. 11 the impact of the adoption threshold $\theta$. As in the case of Fig. 11, when we have a smaller adoption threshold $\theta$, we tend to increase the likelihood of adopting immunization behavior. In addition, due to the characteristic of the three-layer network model and the Heaviside step function, a multi-stage step is presented, especially in the case of a larger basic adoption rate $\alpha$. For two-layer networks, the results using the Heaviside step function tend to be two-stage, while for three-layer networks, we show a multi-stage. This non-monotonicties occurs because when the basic adoption rate is large, there is a situation where the cascade of behavioral adoption overcame the cascade of epidemic transmission, which is consistent with the situation in Ref [21] where the cascade of awareness overcame the cascade of epidemic. Immunization behaviors played a huge role during the COVID-19 epidemic, adopting immunization behaviors can substantially reduce individual's risk of infection. Additionally, a larger adoption proportion tends to promote adoption in other individuals who do not adopt in reality. Therefore, how to conduct scientific campaigns to increase behavior adoption rates has an important impact on epidemic control, individuals should raise their own awareness and acceptance of immunization behaviors.

Fig. 11.

${\rho }^I$ as a function of the infection probability $\beta$ and the adoption threshold $\theta$. Three phase diagrams are obtained by MMCA for each point in the grid 40*40. $\alpha$ is set as follows: (a) $\alpha =0.2$, (b) $\alpha =0.5$, (c) $\alpha =0.8$, from left to right. The remaining parameters are set to be $\delta =0.2$, $\eta =0.4$, $\mu =0.4$, $\phi =0.5$, $m=0.2$, $\xi =0.8$, $\gamma =0.5$, ${\mu }_1=0.2$, ${\mu }_2=0.2$, $\lambda =0.8$, ${\sigma }_1={\sigma }_2=0$.

Then, as shown in Fig. 12 , the epidemic outbreak threshold ${\beta }_c$ is effect with $\theta$. Appropriate lowering of the immunization behavior adoption threshold $\theta$ and increasing individual's acceptance of the immunization behavior are important in delaying epidemic outbreaks and reducing the scale of epidemic transmission. When $\beta >{\beta }_c$, for large $\beta$, there are little impact on the proportion of infected individuals ${\rho }^I$. $\theta$ can be seen as the individual's acceptance of immunization behavior. Different $\theta$ exert a greater effect for smaller $\beta$, while for larger $\beta$, the effect is relatively small. In the early stages of epidemic transmission, it is important to increase citizen acceptance of immunization behaviors in a timely manner to delay epidemic outbreaks and reduce epidemic magnitude.

Fig. 12.

${\rho }^I$ as a function of the infection probability $\beta$ under different attenuation factor $\theta$. The remaining parameters are the same as Fig. 11(a).

In Fig. 13 , we focus on the effect of $\gamma$, the reduction factor of adopted immunization behavior on individual immunity, on the proportion of infected individuals ${\rho }^I$. $\gamma$ can be seen as the effect of adopting immunization behavior on individuals' physical quality. As we can see from the subplots, the epidemic threshold gets significantly increased with the increase of $\gamma$. Since increasing $\gamma$ reduces the risk of infection of individuals indirectly by strengthening their physical quality. The larger $\gamma$ is, the more effective the immunization is in increasing epidemic resistance. While when the epidemic infection possibility is huge, increasing $\gamma$ has little effect on the density of infected individuals at steady state. Furthermore, by comparing Fig. 13(a), (b) and (c), we found that the proportion of infected individuals ${\rho }^I$ was significantly reduced as $\phi$ reduced. This indicates that when individuals take certain measures to protect themselves due to resisting negative information or adopting immunization behaviors, they reduce their risk of contracting the epidemic, which in turn suppresses the epidemic.

Fig. 13.

${\rho }^I$ as a function of the infection probability $\beta$ and the reduction factor $\gamma$. Three phase diagrams are obtained by MMCA for each point in the grid 40*40. $\phi$ is set as follows: (a) $\phi =0.3$, (b) $\phi =0.5$, (c) $\phi =0.7$, from left to right. The remaining parameters are set to be $\delta =0.2$, $\eta =0.5$, $\mu =0.6$, $m=0.2$, $\theta =0.5$, $\xi =0.2$, ${\mu }_1=0.1$, ${\mu }_2=0.1$, $\lambda =0.5$, $\alpha =0.8$, ${\sigma }_1={\sigma }_2=0$.

To further explore the influence of behavior on epidemic, we discuss effect of different $\gamma$ on the epidemic outbreak threshold in Fig. 14 . we can observe that increasing $\gamma$ can increase the epidemic outbreak threshold. This is consistent with the conclusions we obtained in Fig. 13. Therefore, it is important for individuals to adopt immunization behaviors timely to increase epidemic resistance and then inhibit epidemic transmission.

Fig. 14.

${\rho }^I$ as a function of the infection probability $\beta$ under different reduction factor $\gamma$. The remaining parameters are the same as Fig. 13(a).

In the following, we investigate the impact of the attenuation factor $\phi$ and the average physical quality of the whole population ${\mu }_2$ on the proportion of infected individuals ${\rho }^I$. $\phi$ can be seen as the intensity of self-protective measures taken by individuals due to their different states (believing different information or adopting different behaviors), and a smaller $\phi$ means a larger degree of self-protective measures taken. As shown in Fig. 15 , an increase in $\phi$ can promote the epidemic, especially when the ${\mu }_2$ is smaller. This is mainly because when more individuals don't believe the negative information and adopt the immunization behaviors, they will have smaller infection probability in the physical connect network. Moreover, by contrasting Fig. 15(a), (b) with (c), we can observe that the ${\rho }^I$ will be larger when ${\mu }_2$ is lower. In other words, when the whole population have lower average physical quality, their infection risk will increase. This phenomenon suggests that the success of individuals' system to be compromised increases dramatically when their physical fitness is low. For Fig. 15(c), the reason for the non-monotonicties results to occur could be that there may be some numerical artifacts for higher average physical quality of whole population and larger epidemic transmission rate. In addition, we also explore the role of the attenuation factor $\phi$ on the epidemic outbreak threshold ${\beta }_c$ in Fig. 16 . We find that the epidemic outbreak threshold ${\beta }_c$ increases with the decrease of $\phi$, which is in agreement with the conclusions we obtained in Fig. 15. When $\phi$ is larger, individuals often fail to realize the danger of the epidemic in time due to listening to negative information, or fail to adopt immunization behaviors in time, thus exposing themselves to the risk of epidemic. In terms of epidemic control and prevention, the relevant authorities should promptly strengthen the dissemination of relevant information and, if necessary, take mandatory measures to induce individuals to adopt immunization behaviors. In addition, strengthening physical exercise for the general public and improving the average physical quality of the whole population are important for epidemic control.

Fig. 15.

${\rho }^I$ as a function of the infection probability $\beta$ and the attenuation factor $\phi$. Three phase diagrams are obtained by MMCA for each point in the grid 40*40. ${\mu }_2$ is set as follows: (a) ${\mu }_2=0.1$, (b) ${\mu }_2=0.15$, (c) ${\mu }_2=0.2$, from left to right. The remaining parameters are set to be $\delta =0.2$, $\eta =0.5$, $\mu =0.6$, $m=0.5$, $\theta =0.5$, $\xi =0.5$, $\gamma =0.5$, ${\mu }_1=0.1$, $\lambda =0.5$, $\alpha =0.8$, ${\sigma }_1={\sigma }_2=0$.

Fig. 16.

${\rho }^I$ as a function of the infection probability $\beta$ under different attenuation factor $\phi$. The remaining parameters are the same as Fig. 15(a).

In addition, we find an interesting phenomenon. That is, for the influence parameters in the Heaviside step function of the negative information diffusion layer and the immunization behavior adoption layer, our effect of action tends to show a phase transition, as $\xi$ shown in Fig. 9 and $\theta$ in Fig. 11. However, for the effect parameters in the epidemic transmission layer, such as $\gamma$ in Fig. 13 and $\phi$ in Fig. 15, we find that the effect tends to be continuous. To explore this phenomenon in more depth, we discuss the effect of the average self-recognition ability ${\mu }_1$ and physical quality ${\mu }_2$ of the whole population, where the same phenomenon can be found in Fig. 17 .

Fig. 17.

${\rho }^I$ as a function of the infection probability $\beta$ and the average self-recognition ability ${\mu }_1$ and physical quality ${\mu }_2$ of the whole population. Three phase diagrams are obtained by MMCA for each point in the grid 40*40. ${\sigma }_1$ and ${\sigma }_2$ are set as follows: (a) and (d)${\sigma }_1={\sigma }_2=0$, (b) and (e). ${\sigma }_1={\sigma }_2=0.1$, (c) and (f)${\sigma }_1={\sigma }_2=0.2$. ${\mu }_2=0.2$ for (a), (b) and (c). ${\mu }_1=0.1$ for (a), (b) and (c). The remaining parameters are set to be $\delta =0.2$, $\eta =0.4$, $\mu =0.4$, $\phi =0.5$, $m=0.2$, $\theta =0.5$, $\xi =0.8$, $\gamma =0.5$, $\lambda =0.8$, $\alpha =0.5$.

Finally, we analyze the effect of the average self-recognition ability ${\mu }_1$ and physical quality ${\mu }_2$ of the whole population on the epidemic outbreak threshold ${\beta }_c$ and the proportion of infected individuals ${\rho }^I$. In Fig. 17 and Fig. 18 , we can find that as the average self-recognition ability ${\mu }_1$ and physical quality ${\mu }_2$ increase, the epidemic outbreak threshold increases ${\beta }_c$ and the proportion of infected individuals ${\rho }^I$ decreases. Moreover, the heterogeneity of the individuals in the information diffusion layer leads to a two-stage phase transition, while it leads to a continuous phase transition in the epidemic layer. When the average self-recognition ability ${\mu }_1$ reaches a critical point, there has a huge increment, resulting in a sudden increase in epidemic prevalence. While for the average physical quality ${\mu }_2$, its effect on the proportion of infected individuals is continuous. The reason for this phenomenon is that when the average self-recognition ability ${\mu }_1$ accumulates to a certain level, the overall ability of the population to recognize negative information is improved, and negative information are less likely to spread, which in turn has a two-stage effect on epidemic control by influencing immunization behavior and the probability of infection. In contrast, for the average physical quality ${\mu }_2$ acts only on the transmission process on the physical contact network, thus showing a continuous effect. What's more, for the different subplots in Fig. 17, we also consider the effect of the Gaussian distribution standard deviation ${\sigma }_1$ and ${\sigma }_2$ on the epidemic transmission. As can be seen from the figure, as $\sigma$ increases, the epidemic outbreak threshold ${\beta }_c$ decreases, while the proportion of infected individuals ${\rho }^I$ increases, which indicates when the uneven quality of the population, the easier diffusion of negative information. Although the physical quality of some individuals is improved when $\sigma$ is larger, the effect of individuals with lower quality level on the epidemic transmission is more significant, which should prove that economically, the same as in the face of natural disasters, the disadvantaged individuals are more likely to be exposed to danger, more vulnerable to risk, and more likely to cause negative results on the overall development. In summary, improving the average self-recognition ability and physical quality of the whole population and reducing the variability among individuals is important to delay epidemic outbreak and reduce the proportion of infected individuals. Therefore, managers should strengthen scientific education and psychological counseling, rather than promoting the importance of measures related to physical contact.

Fig. 18.

${\rho }^I$ as a function of the infection probability $\beta$ under different average self-recognition ability ${\mu }_1$ and physical quality ${\mu }_2$ of the whole population. ${\mu }_2=0.2$ for (a). ${\mu }_1=0.1$ for (b). The remaining parameters are set to be $\delta =0.2$, $\eta =0.4$, $\mu =0.4$, $\phi =0.5$, $m=0.2$, $\theta =0.5$, $\xi =0.8$, $\gamma =0.5$, $\lambda =0.8$, $\alpha =0.5$.

5. Conclusions

In this paper, we introduced the Heaviside step function to portray the herd effect and used the Gaussian distribution to describe the heterogeneous of individual's self-recognition ability and physical quality. We proposed a three-layer UAU-DKD-SIS model with considering mass media to explore the co-evolution among the negative information diffusion, immunization behavior adoption and epidemic transmission. The main contributions of this paper are summarized as follows:

(1) The diffusion of negative information reduces individuals' adoption of immunization behaviors, and the mass media can reduce this negative effect by clarifying the unconfirmed information timely.

(2) Enhancing individuals' self-recognition ability can promote them to adopt the immunization behavior, thereby reducing their risk of infection, and it leads to a two-stage phase transition.

(3) Improving individuals' physical quality will have a continuous impact on reducing epidemic outbreak threshold and suppressing the epidemic.

(4) The reduction of whole population differences is of positive significance for epidemic prevention.

Moreover, time-varying networks are often more capable of describing the complexities of real society. In addition, except for the influence of information and herd effect when individuals adopt immunization behaviors, individuals' behavioral adoption is also related to their preference for risk in the decision-adoption process. In the future, we will attempt to propose a preferential model to study the impact of these factors.

CRediT authorship contribution statement

Liang'an Huo: Conceptualization, Methodology, Supervision; Writing- Reviewing and Editing, Formal analysis; Funding acquisition; Investigation; Project administration.

Yue Yu: Data curation; Methodology; Writing- Original draft preparation; Visualization; Writing- Reviewing and Editing; Investigation; Software.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Data availability

Data will be made available on request.