A New Coupled Awareness-Epidemic Spreading Model with Neighbor Behavior on Multiplex Networks

In this paper, we propose a nonlinear coupled model to study the two interacting processes of awareness diffusion and epidemic spreading on the same individual who is affected by different neighbor behavior status on multiplex networks. We achieve this topology scenario by two kinds of factors, one is the perception factor that can change interplay between different layers of networks and the other is the neighbors’ behavior status that can change the infection rate in each layer. According to the microscopic Markov chain approach (MMCA), we analyze the dynamical evolution of the system and derive the theoretical epidemic threshold on uncorrelated heterogeneous networks, and then, we validate the analysis by numerical simulation and discuss the final size of awareness diffusion and epidemic spreading on a scale-free network. With the outbreak of COVID-19, the spread of epidemic in China prompted drastic measures for transmission containment. We examine the effects of these interventions based on modeling of the awareness-epidemic and the COVID-19 epidemic case.)e results further demonstrate that the epidemic spreading can be affected by the effective transmission rate of the awareness and neighbors’ behavior status.


Introduction
e outbreak of COVID-19 can involve the diffusion of information about the epidemic, including the officially released authoritative information, rumors, and fears [1][2][3]. In reality, awareness diffusion can stimulate individuals to take spontaneous behavioral responses such as wearing masks or staying at home to reduce the frequency of faceto-face contact [4,5]. erefore, the epidemic-spreading dynamics in complex networks has attracted increasing attention in many disciplines [6,7]. However, the exact impact they can have on the epidemic dynamics is difficult to quantify, so mathematical modelling is used to test hypotheses and identify pivotal parameters in the interplay between awareness diffusion and epidemic spreading [8][9][10][11]. For example, Granell et al. constructed a UAU-SIS (unaware-aware-unaware/susceptible-infected-susceptible) model to study the interaction between epidemics spreading and awareness diffusion on multiplex networks [12]. Funk et al. researched the coevolution of information and epidemic on public and found that the spread of positively oriented information about the epidemic can suppress the epidemic [13]. Wang et al. utilized real data to investigate the coevolution mechanisms between information and disease spreading, and the empirical analysis showed that there is an asymmetric interaction between information and disease spreading [14]. Moreover, Wang et al. further found that the awareness inhibits the epidemic spreading, whereas the epidemic spreading facilitates the awareness diffusion [15]. Zhu et al. thought that an epidemic might spread among multicommunities, so they modeled each community as a multiplex network that included both a virtual awareness layer and a physical layer [16]. ey further presented a coupled UAU-SIRD model, which consisted of a virtual layer sustaining unaware-aware-unaware dynamics and a physical layer supporting the susceptible-infected-recovered-dead process, to investigate the spreading property of epidemics and the relationships between the focused variables and parameters of the epidemic. e result indicated that the incorporation of virtual layers would reduce the range of affected individuals [17]. Kabir and Tanimoto established a two-layer SIR-UA (susceptible-infected-recovered/unaware-aware) epidemic transmission model that comprehensively considered the influence of individual behavior and awareness diffusion on epidemic spreading on heterogeneous networks [18]. Shang studied the effects of global, local, and contact awareness on a discrete-time SIS epidemic dynamics and used the stability theory of matrix difference equation to derive the epidemic threshold [19]. ey further investigated the impact of the three forms of awareness on epidemic spreading by the mean-field approach with heterogeneous transmission rates, and the numerical simulation results showed that both local and contact awareness can raise the epidemic threshold while global awareness can only decrease the final epidemic size [20].
ough the effects of information-based behavioral responses on the epidemic dynamics have been studied by many documents, most of them assumed that individuals are treated equally when they contact different neighbors. In fact, the deviation among different neighbors has a remarkable influence on the individual infection process due to the complex topological structures of networks. In view of this, there has been an increasing focus on the discussion of neighbor behavior in the interacting processes of awareness diffusion and epidemic spreading [21][22][23][24][25]. For instance, Kan and Zhang assumed that susceptible individuals can not only be informed by aware neighbors in the information network but also by self-awareness induced by the infected neighbors in the contact network. Results indicated that the individuals with more neighbors in the information network have higher awareness acquisition and they are hard to be infected in the contact network [26]. Guo et al. introduced a heterogeneous threshold model to converse that a hub with a large number of neighbors is relatively easy to become infected [6]. Moreover, they proposed a heterogeneous spreading model that considers the degree heterogeneity and k-core measures heterogeneity of individuals with the belief that different individuals facing the same epidemic would exhibit distinct behaviors according to their own experiences and attributes [27]. Zhu et al. generated an adaptive strategy of the susceptible/exposed individual that a susceptible/ exposed individual would disconnect with infected neighbors probabilistically and one new connection would be constructed [28]. Chen et al. proposed a resource-epidemic coevolution model to investigate the effects of the heterogeneous distribution of self-awareness and the heterogeneous distribution of node degree on the epidemic dynamics.
ey found that the heterogeneity of selfawareness distribution suppresses the outbreak of an epidemic and the heterogeneity of degree distribution enhances the epidemic spreading [29]. Pan and Yan proposed a coupled awareness-epidemic spreading model incorporating the heterogeneity of neighbors' responses to disease outbreaks, and the result showed that the heterogeneity of neighbors' responses acts on the epidemic threshold in the higher stage [30].
Some research studies assumed that the diffusion of awareness will reduce the infection rate of individuals [12][13][14][15][16]. Considering that the spread rate of awareness is also affected by the spread of the epidemic [15], in this work, the awareness perception factor f 1 (0 ≤ f 1 ≤ 2) stands for the influence of awareness diffusion on the spread rate of epidemic, and we also introduce an epidemic perception factor f 2 (0 ≤ f 2 ≤ 2) to express the impact of epidemic spreading on the diffusion rate of awareness. We define f 1 ≤ 1, f 2 > 1 when authoritative information and epidemic interact and rule f 1 > 1, f 2 ≤ 1 when rumors and epidemic interact, as shown in Table1.
Moreover, Kan and Zhang discussed the effect of infected neighbors on the formation of node self-awareness [26], and some references confirmed that the heterogeneity of degree distribution enhances the epidemic spreading [27,29]. On this basis, we assume that the health of awareness neighbors (susceptible, infected, and recovered) plays different roles in the formation of node self-awareness and propose the health-impact-awareness status factors (∆ρ 1 , ∆ρ 2 , and ∆ρ 3 ) to distinguish this effect in the awareness layer; the heterogeneity of neighbors degree distribution will affect the individual's epidemic infection rate, and the behavior-impact-epidemic status factor (1 − e (− k j / k) , a positive correlation function of neighbors degree) is established to discriminate the difference in the epidemic layer. erefore, we propose a multiplex networks model to comprehend the spreading dynamics between epidemic and awareness on the same population with the influence of different neighbor behavior status. We introduce the behavior status to each node in the multiplex networks. On the one hand, people can acquire awareness from aware neighbors with different health-impact-awareness status in layer 1 of fictitious contacts, such as Microblogs, WeChat, or other social media. On the other hand, in layer 2 of physical contacts, the level of epidemic spreading would be influenced by the infected neighbors with different behaviorimpact-epidemic status. e MMCA theoretical analysis and numerical simulations results reveal that awareness and behavioral changes could have favorable effects on the epidemic spreading. e rest of this paper is organized as follows: in Section 2, we introduce the multiplex networks model. en, we analyze the dynamical processes of the model with the MMCA method and derive the expression of epidemic threshold in Section 3. Next, in Sections 4 and 5, we perform numerical simulations and discuss the Chinese COVID-19 epidemic case to validate theoretical predictions. Finally, we summarize our findings and conclusions in Section 6.

Model Descriptions.
In this work, we generalize a multiplex networks model to formalize and simplify the spread mechanisms, as illustrated in Figure 1. In layer 1, people share epidemic-prevention awareness through social media, while people get infected in layer 2. Neighbor behavior is simplified as the "health-impact-awareness" status of aware neighbors in layer 1 and the "behavior-impact-epidemic" status of infected neighbors in layer 2. e definitions of key parameters are shown in Table 2. We suppose the multiplex propagation process according to the following regulations: 2 Complexity (i) Awareness diffusion in layer 1: the diffusion of awareness satisfies the UAU (unaware-aware-unaware) process. Aware (A) individuals can apply their awareness to decrease the probability of being infected. Unaware (U) individuals do not have any awareness about the epidemic prevention. Unaware individuals can contact with aware neighbors to get awareness with probability λ, while the aware individuals can forget the awareness with probability δ. As we all know, in many cases, individuals who are infected or cured are more aware than those who are far away from the epidemic because once they are exposed to a certain infectious disease (such as SARS or COVID-19), they will be isolated and feel panicked, that is, whether an individual has been physically exposed to the epidemic changes the probability of awareness acquisition. erefore, we assume that unaware infected (UI) and unaware recovered (UR) individuals know about the epidemic and can increase the probability of being aware with the epidemic perception factor f 2 . en, the awareness infection rate can be represented by the susceptible infection rate λ S � λ and infected (recovered) infection rate respectively. (ii) Epidemic spreading in layer 2: the spreading of epidemic satisfies the SIR (susceptible-infected-recovered) process. Susceptible individuals get infected with the probability β, while the infected individuals recover with the probability μ. Yet, aware susceptible (AS) individuals can apply the awareness to decrease the probability of being infected with the awareness perception factor f 1 . en, the epidemic infection rate can be expressed by the unawareness of infection rate β U � β and awareness of infection rate β A � f 1 β, respectively. (iii) e health-impact-awareness status of aware neighbors: the direct application of the UAU model in the two-layer network means that the default importance ratio of aware neighbors is AS : AI : AR � 1/3 : 1/3 : 1/3. We believe that the aware neighbors of different health states play different roles in the process of awareness diffusion and assume that the importance ratio is AS : AI : AR � 1/ 3 + ∆ρ 1 : 1/3 + ∆ρ 2 : 1/3 + ∆ρ 3 (∆ρ 1 + ∆ρ 2 + ∆ρ 3 � 0). ∆ρ 1 , ∆ρ 2 , and ∆ρ 3 are the health-impact-awareness status factors of the model, as shown in Figure 2. (iv) e behavior-impact-epidemic status of infected neighbors: the heterogeneity of degree distribution plays a role in the epidemic spreading, and we believe that infected individuals with a large range of daily activities (a large node degree) are more likely to carry and transmit the virus. We use the degree k j of neighbor j to represent its daily activity range and use k to represent the total degrees of all infected neighbors; as shown in Figure 3, 1 − e − (k j / k) can distinguish the impact of each infected neighbor on epidemic infected probability and also limit the size of this impact (the interval is 0-0.5).

Dynamic
Model. According to these presumptions, there are six primary states: US (unawareness susceptibility); UI (unawareness infection); UR (unawareness recovered); AS (awareness susceptibility); AI (awareness infected); and  Complexity 3 AR (awareness recovered). e reaction procedure of the model can be schematically expressed as follows:

Microscopic Markov Chain Approach
We analyze the dynamical processes of the model with the MMCA method [31,32]. Let x ij and y ij be the adjacency matrices that support the UAU and SIR processes, respectively. e probability of node i in one of the six states at time t is denoted by Under the assumption that the possibilities of becoming aware or infected by any neighbor are independent, we use to represent four basic quantities as follows: Probability of becoming unaware f 1 e awareness perception factor f 2 e epidemic perception factor ∆ρ 1 , ∆ρ 2 , and ∆ρ 3 e health-impact-awareness status influence of aware neighbors  where r * i (t) denotes the probability for susceptible node i not getting aware by any neighbors. r i ′ (t) is the probability for infected (recovered) node i not getting aware by any neighbors. x ui means that node i has u neighbors in state P AS i : where q A * i (q U * i ) denotes the probability of susceptible aware (unaware) node i not being infected by any neighbors and y ji � 1 means that node j is the neighbor of node i at the same layer. We extend Markov's theory to steady state and assume that P AI i + P UI i � P I i � π i ≪ 1. We can obtain the approximations of q A * i and q U * i as follows: e potential state transition process of the UAU-SIR model is shown in Figure 4.
As shown in Figure 4, every time step is divided into six stages, and we can easily get the MMCA equations for node i as follows: e solution of equation (4) is a set of fixed-point equations that satisfy P AI i (t + 1) � P AI i (t) � P AI i , and this relationship is also true for the other five types of nodes. We add the second and the fifth equations in equation (4) as follows: Near the threshold, the probability of nodes being infected is very small, i.e., P AI i + P UI i � P I i � π i ≪ 1. Taking equation (3) into equation (4) and omitting the O(εi) terms, we get Substituting (6) into (5), we get with P AS i ≈ P A i , P US i ≈ 1 − P A i ; then, equation (7) can be rewritten as where τ ij denotes the elements of the identity matrix. We define (2 − e − (k j / k) )y ji , to simplify equation (8). e epidemic threshold is equal to the minimum value of β that satisfies equation (8). We can obtain the epidemic threshold by denoting Λ max (h), the maximum eigenvalue of H, and the threshold formula is Equations (8) and (9) show that the epidemic threshold is based on the structure of layer 2 (y ji ), the parameter f 1 , and the density of awareness P A i . e value of P A i is further determined by the structure of layer 1(x ji ), the transmission rate λ, and the recovery rate σ.

Results
We discuss the effect of f 1 and f 2 through Model I, on which we introduce the neighbor behavior state, and obtain Model II. e comparison between Model 1 and Model 2 reflects the influence of the neighbor behavior state on the propagation process. We perform extensive Monte Carlo numerical simulations [33,34] for the model (run 100 times) and obtain the infection density ρ I and aware density ρ A at steady state.
where N represents the number of all individuals in the model. In our simulations, the size of the two-layer network is N � 1000, the initial number of nodes m 0 � 5, the number of connected edges m � 3, and the average degree 〈k〉 � 6. e default value of the parameter is (the values of Δρ 1 , Δρ 2 , and Δρ 3 in this paper can be set according to actual requirements. We believe that the awarenessspreading intensity of different neighbors is AI > AR > AS, and by the summation product method and consistency test, we obtain the importance ratio AS : AI : AR � 0.067, 0.689, 0.244). Figure 5 plots the density of β c as a decreasing function of f 1 . It is clear that β c obviously decreases as f 1 (f 1 ∈ (0, 1)) increases and basically remains unchanged as f 1 , (f 1 ∈ (1, 2)) increases. Figure 6 presents that β c grows as λ increases (f 1 � 0.2, f 2 � 1.8) and β c decreases as λ increases (f 1 � 1.8, f 2 � 0.2), which means authoritative information can inhibit the outbreak of the epidemic and rumors will contribute to the spread of the epidemic. At the same time, a smaller information forgetting rate, a greater rumor forgetting rate, and a greater infectious diseases recovery rate can raise the epidemic threshold. Moreover, the curve of Model II is lower than that of Model I with equal parameters in both Figures 5 and 6; measures such as locating and isolating infected neighbors are good ways to prevent an outbreak of the epidemic. As shown in Figure 7, λ c is a decreasing function of f 2 . It is clear that λ c obviously decreases as f 2 (f 2 ∈ (0, 1)) increases and basically remains unchanged as f 2 (f 2 ∈ (1, 2)) increases. Figure 8 suggests that λ c grows as β increases (f 1 � 1.8, f 2 � 0.2) and λ c decreases as β increases (f 1 � 0.2, f 2 � 1.8), which means that the rapid spread of the epidemic can inhibit the outbreak of rumors and promote the dissemination of authoritative information. A smaller information forgetting rate, a greater rumor forgetting rate, and a greater infectious disease recovery rate can raise the awareness threshold. Moreover, the curve of Model II is lower than that of Model I with equal parameters in both Figures 7 and 8, publicizing the epidemic information through the infected and cured people will contribute to spreading crisis awareness and preventing the outbreak of the epidemic.

Infection Scale.
e key influencing factors for ρ I and ρ A are f 1 and f 2 , respectively. Figure 9 reveals that a larger epidemic cure rate μ helps control the epidemic diffusion scale and a smaller awareness forgetting rate σ contributes to the spread of awareness.

Authoritative Information and Epidemic Interaction
Model (f 1 < 1, f 2 > 1). Both ρ A and ρ I in Figure 10 show an upward trend with the increase of β, while the curve in Figure 10(a) is less obvious than that in Figure 10(b) because β can only indirectly affect the dissemination of authoritative information. In the meanwhile, a higher prevention degree of the individuals (f 1 � 0.1) corresponds to a smaller ρ I (Figure 10(b)) and a smaller ρ A (Figure 10(a)). Also, a stronger willingness of the individuals to spread information (f 2 � 1.8) led to a larger ρ A and a smaller ρ I . e increase of λ contributes to the spread of awareness (Figure 11(a)) and the reduction of the epidemic scale ( Figure 11(b)), which is in line with the actual situation. When the individuals' willingness to disseminate information gets stronger, (f 2 � 1.8), the value of ρ A will be larger and ρ I will decrease more. A higher prevention degree of the individuals (f 1 � 0.1) corresponds to a small ρ I and a larger ρ A . e comparative experimental results of Figures 10 and  11 show that, under the same conditions, the ρ A obtained by Model II is smaller and ρ I is larger than that obtained by Model I, which suggests that models ignoring neighbors' behavior status would underestimate the epidemic scale and limit the formulation of epidemic control strategies.

Rumor and Epidemic
Interaction Model (f 1 > 1, f 2 < 1). A larger β means a greater risk of individual infection, and it will led to a lager ρ I . However, rumors are not beneficial to epidemic control, and individuals would resist the spread of rumors; then, the curve of ρ A would drop. As shown in Figure 12, a smaller f 2 (f 2 � 0.1) will lead to a smaller ρ A , and a larger f 1 , (f 1 � 1.8) will cause a larger ρ I .
A larger λ means that individuals are more likely to have rumor awareness, which will lead to a lager ρ A . However, individuals in the epidemic layer affected by rumors will take improper behaviors to accelerate the spread of the epidemic, and the ρ I curve will rise, as illustrated in Figure 13. Furthermore, the more resistant an individual is to the rumor (f 2 � 0.1), the more difficult it is for the rumor to spread in the population, and ρ A and ρ I will be smaller; the greater an individual's contribution to epidemic transmission (f 1 � 1.8), the easier it is for the epidemic to spread, and ρ A and ρ I will be larger. Figures 12 and 13 show that when the spread of the epidemic is accompanied by rumors, ρ A obtained by Model II is larger than Model I, and the obtained ρ I is smaller than Model I, which suggests that models ignoring neighbors' behavior status would overvalue the epidemic scale and influence the formulation of epidemic control strategies. As a result, it is necessary and reasonable to consider the influence of neighbors' behavior status when we establish an interactive propagation model.

Epidemic Analysis of COVID-19 in China
Four stages of strict measures for transmission containment were prompted during the early spread of COVID-19 in China, which reflects the change in public awareness. Each awareness stage plays discrepant roles in epidemic control and public response behaviors. We hypothesized that the awareness impact factor f 1 is negatively correlated with the importance of epidemic control measures and observe the assumed value of f 1 as shown in Table 3 [35].
Model I and Model II simulate the density of infected individuals as a function of t for different values of f 1 . It is clear that the large value of f 1 can increase the density of infected individuals, see Figure 14(a), and the neighbor behavior status can narrow the gap between the effects of different f 1 on the infection scale, see Figure 14(b). erefore, the government can actively release positive information to enhance public awareness and restrict the behavior of infected neighbors for the purpose of controlling the spread of the epidemic during the COVID-19 epidemic. Figure 15 is the daily new data released by the Publicity Department of the National Health Commission except for

Complexity
Hubei Province, and Figure 16 is the data fitting curve of infected individuals (except Hubei Province). Figures 15 and  16 indicate that the number of infected individuals is gradually decreasing since January 29 with the strengthening of prevention and control measure [36]. Figure 17 shows the cumulative number of confirmed cases in Hubei Province from January to April. Since February 16, the cumulative number of confirmed cases has been close to 0, indicating that the epidemic has been under control. As we all know, public awareness is affected by the spread of the epidemic, and if the epidemic is effectively controlled, public awareness will become weaker. Public awareness is difficult to quantify, but we found such a phenomenon: the online big data information demonstrates that the amount of comprehensive information has risen rapidly since January 20, 2020, and reached the peak on February 15, and the peak time coincides with the epidemic control time (Feb 16), as shown in Figure 17, which reflect the influence of epidemic spreading on the diffusion of awareness.
As shown in Table 4, the Hubei Provincial Government adjusts information report management at different stages of the epidemic. Each epidemic stage plays discrepant roles in public awareness diffusion. We hypothesized that the epidemic impact factor f 2 is positively correlated with the importance of information report management and observe the assumed value of f 2 .
Model I and Model II simulate the density of aware individuals as a function of t for different values of f 2 . It is clear that the large value of f 1 can increase the density of aware individuals, see Figure 18(a), and neighbor behavior status can expand the gap between the effects of different f 2 on the awareness scale, see Figure 18        High-risk period (1) Provide 4 analysis reports of more than 20 pages per day  data and epidemic prevention process during the spread of COVID-19 can enhance public awareness and effectively control the spread of the epidemic.

Conclusions
In this paper, on the one hand, we discuss the interaction between awareness diffusion (including authoritative information and rumors) and epidemic spreading; on the other hand, we analyze the influence of neighbor status on epidemic spreading when awareness diffusion and neighbor behavior are coupled in multiple networks. Distinct from previous studies, people can get epidemic prevention information to reduce the infection rates from their social communication circles, such as microblogs, as well as increase awareness based on their own health status. Moreover, different neighbor status play distinct roles in the awareness acquisition process and the response behavior change process of an individual. We investigate the impact of these factors on epidemic spreading processes and obtain the epidemic threshold with the MMCA approach. Analysis based on the numerical simulations reveal that the epidemic can be reduced by publishing authoritative information to reduce f 1 , publicizing the epidemic prevention process to increase f 2 , and controlling neighbor behavior to be more reasonable, for example, isolating infected neighbors and encouraging them to propagate information related to the epidemic.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare no conflicts of interest.