# Controlling Virus Spread Using an Intermittent Quarantine Strategy on Multiplex Networks

Authors: Xingguo Li; Xiaoping Luo; Yiwu Wang
Journal: Security and Communication Networks
 (2021)

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## Abstract

Virus spreading on the Internet will negatively affect cybersecurity. An intermittent quarantine immunization strategy to control virus spreading when containing information diffusion is proposed herein. In this model, information and virus spread on different subnetworks and interact with each other. We further develop a heterogeneous mean-field approach with time delays to investigate this model and use Monte Carlo simulations to systematically investigate the spreading dynamics. For a relatively short intermittent period, the optimal information transmission probability of the virus will be significantly suppressed. However, when the intermittent period is extremely long; increasing the probability of information transmission can control the virus spreading as well as suppress the increase in the intermittent period. Finally, it is shown that the average degree of the two subnetworks does not qualitatively affect the spreading dynamics.


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## Body

## 1. Introduction

Computer viruses are spreading widely on the Internet, thereby significantly affecting the cyberspace security [1–4]. For instance, the WannaCry virus, one of the most influential viruses since Panda Burned Incense, has affected more than 100 countries and regions. Researchers from different fields, including computer science, network science, physics, and mathematics, have attempted to eliminate the spread of the virus [5–11]. In this study, we focused on network science to address this problem, in which the ultimate aim is to develop effective measures to control virus spreading using mathematical modeling and network theory.Network science researchers have developed many effective models to investigate the spread of viruses on the Internet. Data analyses revealed that the topology of the Internet exhibits the heavy-tail degree distribution [12–14]; hence, PastorCSatorras and Vespignani proposed a mathematical susceptible-infected-susceptible (SIS) model on scale-free complex networks [5, 15]. They used a heterogeneous mean-field theory to describe the dynamics and revealed that a few hubs resulted in vanishingly low values of infection transmission probability that triggered virus spreading on the Internet. Based on this finding, we can design an effective approach to control virus spread. Target immunization [15–19] is one of the most popular strategies, which involve the immunization of hubs to suppress viruses significantly. To implement the target immunization strategy, the topology of the Internet must be determined, which limits its applications. Therefore, researchers proposed the acquaintance immunization strategy [20–23], which necessitates only information regarding local network structures. The progress in state-of-the-art network immunization has been reported in a recent review [24].Computer users typically communicate with their friends through social networks [25–28]. Hence, the information spread by a virus will diffuse in the social network when the virus is spreading on the Internet. Therefore, researchers have developed models to describe the dynamics of virus-information spread. Scholars typically use epidemic spreading models to model the spread of viruses, such as SIS and susceptible-infected-removed (SIR) models. Therefore, we herein present some of the advancements in the field of epidemic-information spreading dynamics that can be used to investigate the spread of virus information. Funk et al. [29] assume that an epidemic is spread on homogeneous networks, whereas information regarding the epidemic is also spread within the same network. Using a mean-field approach, they discovered that information spreading can significantly suppress the spread of an epidemic but cannot alter the outbreak threshold. Granell et al. [30, 31] used the unaware-aware-unaware (UAU)-SIS model, which assumes UAU information spreading dynamics on social networks and SIS epidemic spreading dynamics on face-to-face contact networks. They used a discrete Markovian approach to describe the interacting dynamics and revealed that the epidemic outbreak threshold was determined by the topologies of social networks, contact networks, and dynamic information spreading parameters. Subsequently, researchers investigated the effects of network topologies and interacting mechanisms on spreading dynamics [32–37]. Wang et al. [38–40] used an asymmetrical interacting SIR-SIRV model and discovered that the interlayer degree correlation facilitated the control of epidemic spread.To immunize virus spreading on the Internet, different strategies can be adopted, e.g., barring Internet access, implementing patches, and turning on the firewall. Additionally, advanced information-epidemic spreading dynamics can be adopted to describe more realistic scenarios. Zhao et al. [41] developed a virus-information spreading model to design an optimal allocation strategy for patches. Once a virus spreads on the Internet, the simplest approach to stop the spreading is to bar Internet access until the virus is eliminated [42–44]. Because mathematical studies pertaining to information spreading dynamics are scarce, we herein propose a mathematical model to describe an interaction model based on multiplex networks. We assume that information regarding the virus spreads in the social network, i.e., the virus spreads on the Internet and then performs extensive numerical simulations to determine the optimal intervention strength.


## 2. Model Descriptions

In this section, an interacting spreading dynamics model based on the intermittent quarantine immunization strategy on multiplex networks withNA=NB=N nodes is described. The multiplex networks comprise two subnetworks, denoted as A and B, and each subnetwork represents a communication platform. Each node in two subnetworks is matched one-to-one, which implies that a node exists in different communication platforms. The two subnetworks were constructed based on an uncorrelated configuration model with degree distributions PAkA and PBkB, separately. When the network size is N⟶∞, no inter- and interlayer degree correlations exist in the multiplex subnetwork (as shown in Figure 1(a)).Figure 1
Illustration of information-virus spreading on communication-computer networks. (a) Communication-computer multiplex network. (b) Virus spreading promotes the information spreading. (c) Information spreading induces the intermittent quarantine.Assume that subnetworksA and B represent the social and virus networks, respectively. Information is typically spread on social network A, whereas virus spreads on subnetwork B. To describe information spreading, we used the classic SIR epidemiological model [45–47], in which each node can be susceptible, informed, or recovered at a specified time. At each time step, each informed node first transmits the information to every susceptible neighbor with probability βA and then recovers with probability γA. It is noteworthy that the susceptible node in subnetwork A is informed once its counterpart in subnetwork B becomes infected. Hence, virus spread promotes information spreading (as shown in Figure 1(b)).For virus spreading on subnetworkB, we used a generalized SIR model. The virus spreading dynamics is the same as the information spreading dynamics on subnetwork A but with different infection and recovery probabilities, i.e., βℬ and γℬ, respectively. We assume that the susceptible node in subnetwork B adopts an intermittent quarantine strategy to control virus spread. Specifically, the susceptible node iB in subnetwork B will be quarantined for a period tb if its corresponding node iA in subnetwork A is in the informed state. After period tb, the susceptible node iB is reconnected its neighbors, i.e., the intermittent quarantine strategy is induced (as shown in Figure 1(c)). From the above descriptions, we know the state of the node iA depends on the information spreading and thus further determines the adoption intermittent quarantine strategy for the node iA. Hence, virus spreading on subnetwork B is suppressed. According to the above descriptions, the differences between the information diffusion and virus spreading are listed as follows: (i) different dynamical parameters, i.e., transmission and recovery probabilities, and (ii) an addition intermittent quarantine state is induced in the virus spreading.We randomly selected one node as the virus seed in subnetworkB; subsequently, we set its counterpart in subnetwork A as the information seed. The remaining nodes were set in a susceptible state. The effective information transmission and virus transmission probabilities are expressed as λA=βA/γA and λB=βB/γB, respectively. The interacting spreading dynamics terminate once no node is available in the informed or infected state. In Table 1, we illustrate the definitions of the parameters in our model.Table 1
Definitions of parameters and abbreviations.
Parameters/abbreviationsDefinitionsβAInformation transmission rateβBVirus transmission rateγAInformation recovery rateγBVirus recovery ratetBIntermittent quarantine periodkAAverage degree of subnetworkAkBAverage degree of subnetworkBΘAtThe probability of a node connects to an infected neighbor in networkAΘBtThe probability of a node connects to an infected neighbor in networkBSkAAtA node with degreekA in network A is in the susceptible stateSkBBtA node with degreekB in network B is in the susceptible stateIkAAtA node with degreekA in network A is in the informed stateIkBBtA node with degreekB in network B is in the infected stateRkAAtA node with degreekA in network A is in the recovered stateRkBBtA node with degreekB in network B is in the recovered state


## 3. Results

The results of this study are presented in this section.


### 3.1. Theoretical Analysis

To analyze the results of this study quantitatively, we used a heterogeneous mean-field approach with time delays. We denoteSkAAt, IkAAt, and RkAAt as the fractions of the susceptible, informed, and recovered nodes with degree kA in subnetwork A at time t, respectively. Similarly, we denote SkBBt, IkBBt, RkBBt, and VkBBt to represent the fraction of nodes in the susceptible, infected, recovered, and vaccinated states with degree kB in subnetwork B at time t, respectively. Once the parameters above are obtained, the fraction of nodes in each state can be determined using the degree distributions of the two networks, denoted as PAkA and PBkB, separately. For instance, the fraction of nodes in the susceptible state in subnetwork A is SAt=∑kASkAAtPAkA.Inspired by [38, 39], we developed a heterogeneous mean-field approach to investigate virus-information spreading dynamics. For the information spreading dynamics on subnetwork A, a susceptible node (e.g., node iA) becomes infected via one of two approaches. (i) Node iA can receive the information from its infected neighbors with probability skAAtβAkAΘAt, where ΘAt is the probability that node iA is connected to an infected neighbor. For uncorrelated networks, ΘAt is expressed as(1)ΘAt=∑kA′kA′−1PAkA′ρkA′AtkA,where kA=∑kAkAPAkA is the average degree of subnetwork A. (ii) Node iA receives the infection once its corresponding node iB becomes infected, and the infection probability is skAAtβBΘBt∑kBkBPBkB, where ΘBt is the probability that a susceptible node is connected to an infected neighbor in subnetwork B. ΘBt is expressed as(2)ΘBt=∑kB′kB′−1PBkB′ρkB′BtkB,where kB=∑kBkBPBkB is the average degree of subnetwork B. Combining the two approaches above, the evolution of SkAAt can be expressed as follows:(3)dSkAAtdt=−SkAAtβAkAΘAt+βBΘBt∑kBkBPBkB.Once the susceptible node is infected by the information, it is regarded as infected. Hence, the evolution ofIkAt is expressed as(4)dIkAAtdt=skAAtβAkAΘAt+βBΘBt∑kBkBPBkB−γAIkAAt,where γAIkAAt is the fraction of nodes recovered from the infected state. Therefore, the evolution of RkAt is expressed as(5)dRkAAtdt=γAIkAAt.For virus spreading on subnetworkB, the situation becomes more complex. A susceptible node iB may be quarantined for a period tb if its corresponding node iA becomes the infected state. Mathematically, we use a new state V to represent the quarantined nodes. We use VkBt to denote the fraction of nodes with degree kB in the quarantined state. The fraction of nodes becomes a quarantined state with probability pβAΘAt∑kASkAAtkAPAkA. Hence, for a randomly selected susceptible node, it becomes an infected state with probability SkBBtβBkBΘBt. The evolution of SkAAt is expressed as(6)dSkBBtdt=−SkBBtβBkBΘBt−−pβAΘAt∑kASkAAtkAPAkA.The evolution ofVkBBt is(7)dVkBBtdt=pβAΘAt∑kASkAAtkAPAkA.The evolution ofIkBt occurs in one of two processes: (i) the susceptible nodes become infected by their infected neighbors with probability SkBBtβBkBΘBt and (ii) the quarantined node experiences more time steps than tb and is infected by neighbors with probability ∑t′=tbVkBBt−t′βBkBΘBt. Therefore, the evolution of IkBBt is expressed as(8)dρkBBtdt=SkBBtβBkBΘBt+∑t′=tbVkBBt−t′βBkBΘBt−γBIkBBt.The evolution ofRkBt is(9)dRkBBtdt=γBIkBBt.


### 3.2. Numerical Simulations

We performed extensive Monte Carlo simulations on uncorrelated configuration networks using the method proposed in [48]. Specifically, we generated an uncorrelated configuration network as follows. (i) The network size was determined based on the average degree and degree distributions of the two subnetworks. We set the network sizes of subnetworks A and B as NA=NB=104. The degree distributions of the two networks were set based on a power law as PAkA=ξAkA−γA and PBkB=ξBkB−γB, respectively, where γA and γB are the degree exponents of the two subnetworks, separately: ξA=∑kAkA−γA and ξB=∑kBkB−γB. In the numerical simulations, we set γDA=γDB=3.0, and the recovery probability γA=γB=0.1. The multiplex networks are denoted as SFCSF multiplex networks. (ii) Degree sequences were generated for each subnetwork based on the degree distributions, and stubs were assigned for each node. (iii) For each, two stubs were selected randomly, and an edge was built until no stubs remained in the subnetwork. (iv) For each node in the two subnetworks, we performed one-to-one matching randomly.We first investigated the final information spreading sizeRA and the virus spreading size RB for a specified quarantine value for period tb, as shown in Figure 2. For a specified value of λB, RA increased monotonically with λA (see Figure 2(a)) because the nodes in subnetwork A had a higher probability of accessing the information. In addition, RA increased with λB because the spread of the virus promoted information spreading, as shown in Figures 2(a) and 2(b). In general, the virus spreading size RB increased with λB for a specified λA. However, we discovered an optimal information transmission probability for which the virus spreading suppressed significantly, as illustrated in Figures 2(c) and 2(d). Specifically, RB first decreased with λA because information was obtained by more nodes, and the quarantine strategy was adopted. However, as λA increased, more nodes adopted the quarantine strategy for period tb. If tb (i.e., the virus spreading terminal time) is sufficiently large, those nodes will be protected from being infected by the virus. Otherwise, the quarantined nodes will participate in virus spreading. Therefore, RB increases with λA. It is noteworthy that when λA is sufficiently large, many nodes are quarantined, and only a fraction of quarantined nodes will participate in virus spreading when their quarantine period is completed. Hence, RB decreases with λA.Figure 2
Virus-information spreading dynamics on SFCSF networks. (a) Final information outbreak sizeRA and (b) epidemic outbreak size RB vs. information transmission probability λB under different epidemic transmission probabilities λB=0.3,0.5, and 0.8. (c) RA and (d) RB vs. λB under λA=0.3,0.5, and 0.8. We set the intermittent breaking period as tb=60, and average degree of two subnetworks are kA=kB=8.
(a)(b)(c)(d)We further investigated the effects of the quarantine period on the virus-information spreading dynamics, as shown in Figure3. For the case of tb=40 (see Figures 3(a)–3(d)), i.e., when the quarantine period was relatively small, a similar phenomenon to that shown in Figure 2 was observed. For a large value of tb, i.e., when tb=90 (as shown in Figures 3(e)–3(h)), a different phenomenon was observed, and the optimal information disappeared. Specifically, RA increased monotonically with λA or λB, whereas RB decreases monotonically. When tb was extremely large, the quarantined nodes did not participate in the virus spreading dynamics when they completed their quarantine period. Hence, we conclude that increasing the information transmission probability can contain virus spreading for a long quarantine period.Figure 3
Virus-information spreading dynamics on SFCSF networks. (a) Final information outbreak sizeRA and (b) epidemic outbreak size RB vs. information transmission probability λA under different epidemic transmission probabilities λB=0.3,0.5, and 0.8 when tb=40. (c) RA and (d) RB vs. λB under λA=0.3,0.5, and 0.8 when tb=40. (e) RA and (f) RB as a function of λA under λB=0.3,0.5, and 0.8 when tb=90. (g) RA and (h) RB vs. λB under λA=0.3,0.5, and 0.8 when tb=90. Average degree of two subnetworks are kA=kB=8.
(a)(b)(c)(d)(e)(f)(g)(h)Next, we investigate whether an optimal quarantine period can significantly suppress virus spreading. Hence, we systematically investigated the spread of virus information for different values ofλA and λB, as shown in Figure 4. We discovered that RA decreased with tb, regardless of the values of λA and λB. In terms of virus spreading, RB decreased monotonically with tb because the larger value of tb resulted in fewer nodes participating in virus spreading when they complete their quarantine period.Figure 4
Virus-information spreading dynamics on SFCSF networks. Average degrees arekA=8 and kB=8. We set the intermittent breaking period as tb=40.
(a)(b)(c)(d)(e)(f)Finally, we investigated the effects of the average degree of the multiplex networks on virus-information spreading, as shown in Figure5). It is noteworthy that varying the values of kA and kB do not qualitatively affect the phenomena presented in Figure 2. In other words, an optimal information transmission probability exists at which the spread of virus will be significantly suppressed.Figure 5
Virus-information spreading dynamics on SFCSF networks.RA vs. λA under different epidemic transmission probabilities λB=0.3,0.5, and 0.8 (a) and different information transmission probabilities λA=0.3,0.5, and 0.8 (b). RB vs. λA under different λB=0.3,0.5, and 0.8 (e) and different λA=0.3,0.5, and 0.8 (f). In (a), (b), (e), and (f), the average degrees are kA=8 and kB=15. RA vs. λA under different λB=0.3,0.5, and 0.8 (c) and different λA=0.3,0.5, and 0.8 (d). RB vs. λA under different λB=0.3,0.5, and 0.8 (g) and different λA=0.3,0.5, and 0.8 (h). In (c), (d), (g), and (h), average degrees are kA=15 and kB=8.
(a)(b)(c)(d)(e)(f)(g)(h)


## 3.1. Theoretical Analysis

To analyze the results of this study quantitatively, we used a heterogeneous mean-field approach with time delays. We denoteSkAAt, IkAAt, and RkAAt as the fractions of the susceptible, informed, and recovered nodes with degree kA in subnetwork A at time t, respectively. Similarly, we denote SkBBt, IkBBt, RkBBt, and VkBBt to represent the fraction of nodes in the susceptible, infected, recovered, and vaccinated states with degree kB in subnetwork B at time t, respectively. Once the parameters above are obtained, the fraction of nodes in each state can be determined using the degree distributions of the two networks, denoted as PAkA and PBkB, separately. For instance, the fraction of nodes in the susceptible state in subnetwork A is SAt=∑kASkAAtPAkA.Inspired by [38, 39], we developed a heterogeneous mean-field approach to investigate virus-information spreading dynamics. For the information spreading dynamics on subnetwork A, a susceptible node (e.g., node iA) becomes infected via one of two approaches. (i) Node iA can receive the information from its infected neighbors with probability skAAtβAkAΘAt, where ΘAt is the probability that node iA is connected to an infected neighbor. For uncorrelated networks, ΘAt is expressed as(1)ΘAt=∑kA′kA′−1PAkA′ρkA′AtkA,where kA=∑kAkAPAkA is the average degree of subnetwork A. (ii) Node iA receives the infection once its corresponding node iB becomes infected, and the infection probability is skAAtβBΘBt∑kBkBPBkB, where ΘBt is the probability that a susceptible node is connected to an infected neighbor in subnetwork B. ΘBt is expressed as(2)ΘBt=∑kB′kB′−1PBkB′ρkB′BtkB,where kB=∑kBkBPBkB is the average degree of subnetwork B. Combining the two approaches above, the evolution of SkAAt can be expressed as follows:(3)dSkAAtdt=−SkAAtβAkAΘAt+βBΘBt∑kBkBPBkB.Once the susceptible node is infected by the information, it is regarded as infected. Hence, the evolution ofIkAt is expressed as(4)dIkAAtdt=skAAtβAkAΘAt+βBΘBt∑kBkBPBkB−γAIkAAt,where γAIkAAt is the fraction of nodes recovered from the infected state. Therefore, the evolution of RkAt is expressed as(5)dRkAAtdt=γAIkAAt.For virus spreading on subnetworkB, the situation becomes more complex. A susceptible node iB may be quarantined for a period tb if its corresponding node iA becomes the infected state. Mathematically, we use a new state V to represent the quarantined nodes. We use VkBt to denote the fraction of nodes with degree kB in the quarantined state. The fraction of nodes becomes a quarantined state with probability pβAΘAt∑kASkAAtkAPAkA. Hence, for a randomly selected susceptible node, it becomes an infected state with probability SkBBtβBkBΘBt. The evolution of SkAAt is expressed as(6)dSkBBtdt=−SkBBtβBkBΘBt−−pβAΘAt∑kASkAAtkAPAkA.The evolution ofVkBBt is(7)dVkBBtdt=pβAΘAt∑kASkAAtkAPAkA.The evolution ofIkBt occurs in one of two processes: (i) the susceptible nodes become infected by their infected neighbors with probability SkBBtβBkBΘBt and (ii) the quarantined node experiences more time steps than tb and is infected by neighbors with probability ∑t′=tbVkBBt−t′βBkBΘBt. Therefore, the evolution of IkBBt is expressed as(8)dρkBBtdt=SkBBtβBkBΘBt+∑t′=tbVkBBt−t′βBkBΘBt−γBIkBBt.The evolution ofRkBt is(9)dRkBBtdt=γBIkBBt.


## 3.2. Numerical Simulations

We performed extensive Monte Carlo simulations on uncorrelated configuration networks using the method proposed in [48]. Specifically, we generated an uncorrelated configuration network as follows. (i) The network size was determined based on the average degree and degree distributions of the two subnetworks. We set the network sizes of subnetworks A and B as NA=NB=104. The degree distributions of the two networks were set based on a power law as PAkA=ξAkA−γA and PBkB=ξBkB−γB, respectively, where γA and γB are the degree exponents of the two subnetworks, separately: ξA=∑kAkA−γA and ξB=∑kBkB−γB. In the numerical simulations, we set γDA=γDB=3.0, and the recovery probability γA=γB=0.1. The multiplex networks are denoted as SFCSF multiplex networks. (ii) Degree sequences were generated for each subnetwork based on the degree distributions, and stubs were assigned for each node. (iii) For each, two stubs were selected randomly, and an edge was built until no stubs remained in the subnetwork. (iv) For each node in the two subnetworks, we performed one-to-one matching randomly.We first investigated the final information spreading sizeRA and the virus spreading size RB for a specified quarantine value for period tb, as shown in Figure 2. For a specified value of λB, RA increased monotonically with λA (see Figure 2(a)) because the nodes in subnetwork A had a higher probability of accessing the information. In addition, RA increased with λB because the spread of the virus promoted information spreading, as shown in Figures 2(a) and 2(b). In general, the virus spreading size RB increased with λB for a specified λA. However, we discovered an optimal information transmission probability for which the virus spreading suppressed significantly, as illustrated in Figures 2(c) and 2(d). Specifically, RB first decreased with λA because information was obtained by more nodes, and the quarantine strategy was adopted. However, as λA increased, more nodes adopted the quarantine strategy for period tb. If tb (i.e., the virus spreading terminal time) is sufficiently large, those nodes will be protected from being infected by the virus. Otherwise, the quarantined nodes will participate in virus spreading. Therefore, RB increases with λA. It is noteworthy that when λA is sufficiently large, many nodes are quarantined, and only a fraction of quarantined nodes will participate in virus spreading when their quarantine period is completed. Hence, RB decreases with λA.Figure 2
Virus-information spreading dynamics on SFCSF networks. (a) Final information outbreak sizeRA and (b) epidemic outbreak size RB vs. information transmission probability λB under different epidemic transmission probabilities λB=0.3,0.5, and 0.8. (c) RA and (d) RB vs. λB under λA=0.3,0.5, and 0.8. We set the intermittent breaking period as tb=60, and average degree of two subnetworks are kA=kB=8.
(a)(b)(c)(d)We further investigated the effects of the quarantine period on the virus-information spreading dynamics, as shown in Figure3. For the case of tb=40 (see Figures 3(a)–3(d)), i.e., when the quarantine period was relatively small, a similar phenomenon to that shown in Figure 2 was observed. For a large value of tb, i.e., when tb=90 (as shown in Figures 3(e)–3(h)), a different phenomenon was observed, and the optimal information disappeared. Specifically, RA increased monotonically with λA or λB, whereas RB decreases monotonically. When tb was extremely large, the quarantined nodes did not participate in the virus spreading dynamics when they completed their quarantine period. Hence, we conclude that increasing the information transmission probability can contain virus spreading for a long quarantine period.Figure 3
Virus-information spreading dynamics on SFCSF networks. (a) Final information outbreak sizeRA and (b) epidemic outbreak size RB vs. information transmission probability λA under different epidemic transmission probabilities λB=0.3,0.5, and 0.8 when tb=40. (c) RA and (d) RB vs. λB under λA=0.3,0.5, and 0.8 when tb=40. (e) RA and (f) RB as a function of λA under λB=0.3,0.5, and 0.8 when tb=90. (g) RA and (h) RB vs. λB under λA=0.3,0.5, and 0.8 when tb=90. Average degree of two subnetworks are kA=kB=8.
(a)(b)(c)(d)(e)(f)(g)(h)Next, we investigate whether an optimal quarantine period can significantly suppress virus spreading. Hence, we systematically investigated the spread of virus information for different values ofλA and λB, as shown in Figure 4. We discovered that RA decreased with tb, regardless of the values of λA and λB. In terms of virus spreading, RB decreased monotonically with tb because the larger value of tb resulted in fewer nodes participating in virus spreading when they complete their quarantine period.Figure 4
Virus-information spreading dynamics on SFCSF networks. Average degrees arekA=8 and kB=8. We set the intermittent breaking period as tb=40.
(a)(b)(c)(d)(e)(f)Finally, we investigated the effects of the average degree of the multiplex networks on virus-information spreading, as shown in Figure5). It is noteworthy that varying the values of kA and kB do not qualitatively affect the phenomena presented in Figure 2. In other words, an optimal information transmission probability exists at which the spread of virus will be significantly suppressed.Figure 5
Virus-information spreading dynamics on SFCSF networks.RA vs. λA under different epidemic transmission probabilities λB=0.3,0.5, and 0.8 (a) and different information transmission probabilities λA=0.3,0.5, and 0.8 (b). RB vs. λA under different λB=0.3,0.5, and 0.8 (e) and different λA=0.3,0.5, and 0.8 (f). In (a), (b), (e), and (f), the average degrees are kA=8 and kB=15. RA vs. λA under different λB=0.3,0.5, and 0.8 (c) and different λA=0.3,0.5, and 0.8 (d). RB vs. λA under different λB=0.3,0.5, and 0.8 (g) and different λA=0.3,0.5, and 0.8 (h). In (c), (d), (g), and (h), average degrees are kA=15 and kB=8.
(a)(b)(c)(d)(e)(f)(g)(h)


## 4. Conclusions

Controlling the spread of virus on the Internet is vital to cyberspace security. The spread of virus on the Internet triggers information spreading on social networks. Hence, we proposed an interacting virus-information spreading dynamics model for multiplex networks, in which a node receiving information is intermittently quarantined for a specified period. The spreading dynamics were described using a time-delay heterogeneous mean-field approach. By performing Monte Carlo simulations, we investigated the spreading dynamics of SFCSF networks and identified two situations. For relatively small intermittent periods, virus spreading suppressed significantly at the optimal information transmission probability. For large intermittent periods, i.e., the intermittent period was larger than the virus spreading time. Therefore, increasing the information diffusion probability will likely facilitate the control of viruses. Furthermore, we discovered that increasing the intermittent period suppressed virus spreading. Finally, we changed the average degrees of the two subnetworks and discovered that the above mentioned phenomena were not qualitatively affected.



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