# Virus-Information Coevolution Spreading Dynamics on Multiplex Networks

Authors: Jian Wang; Xiaolin Qin; Hongying Fang
Journal: Complexity
 (2021)

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## Abstract

Virus and information spreading dynamics widely exist in complex systems. However, systematic study still lacks for the interacting spreading dynamics between the two types of dynamics. This paper proposes a mathematical model on multiplex networks, which considers the heterogeneous susceptibility and infectivity in two subnetworks. By using a heterogeneous mean-field theory, we studied the dynamic process and outbreak threshold of the system. Through extensive numerical simulations on artificial networks, we find that the virus’s spreading dynamics can be suppressed by increasing the information spreading probability, decreasing the protection power, or decreasing the susceptibility and infectivity.


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## Body

## 1. Introduction

Coevolution spreading dynamics, ranging from cyberspace security to epidemic contagions, widely exist in the natural systems, in which there are at least two spreading dynamics evolving and interacting simultaneously [1, 2]. For instance, the virus’s information is always spreading on the social network when a computer virus spreads on the Internet. The users whose computers are not infected by the virus will install antivirus software and patches to protect their computers from being infected by the virus [3–6]. In this way, computer viruses can be prevented from spreading widely. Another example is the spreading of the epidemic in society. When a global pandemic was spreading, various kinds of information about the pandemic, such as protecting healthy individuals from infection, spreading on social networks will suppress the pandemic [7–9].Researchers from the field of computer science and network science have developed some successful mathematical models to model such coevolution spreading dynamics. The state of the art in this field is reviewed in a recent paper by Wang et al. [1]. Historically, Newman [10] studied two viruses spreading on the same computer network in succession, where the two viruses follow the susceptible-infected-recovered model, and the second virus can only infect the remaining susceptible nodes. Using a bond percolation theory, he revealed that a global outbreak of the second virus is possible only if the susceptible nodes form a large cluster of connections and the outbreak threshold of the second virus is much higher than the threshold of the first. Newman and Ferrario [11] further discussed a different situation, i.e., the second virus can only spread on those infected nodes by the first virus. They found that the second virus’s outbreak size can be suppressed by decreasing the spreading probability of the first virus. In reality, the spreading dynamics are always simultaneous. Karrer and Newman [12] proposed a model to include this factor and studied the phase transition by using a competing percolation theory.The two spreading dynamics always evolve on different networks; that is to say, we should use multiplex or multilayer networks to describe the network topology of the coevolution spreading [13–18]. Previous studies have revealed that the topology of multilayer networks markedly affect the dynamics, such as cascading failures [19–21], virus spreading [22, 23], controllability [24], and synchronization [25–27]. For virus-information spreading on multiplex networks, Granell and her colleagues [8] used an unaware-aware-unaware-susceptible-infected-susceptible (UAU-SIS) model. They revealed a metacritical critical point, above which the global virus will break out by using a generalized Markovian approach. Based on the research framework in Reference [8], researchers studied the global information [8], network topology [28, 29], and different interacting mechanisms [30, 31] on the virus spreading. Wang et al. revealed the asymmetric interaction between the virus and information spreading dynamics. They used a susceptible-informed-recovered-susceptible-infected-recovered-vaccination (SIR-SIRV) model. They found that the information can suppress the virus spreading greatly, especially when there is a positive correlation between layers.Many real-world data analyses proved that the spreading dynamics on the network are heterogeneous. There are three aspects. On the one hand, the network topology is heterogeneous, e.g., heterogeneous degree distribution. Scholars revealed that heterogeneous degree distribution could decrease the virus’s outbreak threshold spreading [32–34]. An important result is that Pastor-Satorras and Vespignani [32] used a heterogeneous mean-field theory to describe the computer virus spreading on the Internet and revealed that the existence of some hubs may make the outbreak threshold vanish. On the other hand, infectivity and susceptibility are heterogeneous since different computers have distinct circumstances. Miller [35] revealed that the global virus is more likely to break out for homogeneous infectivity when the average transmissibility is fixed. In addition, he found that the attack rate was highest when the susceptibility was homogeneous and lowest when the variance was maximum. Lastly, the virus and information always transmit through different networks. Generally, the virus spreads on the computer network and the information transmits on the social network. Therefore, the virus-information dynamics spreading on two-layered multiplex networks are more realistic. Previous paragraphs have stated the state-of-the-art progresses for virus-information spreading on multiplex networks. To our best knowledge, systematic study still lacks for the interacting spreading dynamics including the above three aspects. In the paper, we first describe the mathematical model in Section 2. In Section 3, we develop a heterogeneous mean-field theory to describe the spreading dynamics. In Section 4, we perform extensive numerical simulations. Finally, we conclude the paper in Section 5.


## 2. Model Descriptions

In this section, we propose the virus-information coevolution spreading model on computer-social networkM. We first introduce the computer-social network and then present the virus-information spreading model.


### 2.1. Computer-Social Network

We denote the two subnetworks asL1 and L2, respectively. The computer virus spreads on subnetwork L1, and the information spreads on subnetwork L2. In subnetwork L1L2, nodes represent the computers (users), and the edges stand for the relationships among computers (users). To build the two-layered complex networks, we use the following steps: (i) assigning the subnetwork sizes N1=N2=N; (ii) building subnetworks L1 and L2 by using the uncorrelated configuration model [36] (the degree distributions of subnetworks L1 and L2 are P1k1 and P2k2, respectively); and (iii) randomly matching nodes in two subnetworks. Specifically, we build an interlayer connection e for node i1 and i2, which means that the user i2 uses computer i1. By using the above methods, there are node inter- and intradegree correlations. As shown in Figure 1, we illustrate the computer-social network.Figure 1
Illustration of computer-social networks. SubnetworkL1 represents the computer network, and subnetwork L2 stands for the social network.Mathematically, the computer-social networkM can be represented by a adjacency matrix U=A1A12A21A2, where A1 and A2, respectively, stand for the adjacency matrixes of subnetworks L1 and L2. An element Aijx=1 of subnetwork x∈1,2 means that nodes ix and jx are connected. The matrixes A12 and A21 are the adjacency matrixes of interlayer network, where Ai1i212=1 means that node i1 uses computer i2. Note that Ai1i212=Ai2i121 for any values of i1 and i2. The average degrees of the two subnetworks can be denoted as k1=∑i,jAij1=∑k1k1P1k1 and k2=∑i,jAij2=∑k2k2P2k2, respectively.


### 2.2. Virus-Information Spreading Model

We here adopt a susceptible-infected-recovered (SIR) model to describe the virus spreading on subnetworkL1. A node in the susceptible state means that it does not get infected by the computer virus. An infected node represents that it is infected by the virus and can transmit it to one of its neighbors. A node in the recovered state means that it has recovered and does not change its state.For the information spread on subnetworkL2, we consider using the irreversible susceptible-informed-recovered (SIR) model [37]. A susceptible node means that it does not obtain information about the virus. An informed node indicates that it knows information about the virus and is willing to share it with its neighbors. A node in the recovered state means that it loses interest in the information and will not transmit it to its neighbors. In this paper, we denote the virus-information coevolution spreading as a SIR-SIR model.The virus-information coevolution spreading dynamic evolves as follows. Initially, we randomly select a nodei1 in subnetwork L1, that is to say, the computer virus infects node i1. The corresponding node i2, i.e., the user of computer i1, is also set to be the infected state, since the user can release the information of his infection to his neighbors. At every time step t, each infected node v1 in subnetwork L1 tries to transmit the computer virus to one of its neighbors u1, since every infected node usually communicates with one neighbor at a short time interval. In reality, the infection transmission depends on the “source” and “target” nodes [35]. That is to say, the infectivity and susceptibility of the system are distinct for different nodes. To include this factor, we assume that nodes’ infectivity and susceptibility depend on the degree of nodes. More specifically, the infectivity of node v1 with degree k1 is(1)ωk11=1−1−α1k1,where α1 is the unit infectivity for a node with degree 1. Similarly, the susceptibility of node u1 with degree k1′ is(2)ϖk1′1=1−1−α1′k1′,where α1′ is the unit susceptibility for node with degree 1. Varying the values of α1 and α1′, we get different infectivities and susceptibilities of the system. If node u1 is susceptible, two different situations should be considered. (i) If the user u2 of computer u1 is in the susceptible state, the computer u1 infects the virus with probability λ1. Meanwhile, node u2 obtains the information. Otherwise, u2 is in the infected or recovered state, and nothing happens. (ii) If the user u2 has already obtained the information before, the computer u1 is infected by the virus with probability qλ1, where 0≤q≤1. We here use the parameter q to describe the degree of protection when a user knows the virus’s spreading. The smaller the value of q, the stronger the protection against computer viruses. Every infected node recovers with probability γ1.The information about the virus spread on the subnetworkL2 is as follows. At each time step t, every informed node v2 transmits the information to one of its neighbor u2 in subnetwork L2 depending on the infectivity of v2 and the susceptibility of u2. The infectivity of v2 with degree k2 is(3)ωk22=1−1−α2k2,where α2 is the unit infectivity. The susceptibility of u2 with degree k2′ is(4)ϖk2′2=1−1−α2′k2′,where α2′ is the unit susceptibility. The infection probability is λ2. Finally, every informed node loses interest in transmitting the information with probability γ2. The spreading ends when there are no nodes in the infected or informed state. In Table 1, we present the definitions of parameters and abbreviations.Table 1
Definitions of parameters and abbreviations.
ParameterDefinitionN1Subnetwork size of computer networkL1N2Subnetwork size of social networkL2P1k1Degree distribution of computer networkL1P2k2Degree distribution of social networkL2k1Average degree of computer networkL1k2Average degree of social networkL2ωk11Infectivity of nodev1 with degree k1 in subnetwork L1ωk22Infectivity of nodev2 with degree k2 in subnetwork L2ϖk1′1Susceptibility ofu1 with degree k1′ in subnetwork L1ϖk2′2Susceptibility ofu2 with degree k2′ in subnetwork L2α1Unit infectivity in subnetworkL1α2Unit infectivity in subnetworkL2α1′Unit susceptibility in subnetworkL1α2′Unit susceptibility in subnetworkL2qProtection powerλ1Computer virus transmission probabilityλ2Information transmission probabilityγ1Computer virus recovery probabilityγ2Information recovery probabilitysk11tProbability of node with degreek1 in the susceptible state at time t in subnetwork L1ρk11tProbability of node with degreek1 in the infected state at time t in subnetwork L1rk11tProbability of node with degreek1 in the recovered state at time t in subnetwork L1sk22tProbability of node with degreek2 in the susceptible state at time t in subnetwork L2ρk22tProbability of node with degreek2 in the infected state at time t in subnetwork L2rk22tProbability of node with degreek2 in the recovered state at time t in subnetwork L2Θ1tAverage probability that a node connects to an infected neighbor through an edge in subnetworkL1Θ2tAverage probability that a node connects to an informed neighbor through an edge in subnetworkL2


## 2.1. Computer-Social Network

We denote the two subnetworks asL1 and L2, respectively. The computer virus spreads on subnetwork L1, and the information spreads on subnetwork L2. In subnetwork L1L2, nodes represent the computers (users), and the edges stand for the relationships among computers (users). To build the two-layered complex networks, we use the following steps: (i) assigning the subnetwork sizes N1=N2=N; (ii) building subnetworks L1 and L2 by using the uncorrelated configuration model [36] (the degree distributions of subnetworks L1 and L2 are P1k1 and P2k2, respectively); and (iii) randomly matching nodes in two subnetworks. Specifically, we build an interlayer connection e for node i1 and i2, which means that the user i2 uses computer i1. By using the above methods, there are node inter- and intradegree correlations. As shown in Figure 1, we illustrate the computer-social network.Figure 1
Illustration of computer-social networks. SubnetworkL1 represents the computer network, and subnetwork L2 stands for the social network.Mathematically, the computer-social networkM can be represented by a adjacency matrix U=A1A12A21A2, where A1 and A2, respectively, stand for the adjacency matrixes of subnetworks L1 and L2. An element Aijx=1 of subnetwork x∈1,2 means that nodes ix and jx are connected. The matrixes A12 and A21 are the adjacency matrixes of interlayer network, where Ai1i212=1 means that node i1 uses computer i2. Note that Ai1i212=Ai2i121 for any values of i1 and i2. The average degrees of the two subnetworks can be denoted as k1=∑i,jAij1=∑k1k1P1k1 and k2=∑i,jAij2=∑k2k2P2k2, respectively.


## 2.2. Virus-Information Spreading Model

We here adopt a susceptible-infected-recovered (SIR) model to describe the virus spreading on subnetworkL1. A node in the susceptible state means that it does not get infected by the computer virus. An infected node represents that it is infected by the virus and can transmit it to one of its neighbors. A node in the recovered state means that it has recovered and does not change its state.For the information spread on subnetworkL2, we consider using the irreversible susceptible-informed-recovered (SIR) model [37]. A susceptible node means that it does not obtain information about the virus. An informed node indicates that it knows information about the virus and is willing to share it with its neighbors. A node in the recovered state means that it loses interest in the information and will not transmit it to its neighbors. In this paper, we denote the virus-information coevolution spreading as a SIR-SIR model.The virus-information coevolution spreading dynamic evolves as follows. Initially, we randomly select a nodei1 in subnetwork L1, that is to say, the computer virus infects node i1. The corresponding node i2, i.e., the user of computer i1, is also set to be the infected state, since the user can release the information of his infection to his neighbors. At every time step t, each infected node v1 in subnetwork L1 tries to transmit the computer virus to one of its neighbors u1, since every infected node usually communicates with one neighbor at a short time interval. In reality, the infection transmission depends on the “source” and “target” nodes [35]. That is to say, the infectivity and susceptibility of the system are distinct for different nodes. To include this factor, we assume that nodes’ infectivity and susceptibility depend on the degree of nodes. More specifically, the infectivity of node v1 with degree k1 is(1)ωk11=1−1−α1k1,where α1 is the unit infectivity for a node with degree 1. Similarly, the susceptibility of node u1 with degree k1′ is(2)ϖk1′1=1−1−α1′k1′,where α1′ is the unit susceptibility for node with degree 1. Varying the values of α1 and α1′, we get different infectivities and susceptibilities of the system. If node u1 is susceptible, two different situations should be considered. (i) If the user u2 of computer u1 is in the susceptible state, the computer u1 infects the virus with probability λ1. Meanwhile, node u2 obtains the information. Otherwise, u2 is in the infected or recovered state, and nothing happens. (ii) If the user u2 has already obtained the information before, the computer u1 is infected by the virus with probability qλ1, where 0≤q≤1. We here use the parameter q to describe the degree of protection when a user knows the virus’s spreading. The smaller the value of q, the stronger the protection against computer viruses. Every infected node recovers with probability γ1.The information about the virus spread on the subnetworkL2 is as follows. At each time step t, every informed node v2 transmits the information to one of its neighbor u2 in subnetwork L2 depending on the infectivity of v2 and the susceptibility of u2. The infectivity of v2 with degree k2 is(3)ωk22=1−1−α2k2,where α2 is the unit infectivity. The susceptibility of u2 with degree k2′ is(4)ϖk2′2=1−1−α2′k2′,where α2′ is the unit susceptibility. The infection probability is λ2. Finally, every informed node loses interest in transmitting the information with probability γ2. The spreading ends when there are no nodes in the infected or informed state. In Table 1, we present the definitions of parameters and abbreviations.Table 1
Definitions of parameters and abbreviations.
ParameterDefinitionN1Subnetwork size of computer networkL1N2Subnetwork size of social networkL2P1k1Degree distribution of computer networkL1P2k2Degree distribution of social networkL2k1Average degree of computer networkL1k2Average degree of social networkL2ωk11Infectivity of nodev1 with degree k1 in subnetwork L1ωk22Infectivity of nodev2 with degree k2 in subnetwork L2ϖk1′1Susceptibility ofu1 with degree k1′ in subnetwork L1ϖk2′2Susceptibility ofu2 with degree k2′ in subnetwork L2α1Unit infectivity in subnetworkL1α2Unit infectivity in subnetworkL2α1′Unit susceptibility in subnetworkL1α2′Unit susceptibility in subnetworkL2qProtection powerλ1Computer virus transmission probabilityλ2Information transmission probabilityγ1Computer virus recovery probabilityγ2Information recovery probabilitysk11tProbability of node with degreek1 in the susceptible state at time t in subnetwork L1ρk11tProbability of node with degreek1 in the infected state at time t in subnetwork L1rk11tProbability of node with degreek1 in the recovered state at time t in subnetwork L1sk22tProbability of node with degreek2 in the susceptible state at time t in subnetwork L2ρk22tProbability of node with degreek2 in the infected state at time t in subnetwork L2rk22tProbability of node with degreek2 in the recovered state at time t in subnetwork L2Θ1tAverage probability that a node connects to an infected neighbor through an edge in subnetworkL1Θ2tAverage probability that a node connects to an informed neighbor through an edge in subnetworkL2


## 3. Heterogeneous Mean-Field Theory

In this section, we develop a heterogeneous mean-field approach to describe the evolution of the virus-information spreading dynamics. In theory, we assume that nodes with same degrees have the same infection probability in statistical [32, 33, 38, 39]. In other words, the probability of nodes with the degree k is the same as each other.We use the following parameters to describe the coevolution process. Denotesk11t, ρk11t, and rk11t as the probability that a node with degree k1 is in the susceptible, infected, and recovered states at time t in subnetwork L1, respectively. Similarly, we denote sk22t, ρk22t, and rk22t as the probability of node with degree k2 in the susceptible, informed, and recovered states at time t in subnetwork L2, respectively. Considering the degree distributions of subnetworks L1 and L2, we know the fraction of nodes in each state. For instance, the fraction of nodes in the susceptible state at time t is S1t=∑k1P1k1sk11t. In the final state, i.e., t⟶∞, the fraction of nodes in the susceptible state is S1∞≡S1.Now we derive the expressions of the probability of nodes in each state. We know thatsk11t decreases with time t when nodes are infected by the virus. A susceptible computer u1 with degree k1 infected by the virus has two situations. (1) The corresponding node (i.e., the user) u2 of node u1 is in the susceptible state. In this situation, the infection transmitted to node u1 should fulfill two necessary conditions.(i) An infected neighborv1 of node u1 contacts node u1. In uncorrelated complex networks, the probability of node u1 connecting to an infected neighbor v1 with degree is k1′ is k1′−1/∑k1k1P1k1, where k1′ is the degree of node v1. Considering the degree distribution of subnetwork L1, the average probability that a node connects to an infected neighbor through an edge is(5)Θ1t=1k1∑k1′ωk1′1k1′−1P1k1′ρk1′1t.(ii) The infection is transmitted successfully with probabilityλ1. According to the description of the model, for a node u1 in the susceptible state, its corresponding node must also be in the susceptible state. However, the opposite situation does not always exist. Combining conditions (i) and (ii), we know situation (1) happens with probability λ1k1ϖk11sk11tΘ1t. Situation (2) indicates that node u2 of node u1 is in the informed state. Using a similar discussion with situation (1), we obtain the probability of situation (2) as follows: qλ1k1ϖk11sk11tΘ1t∑k2P2k2λ2k2ϖk22Θ2t, where ∑k2P2k2λ2k2ϖk22Θ2t is the probability that the corresponding node u2 of u1 is informed by neighbors in subnetwork L2 at time t. Θ2t will be defined later. The rate equation of sk11t is(6)dsk11tdt=−λ1k1ϖk11Θ1tsk11t+q∑k2P2k2λ2k2ϖk22Θ2t.The evolution ofρk11t is(7)dρk11tdt=λ1k1ϖk11Θ1tsk11t+q∑k2P2k2λ2k2ϖk22Θ2t−γ1ρk11t,where γ1ρk11t is the fraction of nodes recovered at time t. Finally, the evolution of rk11t is(8)drk11tdt=γ1ρk11t.According to equations (6)–(8), we obtain the evolution of computer virus spreading on subnetwork L1.Now, we study the rate equations of the information about the virus spreading on social networkL2. There are two different situations for the reduction of sk22t. For the first situation, the susceptible node u2 with degree k2 is infected by its informed neighbor v2. The infection probability is λ2k2ϖk22sk22tΘ2t, where Θ2t denotes the probability of an edge connecting to an informed neighbor in subnetwork L2. The expression of Θ2t can be expressed as(9)Θ2t=1k2∑k2′ωk2′2k2′−1P2k2′ρk2′2t.The second situation of nodeu2 obtaining the information is that the corresponding node u1 of u2 is infected by the computer virus through an edge of u1 with probability λ1k1ϖk11sk22tΘ1t. Since u2 and u1 are randomly coupled, the averaged infection probability of u1 is λ1k1ϖk11sk22tΘ1t, where ϖk11=∑k1P1k1ϖk11. Combining the two situations, we obtain the rate equation of s2k2t as(10)dsk22tdt=−sk22tλ2k2ϖk22Θ2t+λ1k1ϖk11Θ1t.With the similar discussion about the virus spreading on subnetworkL1, we have(11)dρk22tdt=sk22tλ2k2ϖk22Θ2t+λ1k1ϖk11Θ1t−γ2ρk21t,(12)drk22tdt=γ2ρk22t.With the above equations, we know the fraction of nodes in each state at subnetworksL1 and L2.In the following, we study the global outbreak conditions of the computer virus and information spreading. For global information outbreak condition, we can linearize equations (7) and (11) around the initial conditions, i.e., sk11≈1, sk22≈1. We know that sk11=1, sk22=1, ρk11=0, and ρk22=0 are trivial solutions. Denote a vector ρ⟶=ρ1⟶,ρ2⟶T, where ρ1⟶=ρk1=11,…,ρk1,max1, ρ2⟶=ρk2=12,…,ρk2,max2, and k1,max1 and k2,max2 represent the maximal degrees of subnetworks L1 and L2, respectively. We perform a Taylor expansion for equations (7) and (11) at sk11=1, sk22=1, ρk11=0, and ρk22=0 and neglect the high order of ρ⟶. We have(13)dρ⟶dt=Jρ⟶,where J is the Jacobian matrix. The expression of J is(14)J=∂ρ11∂ρ11∂ρ11∂ρ21…∂ρ11∂ρ1,max1∂ρ11∂ρ12∂ρ11∂ρ22…∂ρ11∂ρ2,max2∂ρ21∂ρ11∂ρ21∂ρ21…∂ρ21∂ρ1,max1∂ρ21∂ρ12∂ρ21∂ρ22…∂ρ21∂ρ2,max2⋮⋮⋮⋮⋮⋮⋮⋮∂ρ1,max1∂ρ11∂ρ1,max1∂ρ21…∂ρ1,max1∂ρ1,max1∂ρ1,max1∂ρ12∂ρ1,max1∂ρ22…∂ρ1,max1∂ρ2,max2∂ρ12∂ρ11∂ρ12∂ρ21…∂ρ12∂ρ1,max1∂ρ12∂ρ12∂ρ12∂ρ22…∂ρ12∂ρ2,max2∂ρ22∂ρ11∂ρ22∂ρ21…∂ρ22∂ρ1,max1∂ρ22∂ρ12∂ρ22∂ρ22…∂ρ22∂ρ2,max2⋮⋮⋮⋮⋮⋮⋮⋮∂ρ1,max2∂ρ11∂ρ1,max2∂ρ21…∂ρ1,max2∂ρ1,max1∂ρ1,max2∂ρ12∂ρ1,max2∂ρ22…∂ρ1,max2∂ρ2,max2.The Jacobian matrixJ can be further expressed as block matrix as(15)J=C1D2D1C2,where dimensions of C1, C2, D1, and D2 are k1,max×k1,max, k2,max×k2,max, k2,max×k1,max, and k1,max×k2,max, respectively. When the global information on subnetwork L2 breaks out, the largest eigenvalue of J is larger than zero. The global information outbreak condition is(16)Λ1J=0,where Λ1J is the largest eigenvalue of J. For the virus outbreak condition, we cannot solve it directly. When the network is extensive, we can use the competing percolation theory [40]. That is to say, we can process the information spreading on subnetwork L2 first and then the virus spreading on subnetwork L1.


## 4. Simulation Results

In this section, we perform numerical simulations to study the virus-information spreading dynamics on computer-social network. To build the computer-social network, we use the uncorrelated configuration model [36]. We set the degree distributions of subnetworks L1 and L2 as P1k1∼k−χ1 and P2k2∼k−χ2, respectively, where χ1 and χ2, respectively, represent the degree exponents. There is no inter- and intralayer degree-degree correlations. In numerical simulations, we set the average degrees of the two subnetworks as k1=k2=8, the network sizes as N1=N2=104, and the degree exponent as χ1=χ2=3.0. We set α1=α1′ and α2=α2′. Initially, we randomly select 5 seeds in subnetwork L1. All results presented in this paper are averaged over at least 2000 times.We first study the virus and information spreading sizes, respectively, denoted asR1=1−S1 and R2=1−S2, versus virus transmission probability λ1 as shown in Figure 2. We find that R1 increases with α1=α2, i.e., the virus is more likely to spread when the infectivity and susceptibility are large. Specifically, we note that the virus cannot spread for any values of λ1 when α1=α2 are small, e.g., α1=α2=0.0 and 0.2. When α1=α2 are large enough, enlarging their values cannot promote the virus spreading. When comparing the effects of information spreading on virus spreading, i.e., increasing λ2, the virus spreading can be suppressed, as shown in Figures 2(a)–2(d). That is to say, to contain the virus spreading, we can transmit more information about the virus on the social network. For the information spreading on subnetwork L2, i.e., the social network, we find that R2 increases with λ1, λ2, and α1=α2, since the users have more chances to obtain the information.Figure 2
Virus spreading sizeR1 and information spreading size R2 versus computer virus transmission probability λ1 with q=0.5. R1 versus λ1 with (a) λ2=0.2, (b) λ2=0.4, (c) λ2=0.6, and (d) λ2=0.8. R2 versus λ1 with (e) λ2=0.2, (f) λ2=0.4, (g) λ2=0.6, and (h) λ2=0.8. Other parameters are set to be γ1=γ2=0.2 and λ2=0.8.
(a)(b)(c)(d)(e)(f)(g)(h)In Figure3, we further investigate the effects of protection power q on the virus-information spreading for different values of α1=α2 and λ2. Generally speaking, we find similar results with that discussed in Figure 2. When the protection power is large, we find that the virus spreading size is relatively smaller, i.e., R1q=0.8≥R1q=0.5, since the susceptible nodes are less likely to be infected by neighbors. We also note that R2q=0.8≥R2q=0.5 since the promotion of virus spreading on information spreading is decreased.Figure 3
Virus spreading sizeR1 and information spreading size R2 versus computer virus transmission probability λ1 with q=0.8. R1 versus λ1 with (a) λ2=0.2, (b) λ2=0.4, (c) λ2=0.6, and (d) λ2=0.8. R2 versus λ1 with (e) λ2=0.2, (f) λ2=0.4, (g) λ2=0.6, and (h) λ2=0.8. Other parameters are set to be γ1=γ2=0.2 and λ2=0.8.
(a)(b)(c)(d)(e)(f)(g)(h)In Figure4, we study the effects of susceptibility and infectivity in detail. We find that R1 decreases with the increase of susceptibility and infectivity of L2. That is to say, the virus spreading can be suppressed by increasing the susceptibility and infectivity. We can explain the results as follows. Increasing susceptibility and infectivity, the information will be widely spread on social network (see Figures 4(e)–4(g)), and more susceptible nodes in subnetwork L1 will take measures to protect themselves from being infected. As a result, R1 decreases with α2.Figure 4
Virus spreading sizeR1 and information spreading size R2 versus computer virus transmission probability λ1 with q=0.5. R1 versus α2 with (a) λ1=0.2, (b) λ1=0.4, (c) λ1=0.6, and (d) λ1=0.8. R2 versus α2 with (e) λ1=0.2, (f) λ1=0.4, (g) λ1=0.6, and (h) λ1=0.8. Other parameters are set to be γ1=γ2=0.2 and λ2=0.5.
(a)(b)(c)(d)(e)(f)(g)(h)Finally, we studied the virus-information spreading as a function ofα2 when the protection power is lower with q=0.8 in Figure 5. We reveal similar phenomena as shown in Figure 4. We note that R1q=0.5≥R1q=0.8 and R2q=0.5≥R2q=0.8 since the protection power is decreased.Figure 5
Virus spreading sizeR1 and information spreading size R2 versus computer virus transmission probability λ1 with q=0.8. R1 versus α2 with (a) λ1=0.2, (b) λ1=0.4, (c) λ1=0.6, and (d) λ1=0.8. R2 versus α2 with (e) λ1=0.2, (f) λ1=0.4, (g) λ1=0.6, and (h) λ1=0.8. Other parameters are set to be γ1=γ2=0.2 and λ2=0.5.
(a)(b)(c)(d)(e)(f)(g)(h)


## 5. Discussion

In this paper, we studied the virus-information spreading dynamics on computer-social multiplex networks. We first proposed a mathematical model to describe the coevolution spreading dynamics. In this model, we assumed that nodes’ susceptibility and infectivity are heterogeneous and positively correlated with the node’s degree. To describe the spreading dynamics, we adopt a generalized heterogeneous mean-field approach. Using extensive numerical simulations, we revealed that the virus spreading dynamics can be significantly suppressed by promoting the information spreading on the computer network or decreasing the susceptibility and infectivity of nodes. Our results provide some insight into containing the virus spreading.



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