Dynamics and Control Strategies for SLBRS Model of Computer Viruses Based on Complex Networks

. Te proliferation of computer viruses has escalated in recent years, posing threats not only to individuals’ safety and property but also to societal well-being. Consequently, efectively curtailing virus spread has become an urgent imperative. To address this issue, our paper introduces a new virus propagation model and associated control strategy. First, diverging from conventional approaches in network virus literature, we propose a susceptible-latent-breaking-out-recovered-susceptible (SLBRS) virus propagation model tailored to the topological characteristics of scale-free networks, thus comprehensively incorporating network structure’s impact on virus propagation. Second, we analyze the model’s foundational properties, derive the basic reproduction number, and demonstrate the existence and global asymptotic stability of disease-free equilibrium. Finally, leveraging global stability of the model at the disease-free equilibrium, we integrate the target immunization strategy (TIS) and the acquaintance immunization strategy (AIS) to devise an optimal control strategy. Te paper’s fndings ofer fresh insights into disease-free equilibrium existence and stability, furnishing a more dependable approach to curbing network virus dissemination. Te simulation results demonstrate the persistent presence of network viruses in the absence of control measures and the instability of the disease-free equilibrium. However, efective control is achieved after implementing immunization measures.


Introduction
Human and many biological populations have been facing the threat of viral epidemics.Te spread of epidemics usually causes large numbers of deaths and signifcant economic losses.In particular, the novel coronavirus pneumonia (COVID-19) has rapidly disseminated across numerous countries and regions worldwide due to its high contagiousness [1].Te new "COVID-19" outbreak in 2019, in the current, its global impact is still very severe [2,3].Similar to the spread of biological virus, the spread of computer virus plays a vital role in the development and stability of human society [4].Well-known computer viruses, including the Stuxnet virus (2010) [5], Wannacry (2017) [6], and Phishing e-mail (2020) [7], have caused billions of dollars in economic losses to many countries and regions around the world.Nowadays, hackers use computer viruses to illegally collect a large amount of private data and hijack the use rights of computers, making the virus spread quickly in cyberspace and diffcult to control [8].For example, ransomware wannacry (2017) [6] is a software virus that hijacks user data, spreads via the web or e-mail, replicates itself and spreads quickly.Ransomware wannacry locks down systems in a way that cannot be reversed by knowledgeable people, and its victims are mostly industry organizations and large corporations.Once infected, users have to pay a ransom to decrypt it, which causes instability of computer data.Although much has been done to prevent the spread of computer viruses, the human fght against viruses is still in its infancy [9,10].Due to the intricate nature of computer networks, the investigation of computer virus models and their corresponding control strategies has emerged as a focal point in contemporary research, concurrently bearing signifcance in the realm of biological virus prevention and control.
1.1.Related Works.Over the past few decades, numerous scholars have endeavored to explore the impact of complex network structures on virus propagation models, aiming to formulate a virus transmission model that aligns with realworld conditions [11,12].In terms of network structure, models s like regular networks, small-world networks [13], and scale-free networks [14] provide some new directions for the study of infectious diseases.For virus transmission models on diferent network structures, Kleczkowski and Grenfell [15] and Moore and Newman [16] established diferent infectious disease models on the small-world networks and provided the critical threshold of infectious disease.Subsequently, Pastor-Satorras, and Vespignani introduced the earliest infectious disease models like the susceptible-infected-recovered (SIR), susceptible-infectedrecovered-susceptible (SIRS), and susceptible-exposed-infected-recovered (SEIR) models on scale-free networks [17,18].Inspired by these researches, Liu and Zhang [19] investigated the transmission threshold and stability of the SEIRS model based on the research nodes of Olinky and Stone [20] and Joo and Lebowitz [21], and they have obtained many valuable conclusions.To quantitatively assess the security of virus systems, Tang et al. [26] employs safety entropy to analyze the security situation of the susceptiblelatent-break-out-recovered-susceptible (SLBRS) model based on the work of Yang et al. [22] and Zhang and Yang [23].Take some novel modeling methods for example, Naik [24] and Ahmad et al. [25] delved into a fractional-order SIR epidemic model incorporating memory, conducting modeling and numerical research on bovine babesiosis disease.Very recently, Sun, Ghori, and del Rey built diferent virus models to analyze the key problems such as periodic infection rate [27], global dynamics and bifurcation analysis [28], and mutation transmission [29].However, these models overlook the infuence of virus spread within complex network structures, particularly the dynamics of computer virus propagation in scale-free networks, which brings an opportunity to our work.
In addition to exploring the various modes of virus propagation on complex networks, it is imperative to consider efective control strategies for mitigating the spread of network viruses.Regarding the control strategy for virus models, Pastor-Satorras and Vespignani [30] provided an idea and theoretical framework for virus control on complex networks.Bai et al. [31] compared the diferences in control efect between random immunity and target immunity.As research progressed, people began to discuss the role of network structure and infectivity in virus control, Masuda [32], Zhang and Fu [33], Wu et al. [34], and Buono and Braunstein [35] discussed the relationship between network immunity and network structure, and these studies provide more perspectives and ideas for network virus immunization.To improve the efectiveness of immune control, Yang et al. [36] proposed optimal dynamic immunization of controllable heterogeneous nodes under the SIRS model, Cao et al. [37] compared the control efects of the Susceptible-Infectious-Carriers-Recovered (SICR) model across various immunization strategies by examining the stability of the viral system dynamics, and Xia et al. [41] proposed an improved target immunization strategy based on two rounds of selection.Additionally, to combat malicious attacks in the Internet of Tings and sensor networks, numerous efective control methods integrate machine learning theory [38], knowledge-driven [39], and data-driven methods [40], resulting in signifcant control efcacy.Compared with the control efect of the biological virus COVID-19, Li and Guo [42] analyzed the optimal control and efectiveness of the novel COVID-19 model of the Omicron strain.Guo and Li [43] conducted fractional-order modeling and optimal control of a new online game addiction model based on real data.Te above work has a good enlightening signifcance for us to carry out the control of network viruses.
Although computer viruses and biological viruses have great diferences in the way of transmission and infuence, the control of the two kinds of viruses has many similar rules [44,45].For example, the control of the new corona virus mainly adopts the idea of isolation control and acquaintance immunization.Tis method focuses on the key populations in the infected population to efectively control the spread of the virus.For scale-free computer networks, the network structure not only afects virus propagation but also plays a key role in virus control.Terefore, how to design an efective control strategy based on the virus propagation dynamics is an important research topic.Tis paper designs an improved target immune control strategy based on the transmission law of computer viruses, which also provides a meaningful reference for solving the transmission of biological viruses.

Motivations and Contributions.
Inspired by the above literature, we studied the SLBRS model and its control strategy on scale-free networks.Te contributions are as follows: (1) We present a novel SLBRS model embedded within a scale-free network framework and proceed to analyze its fundamental dynamic characteristics.Initially, we delineate the structural attributes and parameters of the scale-free network environment serving as the habitat for viruses.Subsequently, we devised a model for computer virus propagation on scale-free networks, with the aim of deepening our understanding of virus dissemination dynamics.Furthermore, we innovate the node state transition mode and system parameters.Our study extends classical models (SI, SIR, and SIS) on complex networks to address the defciency in parameter design observed in prior research.(2) We calculate the basic reproduction number and conduct an analysis of the stability of this model at the disease-free equilibrium.Furthermore, we establish the stability of the disease-free equilibrium and demonstrate the persistence of the disease for a fnite size of the scale-free network.

Proposed the SLBRS Model
In this section, we introduce the parameters of the model and scale-free network, and propose a new SLBRS computer virus model on scale-free networks.

Scale-Free Networks and Model Assumptions.
A complex network is a topological structure composed of many nodes and intricate relationships among nodes.One of the most important types of complex networks, scale-free networks, has attracted a great deal of research.Because the dynamic behavior of the network is afected by the network topology, statistical characteristics and formation mechanism, we divide the network into diferent types, including regular networks, random networks, small-world networks, and scale-free networks, etc.In order to investigate the propagation patterns of computer network viruses in real-world scenarios, it is imperative to conduct a comprehensive analysis of the topological characteristics of four fundamental network models.An intriguing observation is the prevalence of power-law degree distributions in many real networks, indicating that the probability distribution function of degree approximately follows the form As indicated in Table 1, it is evident that the scale-free network conforms to a power-law distribution.Scale-free networks exhibit a notable presence of highly connected nodes alongside numerous nodes with relatively few connections.Te probability of connecting new nodes to existing nodes in a scale-free network is proportional to the degree of existing nodes, and repeated links are not allowed in the process of network generation.In computer networks, the scale-free network is used to describe the growing and preferentially open real world.Te topological characteristic parameters of the scale-free network are set as shown in Table 2.
In a scale-free network, virus nodes are classifed into four distinct types: susceptible (S), latent (L), breaking-out (B), and recovered (R).Susceptible and recovered nodes are denoted as healthy, while latent and breaking-out nodes are referred to as diseased.In practical computer networks, each host is regarded as a network node, with the ability to transition between these four states.Te detailed descriptions of these parameters are listed in Table 3.
In reality, computer networks are regarded as scale-free network, because a few computer nodes have a large number of linked nodes, while a large number of computer nodes only connect to a few neighbor nodes.Te schematic diagrams of the scale-free network are shown in Figure 1.
In the SLBRS model, computers connected to the Internet are categorized into four compartments: uninfected computers lacking immunity (S computers), latent infected computers (L computers), breaking-out infected computers (B computers), and computers with temporary immunity (R computers).Figure 2 shows the transition between these four states.

Model Description.
In the SLBRS model of a computer network, where each host is represented by a network node and hosts communicate with each other through connecting edges, viruses spread along these connections.Let S k (t), L k (t), B k (t) and R k (t) be the densities of S, L, B and R node of degree k at time t, respectively.Te SLBRS model on a scale-free network can be described as follows: Te transmission diagram of SLBRS model is shown in Figure 2 In the model SLBRS, each individual has the same probability of being in contact with its neighbors and the ability of each node to be infected by the virus is proportional to its degree, i.e., ϕ(m) or φ(m) is equal to αm, where α is a positive constant and 0 ≤ α ≤ 1.Further, to simplify the complexity of the model, we assume that the Based on the above model assumptions, the dynamic system (1) can be simplifed as It is evident that these variables adhere to the normalization condition, namely, (3) In the system (2), the global densities of all nodes are Te initial conditions of system (2

Dynamical Behavior of the SLBRS Model
According the normalization condition, the system (5) simplifes as We calculate the basic reproduction number R 0 by system (6).
Proof.According to system (6), we have International Journal of Intelligent Systems Since then Ten To calculate the basic reproduction number R 0 of this model, let θ �  M m�1 mp(m)ρ(m)/〈k〉 denote the probability that a susceptible node of degree k is infected by a Breakingout node each time, where ρ(m) � B m /N m .We construct the auxiliary function F(θ), then we have And, we calculate the monotonicity of the function because So, the only necessary and sufcient condition that F(θ) has a unique positive solution in 0 ≤ θ ≤ 1 is Tus, we get the basic reproduction number R 0 from (15), i.e.,

□
Consequently, when R 0 > 0, system (1) has a unique endemic equilibrium E 1 .Te endemic equilibrium indicates that virus nodes in a network system will persist over time, creating a stable state.Te endemic equilibrium is a relatively stable state, once this state is broken, it may lead to the outbreak of endemic diseases.

Te Existence and Stability of Disease-Free Equilibrium.
In order to analyze the existence of disease-free equilibrium point in the given computer virus network and to determine the global stability of the system, we reduce the system (2) as follows: 6 International Journal of Intelligent Systems
Proof.Te system (2) at the disease-free equilibrium point E 0 (1, 0, 0, 0) { } be written as Te Jacobian is a 3M × 3M matrix, where Its characteristic polynomial is Terefore, all characteristic roots are According to Hurwitz criterion, no matter whether R 0 is greater than 1, the characteristic equation has roots of positive real part, which indicates that the disease-free equilibrium point E 0 (1, 0, 0, 0) { } of the system is unstable.

Simulation Results and Discussion.
To verify the correctness of the theory proposed in Section 3, simulation experiments are carried out from three perspectives.
(1) When the model parameters and the initial values of the nodes are the same, we analyze the efect of diferent network average degree 〈k〉 on virus propagation.As shown in Figure 3, the greater the average degree of network nodes, the stronger the ability of computer virus to spread on the network at the initial moment.Figure 3 shows that if the computer virus preferentially infects nodes with higher than average degree in the network, the virus will spread rapidly at the initial moment and the network is under severe security threat.(2) When model parameters and node degrees of the SLBRS model are the same, we analyze the infuence of virus propagation at diferent initial values.In our experiment, we divided 1000 nodes into four different states, including S nodes, L nodes, B nodes, and R nodes.As shown in Figure 4, we analyzed the infuence of initial values of four types of nodes in diferent states on the dynamics of network evolution.Figure 4 shows that the more network nodes are infected at the initial moment, the faster the network virus spreads and the greater the pressure to carry out virus control.(3) When the initial value and node degree of the model are the same, we tried to study the infuence of International Journal of Intelligent Systems diferent parameters on virus transmission.Figure 5 shows the dynamic behaviors under diferent model parameters.By selecting diferent model parameters to carry out a large number of data experiments, the following conclusions are drawn.First, the virus always persists in the system regardless of the values taken for the model parameters.Second, the proportion of infected nodes in the system becomes larger as the parameter α 1 , α 2 become larger, and becomes smaller as the parameter c 1 , c 2 become larger.
To further analyze dynamical behavior of viruses on scale-free networks, we simulate the propagation characteristics of viruses using the NetLogo simulation tool.Te parameter of the model selected in Figure 6 is [α 1 , α 2 , c 1 , c 2 , β, μ] � [0.55, 0.45, 0.35, 0.25, 0.35, 0.65].As can be seen from Figure 6, although the number of recovered nodes in the system increases over time, the virus nodes will always exist in the system.Based on the above discussion, we verify the existence of disease-free equilibrium of SLBRS model and the global stability of the system, and the simulation results agree with the life circumstances.
From Figures 3-6, we get three conclusions.Firstly, as the degree of nodes in the network increases, the faster the virus will infect the network system, and the system will soon be occupied by the virus nodes.Tis shows that the key nodes in the network system with a high degree play an important role in the spread of the virus.Secondly, the growth trend of network viruses in diferent initial value states remains relatively stable, and the number of L nodes and B nodes in the network remains between 200 and 400.Finally, the network system is sensitive to diferent model parameters.Te greater the infectivity of the virus is, the faster the outbreak speed of nodes will be, and the lower the network security it gets.
In conclusion, if there is no strengthening of immune control measures, the viruses will persist the entire cyberspace.Meanwhile, the disease-free equilibrium point of the network system is unstable, and the disease tends to be in positive equilibrium.

The Immunization Strategy for the SLBRS Model
In complex networks, the main channels for virus propagation include emails, links, mobile hard drives, wearable devices, etc. Te targets of virus attacks are mainly core nodes and groups in critical positions, which pose a huge security risk in cyberspace.To ensure network availability and robustness, infectious disease control requires full consideration of network topology and virus

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International Journal of Intelligent Systems transmissibility, and the design of targeted immunization strategies to control the spread of viruses.
For the dynamics model of infectious diseases on scalefree networks, since the spread of network viruses is highly dependent on the structure of the network, the network nodes are classifed according to the node state and degree distribution characteristics, which helps to carry out immune control.Te core nodes within the network are typically prime targets for immune control.Immunization extends to the neighboring nodes of these core nodes, effectively targeting nodes with a degree greater than k within this vicinity.
In this section, we further discuss the immunization efect of the SLBRS model on target immunization and its improved strategy.First, we introduce the specifc approach and immunization efect of the target immunization strategy.Second, We propose a new target immunization strategy to improve the efciency of global immunization, i.e., immunizing key nodes while simultaneously immunizing their neighboring nodes.

Target Immunization
Strategy.Among the existing immunization strategies, the classical immunization strategies containing proportional immunization, targeted immunization, acquaintance immunization, and active immunization are widely known.Since diferent immunization strategies have diferent control efects on diferent virus models, the selection of immunization strategies is very important.For example, the proportional immunization scheme needs to immunize a large number of network nodes, so it is difcult to achieve herd immunity.
In a computer virus network, because scale-free networks are heterogeneous and native, target immunization of nodes in the network with a degree greater than a certain value may be a more efective scheme, but target immunization requires global information of the network and it is difcult in practical applications.For network attacks, choosing acquaintance immunization or active immunization strategy is a more desirable strategy.On the one hand, acquaintance immunization does not need to grasp the global information of the complex network and also has a better immunization efect on the system.On the other hand, the immunization cost of acquaintance immunization is low, and the efciency of immunization can also meet the needs of practical work.
To improve the efectiveness of immune control, we propose a novel active immunization strategy that combines target immunity and acquaintance immunity.
We initially introduce an upper threshold κ, where all nodes with connectivity k > κ are immunized.In other words, we defne the immunization rate δ k as follows:     International Journal of Intelligent Systems where 0 < c ⩽ 1, and  k δ k P(k) � δ, and δ is the average immunization rate.P(k) is the degree distribution of the nodes with degree k.Te immunized SLBRS model is expressed as When the computer virus spreads in the network, we select the nodes whose node degree is greater than the threshold κ for target immunization.Te efect of immunization control on system (2) is analyzed by simulation experiments, as shown in Figure 7.
As we all know, when controlling the spread of network viruses using a target immunization strategy, we should not only consider the initial state of the system but also strengthen the control of susceptible and recovered nodes.It can be seen from Figure 7 that the larger the proportion of target immunization of nodes, the more efective the virus control is.If the upper threshold of the node degree value is smaller, the success rate of immunization is higher.However, the cost of target immunization is large, so target immunization is generally used at the beginning of a virus outbreak.To further reduce the impact of the limitations of the target immunization strategy, improve its efectiveness and reduce the cost of the target immunization strategy, we propose the following improvement strategy.

Improved Target Immunization Strategy.
When dealing with virus events in large-scale computer networks, people usually target immunization of diseased nodes, but it is difcult to ensure the efciency and efectiveness of immunization.For this reason, we can frst select some of the key nodes from all the network nodes as the immunization object, and then classify the selected nodes into healthy nodes and diseased nodes, and fnally take diferent immunization measures for these two types of nodes for system control.For example, controlling server nodes can avoid the risk of a large number of associated hosts going down.At the same time, the improved immunization strategy has merit for the control of the new coronavirus pneumonia (COVID- 19).
Te idea of this immunization strategy mainly consists of the following steps.First, the nodes with proportion f are randomly selected from N nodes, then the number of selected nodes is fN, where the proportion of f is determined according to the security situations of the network system and the correlation characteristics among the nodes.Te goal is to make immunization as efcient as possible without knowing the global information of the network.Second, in order to improve the network immunization efect, the network administrator needs to identify the status of the elected nodes and classify these nodes into healthy and diseased nodes.Tird, the acquaintance immunization strategy is applied to the neighboring nodes of the healthy nodes, i.e., immunizing the nodes with degree values greater than κ among the neighboring nodes to reduce the risk of virus transmission.Finally, since the diseased nodes with node degree greater than k have extremely strong virus infectivity in the network, it is necessary to utilize the target immunization strategy to target immunization of these nodes to block the spread of viruses from the source of virus outbreaks.Tat is to say, the target immunization strategy is used to control the nodes with degree greater than k among the diseased nodes, so that the virus cannot be transmitted to the neighboring nodes through these key nodes.In order to achieve the expected immunization efect, it is necessary to adjust the immunization ratio and target according to the actual situation in the computer network.

International Journal of Intelligent Systems
Te immunized SLBRS model is expressed as where δ k is given by the piecewise function above, f k refers to the proportion of nodes with degrees greater than k selected for immunization among Breaking-out nodes.In general, it is easy to fnd the key nodes in the network system where security faults occur, and the harm of these nodes to the whole system is closely related to the number of its neighbor nodes.Consequently, we get a better immune efect by selecting a few Breaking-out nodes with a degree greater than k for target immunity.Te simulation results are shown in Figure 8.
To ensure the security of the network system, it is imperative to enhance the efcacy of virus control while effectively containing the spread of network viruses.In the context of the new SLBRS virus propagation model on scalefree networks, we implement a target control strategy and propose an enhanced target immune control strategy.After

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International Journal of Intelligent Systems comparing the control efects of the two strategies, two comments can be given: on the one hand, the two immune control strategies have high control efects on the SLBRS virus model, but they difer in the efciency of control.On the other hand, the improved target immunization strategy can improve the timeliness of control, and it is suitable for network security managers to adopt this strategy for security control.
To test the control efect of the immunization strategy in this paper, we give the control efect of the system in the initial and stable periods of virus transmission.
As shown in Figures 9 and 10, the control of network viruses not only requires the selection of key nodes as immune objects but also reduces the infection of key nodes to their neighbors, which is determined by the characteristics of scale-free networks.Terefore, the combination of target immunization and  International Journal of Intelligent Systems acquaintance immunization strategies can efectively control the spread of the virus.As an application, the security of a computer network requires the control of core nodes, especially the key nodes in the center of the network.At the same time, it is necessary to strengthen the protection mechanism of these main nodes and their associated nodes, such as enhancing the deployment of intrusion detection.

Conclusions
In this paper, we investigate a novel SLBRS computer virus model building upon previous literature models such as [19,26,36,41], focusing predominantly on virus propagation characteristics and control strategies within scale-free networks.It is observed that system (2) displays two equilibria, namely the disease-free equilibrium E 0 and the endemic equilibrium E 1 .Te global stability analysis was conducted for both equilibria by examining the basic reproduction number R 0 .Te disease-free equilibrium E 0 has been demonstrated to be unstable, while the endemic equilibrium E 1 exists under certain conditions.Furthermore, a target immunization strategy was implemented to achieve efective control of the proposed SLBRS epidemic model based on the scale-free network.As far as the author knows, this represents an innovative approach to epidemic modeling in the literature, which combines the target immunization strategy and acquaintance immunization strategy.Te results show that the improved targeted immunization scheme is an accurate and efective method, which can solve the problem that the virus will persist in the system in the SIR Model, and efectively improve the system security.
In the realm of infectious diseases, uncertainties and unknown variables pose signifcant challenges in developing accurate models for computer virus transmission based on complex networks.Generally, scale-free network structures are better suited for such scenarios, as they can capture network heterogeneity and preferential linking.Terefore, the application of the fndings in this paper within the realm of cyberspace could potentially lead to the development of efective strategies for the prevention, treatment, and control of severe infectious diseases.Te insights from this study are relevant to network security experts and biomedical scientists alike, aiding in the formulation of comprehensive evaluation and treatment protocols for diseases such as Wannacry, phishing e-mail attacks, HIV, COVID-19, and beyond.Compared with the existing research work, the scheme in this paper has a wider scope of application in terms of modeling methods and virus control efects, and efectively compensates for the shortcomings of past studies.
Recent computer viruses, including ransomware, phishing emails, and trojans, are a major threat to humanity in cyberspace.Faced with the major threat posed by viruses to cyberspace, the application of the proposed method in network security situation awareness and other network virus outbreaks needs to be further studied.Similar to the transmission and control of biological viruses, the application of this research method to the transmission of biological viruses is a new idea for future research, which will provide a valuable reference for the control of biological viruses.
(a).Let S(L, B, R) denote a node state as a susceptible (latent, breaking-out, recovered) node, then S k (L k , B k , R k ) is a susceptible (latent, breaking-out, recovered) node with a node degree of k.Obviously, α 1 and α 2 are associated with infection rates, and c 1 and c 2 are related to the recovery rate.p(m | k) denotes the degree correlation between a node of degree k and a node of degree m.Considering uncorrelated networks, p(m | k) � mp(m)/〈k〉.ϕ(m) and φ(m) are the total efective contact time between the L node and the B node with degree m and its neighbor node in unit time.To International Journal of Intelligent Systems simplify the calculation of the model, assuming that each contact time is equal, then ϕ(m)/m and φ(m)/m are the contact times of the L node and the B node with each edge.At this point, α 1 ϕ(m)/m and α 2 φ(m)/m are the probability of S nodes contacting L nodes or B nodes and being infected within unit time.

Figure 1 :
Figure 1: (a) Te network is referred to as a BA scale-free network.Using the preferential attachment algorithm, we generate a BA scale-free network with 100 vertices; (b) degree distribution diagram of the scale-free networks.

Figure 2 :
Figure 2: (a) State transition diagrams for the four computer states in the SLBRS model; (b) schematic illustration of the propagation of a cyber virus on a scale-free network.

Figure 6 :
Figure 6: (a) Te spread of viruses on scale-free networks; (b) dynamical behavior of four types of nodes on the scale-free network.Te number of recovered and infected nodes tends to stabilize over time.

Figure 9 :
Figure 9: (a) In the initial stage of virus propagation, the network has only a few L and B nodes.At this time, the network virus spreads rapidly and the security of the system decreases rapidly; (b) the fgure shows the trend of the four types of nodes in the network at virus propagation time 73.It can be seen that the number of L nodes, B nodes, and R nodes increases rapidly over time and the number of S nodes decreases rapidly.

Table 2 :
Parameters description of scale-free network.

Table 3 :
Description of parameters.
k to breaking-out node B k μ Conversion rate of susceptible node S k to recovered node R k β Conversion rate of latent node L k to breaking node B k c 1 Conversion rate of latent node L k to recovered node R k c 2 Conversion rate of breaking-out node B k to recovered node R k ϕ(m) Total efective contact time between latent node L k and each edge φ(m) Total efective contact time between breaking-out node B k and each edge

Table 1 :
Te main topological features of the network models.