# Dynamical Behavior of Hybrid Propagation of Computer Viruses

Authors: Qingyi Zhu; Pingfan Xiang; Xuhang Luo; Chenquan Gan
Journal: Security and Communication Networks
 (2022)

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## Abstract

Considering the horizontal and vertical propagation of computer viruses over the Internet, this article proposes a hybrid susceptible-latent-breaking-recovered-susceptible (SLBRS) model. Through mathematical analysis of the model, two equilibria (virus-free and virose equilibria) and their global stabilities are both proved depending on the basic reproduction numberR0, which is affected by the vertical propagation of infected computers. Moreover, the feasibility of the obtained results is verified by numerical simulations. Finally, the dependence of R0 on system parameters and the parameters affecting the stability level of infected computers are both analyzed.


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## Body

## 1. Introduction

Computer virus is a malevolent network code designed to disseminate from one device to another [1]. Even minor computer virus can wreak havoc on a system performance, consume computer memory, and cause frequent computer crashes. With all the technological advances of the 21th century, computer virus grew at a breakneck rate. Mail viruses and macroviruses that rely on the Internet to spread have emerged in large numbers. Due to the characteristics of fast-spreading, strong concealment, and great destructiveness of computer viruses, the work of anticomputer virus becomes very difficult [2], which has brought immeasurable losses to people. At the beginning of the twenty-first century, email was an important way for computer viruses to spread. The Medsa virus and the love letter virus spread rapidly around the world via email. In 2017, the WannaCry ransomware quickly infected extensive computers in a short period, causing incalculable damage [3]. On April 12, 2021, KrebsonSecurity, an international information network security media, reported that hackers were selling the personal details of tens of millions of users of ParkMobile, a North American mobile parking application, on a Russian-language cybercrime forum along with screenshots of the data [4]. Later, Code Red and Nimda virus appeared one after another. These two viruses used a combination of mail transmission and active attack on server vulnerabilities, creating a new way of virus transmission and greatly increasing the speed of virus transmission in the network. Owing to the enormous harm brought by the malevolent virus attack, it is urgent to study the spreading behavior of viruses among network nodes and propose effective prevention and control strategies.Seeing the resemblance between computer viruses and biological viruses, many researchers use epidemiological models to study the spread of network viruses. In the twentieth century, the author introduced the susceptibility-infection-susceptibility (SIS) model in the field of network viruses [5]. Thereafter, the susceptible-exposed-infected (SEI) model was proposed by introducing latency bins. Xie [6] proposed the SEI model by considering the effect of heterogeneity of email networks on virus propagation. On this basis, increasingly popular models are being used and studied in the field of virus transmission, for example, susceptible-infected-recovered (SIR) models [7–9], susceptible-infected-recovered-susceptible (SIRS) models [10–13], susceptible-exposed-infected-recovered (SEIR) models [14–17], susceptible-exposed-infected-recovered-susceptible (SEIRS) models [18–21], susceptible-exposed-infected-quarantined-vaccinated (SEIQV) models [22, 23], and susceptible-exposed-infected-quarantined-recovered-susceptible (SEIQRS) models [24–26]. To explicitly compare the individual characteristics of computer virus propagation models, we have made tabular discussions in Table 1.Table 1
Individual characteristics and stability of computer virus propagation models.
ContributorsModelCharacteristicsStabilityVirus-free equilibrium stabilityVirose equilibrium stabilityEssouifi et al. [27]SIR-SISTwo-degree networks and stochastic methodn/aLocally asymptotically stableUpadhyay et al. [28]SVEIRNonlinear incident rateLocally and globally asymptotically stableLocally and globally asymptotically stableYang et al. [29]SEIR-KSHopf bifurcationn/aLocally asymptotically stableAníbal et al. [30]SEIR-KSPeriodic environmentn/an/aZdemir et al. [31]SEIR-KSThe kill signalsLocally asymptotically stableLocally asymptotically stableTang et al. [32]SLBRSSafety entropyn/aGlobally asymptotically stableThis workSLBRSVertical transmissionLocally and globally asymptotically stableLocally and globally asymptotically stableBased on the direction of computer virus transmission, network viruses can be divided into horizontal transmission and vertical transmission. Horizontal spreading refers to the copying and spreading of malicious programs through the spreading medium between nodes. Specifically, after a network virus infects a computer, it infects other computers associated with it through emails, web browsing, disk media, and mobile media. After these infected computers become new sources of infection, they can infect other computers associated with them in the same way as described above. Vertical transmission refers to the virus attacking the main server, making it a source of infection, and then passing it from the main server to any node. For example, Code Red and Nimda viruses actively attack server vulnerabilities through vertical transmission.Generally speaking, in real life, once a computer is infected, it is immediately infectious. However, preceding scholars did not notice this difference between computer viruses and biological viruses, and they believed that latent computers are not infectious. The author [33] first noticed this deficiency of the previous computer virus model and proposed the susceptible-latent-breaking-susceptible (SLBS) model that was more consistent with the actual situation in 2012, and then proposed a series of improved models. Subsequently, a four-compartment model called the primitive SLBRS model was proposed by Yang [34]. By studying the safety tendency of a virus system based on security entropy, Tang proposed a new application scenario SLBRS computer virus model [32]. However, these models always focus on the horizontal propagation behavior of computer viruses.In a computer network, some computer viruses may be transmitted vertically from the master server to any node, such as worms [35]. On the Internet, we have to pay attention to the influence of vertical transmission of computer virus on uninfected new nodes, but the literature on the vertical analysis of computer virus transmission behavior is not extensive. Considering some computer viruses may be vertical transmission from the main server to any node, Kumar [35] proposed the worms in the computer network SEIRS model of vertical transmission. However, this model ignores the fact that computer viruses might be infectious during both the latent and outbreak phases.Given the vertical spread of computer viruses and the fact that latent computers are still infectious, a new SLBRS model is introduced to discuss the spread of network viruses in both horizontal and vertical directions from a macroperspective. Admittedly, this model is more sophisticated and reasonable than previous models, with the rapid growth of master servers. We deeply study the dynamics of this model. Qualitative analysis of the model has obtained the virus-free equilibrium and the virose equilibrium. Global stability of the SLBRS model is verified by using the global geometric method. In addition, numerical experiments illustrate the feasibility of the theoretical results, and the parameters of the system are discussed and analyzed.The rest of this article is organized as follows: In Section2, we describe the SLBRS model with vertical propagation. In Section 3, we confirm a globally asymptotically stable virus-free equilibrium, and the virose equilibrium is obtained. In Section 4, we perform detailed numerical calculations to verify the local and global stability of the virose equilibrium. In Section 5, the influence of system parameters on R0 is discussed. In Section 6, we present the numerical simulation results. Finally, Section 7 summarizes this article.


## 2. Mathematical Model Expression

In the SLBRS model with vertical propagation, the networked computers are defined as four types: susceptible nodesS, latent nodes L, breaking-out nodes B, and recovering nodes R.LetS, L, B, and R represent the proportion of the number of nodes in each of the above four categories among all nodes at a certain time, so we have(1)S+L+B+R=1.Figure1 shows the transformation process of nodes in various states, and parameters used in the model are given in Table 2. The basic terms and assumptions given are used for the rest of this article.Figure 1
Migration blueprint of the SLBRS model.Table 2
Parameters involved in the model.
ParametersDescriptionξThe ratio of computers connected to the Internet.βThe ratio at which eachS computer becomes L computer by contact with either L computer or B computer.εThe rate at whichB computers are cured.γThe rate at whichB computers are cured.ηThe rate at whichR computers lose immunity.pThe rate at which the descendants of theL computers become the S computers.qThe rate at which the descendants of theB computer become the S computers.


### 2.1. Model Assumptions

(1)
Once newly added nodes are infected, they become contagious(2)
Within the network, nodes are connected to each other(3)
At some point, the bilinear infection rate ofS nodes becoming L nodes is βSL+B(4)
Since the probabilities that the descendants ofL and B nodes are L nodes are p and q, respectively, satisfying 0≤p≤1, 0≤q≤1, the probability that the newly accessed computers are S nodes is ξ−pξL−qξB


### 2.2. Model Expression

Based on the previous description and assumptions, the state transition differential system of the model is(2)dSdt=ξ−βSL+B+ηR−pξL−qξB−ξS,dLdt=βSL+B−εL+pξL+qξB−ξL,dBdt=εL−γB−ξB,dRdt=γB−ηB−ξB.Based on the above theoretical analysis, we know thatR=1−S−L−B, and differential equations of system (2) state can be converted into(3)dSdt=ξ−βSL+B+η1−S−L−B−pξL−qξB−ξS,dLdt=βSL+B−εL+pξL+qξB−ξL,dBdt=εL−γB−ξB.The initial states of all kinds of computers in system (3), respectively, satisfy: L0≥0, S0≥0, and B0≥0. The initial value belongs to the positive invariant set:(4)Ψ=S,L,B:S+L+B≤1,S≥0,L≥0,B≥0.The basic regeneration number is the threshold at which a disease tends to die out or persist. We calculated the basic reproductive number is as follows:(5)R0=βξ+γ+ε+ξ+γpξ+εqξξ+γε+ξ.For one thing, from the above description, the virus-free equilibrium is calculated asE0=S0,L0,B0=1,0,0. In addition, data analysis of system (3) was performed to determine the unique endemic equilibrium point E∗=S∗,L∗,B∗. Specific results are as follows:(6)S∗=ξ+γϵ+ξ−ξ+γpξ−ϵqξβξ+γ+ϵ,L∗=ξ+γ2ξ+ηϵ+ξR0−1ξ+γ+ϵξ+γξ+η+ξ+η+γϵ,B∗=ϵξ+γξ+ηϵ+ξR0−1ξ+γ+ϵξ+γξ+η+ξ+η+γϵ.


## 2.1. Model Assumptions

(1)
Once newly added nodes are infected, they become contagious(2)
Within the network, nodes are connected to each other(3)
At some point, the bilinear infection rate ofS nodes becoming L nodes is βSL+B(4)
Since the probabilities that the descendants ofL and B nodes are L nodes are p and q, respectively, satisfying 0≤p≤1, 0≤q≤1, the probability that the newly accessed computers are S nodes is ξ−pξL−qξB


## 2.2. Model Expression

Based on the previous description and assumptions, the state transition differential system of the model is(2)dSdt=ξ−βSL+B+ηR−pξL−qξB−ξS,dLdt=βSL+B−εL+pξL+qξB−ξL,dBdt=εL−γB−ξB,dRdt=γB−ηB−ξB.Based on the above theoretical analysis, we know thatR=1−S−L−B, and differential equations of system (2) state can be converted into(3)dSdt=ξ−βSL+B+η1−S−L−B−pξL−qξB−ξS,dLdt=βSL+B−εL+pξL+qξB−ξL,dBdt=εL−γB−ξB.The initial states of all kinds of computers in system (3), respectively, satisfy: L0≥0, S0≥0, and B0≥0. The initial value belongs to the positive invariant set:(4)Ψ=S,L,B:S+L+B≤1,S≥0,L≥0,B≥0.The basic regeneration number is the threshold at which a disease tends to die out or persist. We calculated the basic reproductive number is as follows:(5)R0=βξ+γ+ε+ξ+γpξ+εqξξ+γε+ξ.For one thing, from the above description, the virus-free equilibrium is calculated asE0=S0,L0,B0=1,0,0. In addition, data analysis of system (3) was performed to determine the unique endemic equilibrium point E∗=S∗,L∗,B∗. Specific results are as follows:(6)S∗=ξ+γϵ+ξ−ξ+γpξ−ϵqξβξ+γ+ϵ,L∗=ξ+γ2ξ+ηϵ+ξR0−1ξ+γ+ϵξ+γξ+η+ξ+η+γϵ,B∗=ϵξ+γξ+ηϵ+ξR0−1ξ+γ+ϵξ+γξ+η+ξ+η+γϵ.


## 3. Stability of the Virus-Free Equilibrium

Global stability of steady state revealed the basic rule of the spread of the virus. Subsequently, the global stability of the virus-free equilibrium will be derived.Theorem 1.
The virus-free equilibriumE0 is local asymptotically stable if R0≤1, but unstable if R0>1.Proof.
We obtained the Jacobian matrix of system (3) at disease-free equilibrium as follows:(7)JE0=−η−ξ−β−η−pξ−β−η−qξ0β−ε+pξ−ξβ+qξ0ε−γ−ξ,Accordingly, the characteristic equation is easily obtained by the matrix (7) as(8)λ+η+ξλ2+λγ+ξ−β+ε+ξ−pξ+γε+εξ+γξ+ξ2−γβ−ξβ−εβ−γpξ−εqξ−pξ2=0,λ1=−η−ξ,λ2=β−γ−2ξ−ε+pξ2+γ+2ξ−β+ε−pξ22−4∗γε+εξ+γξ+ξ2−γβ−ξβ−εβ−γpξ−εqξ−pξ22,λ3=β−γ−2ξ−ε+pξ2−γ+2ξ−β+ε−pξ22−4∗γε+εξ+γξ+ξ2−γβ−ξβ−εβ−γpξ−εqξ−pξ22.
Clearly,λ1<0, λ3<0, and λ2<0 if R0≤1, which intimates βξ+ε+γ+pξγ+ξ+εqξ−γ+ξε+ξ<0. The correctness of Theorem 1canbe proved according to stability theory in [36].Theorem 2.
IfR0≤1, E0 is globally asymptotically stable relative to Ψ.Proof.
Give a Lyapunov function,(9)Vt=L+ε−pξ+qξ+ξξ+γ+εB,Obviously, the Lyapunov function is positive definite. Through direct computing, the differential equation of the function is obtained:(10)V′t=dLdt+ε−pξ+qξ+ξξ+γ+εdBdt,=βS−ε+ξξ+γ−pξξ+γ−εqξξ+γ+εL+βS−ε+ξξ+γ−pξξ+γ−εqξξ+γ+εB.
WhenR0≤1, it is calculated that βξ+γ+ε+pξξ+γ+εqξ−ξ+γε+ξ<0. It is observed that V′t≤0. Furthermore, V′t=0 when and only when L=0 and B=0 are satisfied. In addition, when L tends to be infinite or B tends to infinity, one can get Vt to tend to be infinite. LaSalle’s invariance principle [37] is an important basis for proving global stability. According to the theorem, we can see that when R0≤1, the global stability of E0 in Ψ has been verified. Theorem 2 has been proved.


## 4. Stability of the Virose Equilibrium

Local stability and global stability analysis of endemic equilibrium is the important approach to analyze the propagation of the virus. At this stage, we put forward two theories to analyze the stability ofE∗ and gave analysis and proof.Theorem 3.
In the case ofR0>1, the virose equilibrium is locally asymptotically stable.Proof.
We obtained the Jacobian matrix of system (3) at endemic equilibrium as follows:(11)JE∗=−βL∗+B∗−η−ξ−βS∗−η−pξ−βS∗−η−qξβL∗+B∗βS∗−ε+pξ−ξβS∗+qξ0ε−γ−ξ,The characteristic equation of JE∗ is λ3+n1λ2+n2λ+n3=0, where(12)n1=η+2ξ+γ+H+A+B,n2=ε+η+2ξ+γH+ηγ+ξγ+ηξ+ξ2+η+ξA+B,n3=ξγ+ξ2+εξ+ηγ+ηξ+εη+γεH,A=qξεγ+ξ,B=βS∗εγ+ξ,H=βL∗+βB∗,βS∗=ε+pξ−A−B,we have(13)n1n2−n3=ε+γ+η+2ξH2+1+ε+γ+η+2ξA+BH+Hγ+ξ2+Hεξ+Hξ+η2γ+ξ+A+B2η+ξ+A+Bξ+ηγ+ξ1+ξ+η+ξ+ηγ+ξγ+η+2ξ.By calculation, n1>0, n2>0, n3>0, and n1n2−n3>0 if R0>1. On the basis of the Routh–Hurwitz criterion [38], Theorem 3 is proved.
Theorem1 proves that the virus-free equilibrium is unstable under the condition of R0>1. Besides that, as a result of E0 belonging to the boundary of the feasible region Ψ, state variables of the system are uniformly persistent [39, 40]. Here, we advance the proposition as follows.Proposition 1.
System (3) is consistent and durable under the condition of R0>1. In other words, there is a positive constant ω independent of the initial state of system (3), which satisfies(14)limt⟶∞infSt>ω,limt⟶∞infLt>ω,limt⟶∞infBt>ω,limt⟶∞inf1−St−Lt−Bt>ω.Theorem 4.
The virus equilibrium is globally asymptotically stable under the condition ofR0>1.Proof.
Now we will use a geometric method [41] to illustrate that E∗ is globally stable. From the above analysis, we know that system (3) has a compact attractive set U∈Ψ∗, and the only viral equilibrium exists in system (3).
Seta=S,L,B, and express the vector field of system (3) by fa. Find the Jacobian determinant, the Jacobian of the general solution at of system (3), that is,(15)Jat=−βL+B−η−ξ−βS−η−pξ−βS−η−qξβL+BβS−ε+pξ−ξβS+qξ0ε−γ−ξ.By calculation, we obtain the second additive compound Jacobian matrix [42, 43] as(16)Jat2=−βL+B−S−η+ξ+ε+pξβS+q′ξβS+η−qξε−βL+B−η+2ξ+γ−βS−η+pξ0βL+BβS−γ+ξ+ε+pξ.
Now, set the functionΓa=ΓS,L,B as(17)ΓS,L,B=q10001−q2LB00q2LBLB,where(18)1<q1<2βω2β+ξ+2η+1,(19)q2=0ε≥β+η+pξ,1−εβ+η+pξε<β+η+pξ.Note that ω is the positive constant given in Proposition 1. Then, there are(20)ΓfΓ−1=0000L˙L−B˙B000L˙L−B˙B.We rewrite the matrix B=ΓfΓ−1+ΓJat2Γ−1 used in the geometric method of global stability [41]:(21)B=B11B12B21B22,where(22)B11=−βL+B−S−η+2ξ+ε−pξ,B12=q1BβS+qξ−q2q1BβS+η+qξ1−q2Lq1βS+η+qξBL,B21=ε1−q2Lq1Bεq2Lq1B,B22=L˙L−B˙B−βL+B−η+2ξ+γ+q2βS+η+pξq2−1βS+η+pξβL+B+q2ε1−q2−q2βS+η+pξL˙L−B˙B−γ+2ξ+ε+η+1−q2βS+η+pξ.
The vectors inR3 are denoted by y=ξ,δ,ψ, where ξ,δ,ψ=maxξ,δ+ψ is selected as the norm of R3. Suppose that the Lozinskii measure about this norm is represented as u, we can obtain the following estimates [44]:(23)uB≤supκ1,κ2=supu1B11+B12,u1B22+B21,where(24)κ1=u1B11+B12,κ2=u1B22+B21.Accordingly, u1 represents the Lozinskii measure of the norm of the L1 vector, and B21, B12 are the matrix norms about L1 vector norm.
Therefore, we obtain(25)κ1=−βL+B−S−η+2ξ+ε−pξ+q1βS+qξBL+max−q1q2ηB1−q2L,q1ηBL,κ2=L˙L−B˙B−η+γ+2ξ+εLq1B+maxq2βS+η+pξ,−ε+21−q2βS+η+pξ,From (18) and (19), we can obtain(26)κ1=−βL+B−S−η+2ξ+ε−pξ+q1βS+qξBL+q1ηBL,(27)κ2≤L˙L−B˙B−η+γ+2ξ+η+pξ+εLq1B.Now rewriting system (3), we have(28)L˙L=βS+βSBL−ε+pξ−ξ+qξBL,(29)B˙B=εLB−γ−ξ.Adjust the consistent continuous constant ω in 4.1 so that there is a constant T greater than zero in K that is independent of the initial value x0 and satisfies(30)St≥ω,Lt≥ω,Bt≥ωfort>T.Substitute (28) into (26), and (29) into (27). From (18) and (19), using formula (27), we have, for t>T,(31)κ1=L˙L+q1−1βS+qξBL−βL+B−ξ−η+q1ηBL≤L˙L+q1−1βS+qξω−2βω−ξ−η+q1ηω≤L˙L−ξ−η≤L˙L−ξ,(32)κ2≤L˙L−εLB+εLq1B≤L˙L−q1−1εLq1B.Consequently, uB≤L˙/L−bfort>T by (23), and (31) and (32), where b=minξ,q1−1εL/q1B>0. For each answer xt,x0 of system (3) satisfying x0∈K and t>T, there is(33)1t∫0TuBds≤1t∫0TuBds+1tlogLtLT−bt−Tt,meaning that q¯≤−b/2<0 from (23), proving Theorem 4. □


## 5. Further Discussion

Combined with the analysis in the previous sections, we first analyze the sensitivity ofR0 to system parameters and give a visualization diagram. Then, we studied the influence of relevant parameters on R0 and drew a series of visualization graphs.As described earlier,β denotes the infection rate of uninfected computers, and p and q are part of infected newborns in the latent internal computers and breaking-out internal computers classes, respectively. In addition, γ represents the healing rate of disconnected nodes. To better analyze the spread of computer viruses, it is necessary to carry out sensitivity analysis of these system parameters for R0. Next, calculate the normalized forward sensitivity index of β, p, q, and γ [45] as follows, respectively:(34)∂R0/R0∂β/β=βR0∂R0∂β=βξ+γ+εβξ+γ+ε+ε+γpξ+εqξ>0,∂R0/R0∂p/p=pξ+γβξ+γ+ε+ε+γpξ+εqξ>0,∂R0/R0∂q/q=qξεβξ+γ+ε+ε+γpξ+εqξ>0,∂R0/R0∂γ/γ=−γβε+εqξξ+γβξ+γ+ε+ε+γpξ+εqξ<0.From these numerical results, we can draw the following conclusions:(1)
If the infection rateβ is reduced, the spread of the virus can be controlled.(2)
If the birth rate of nodes in the vertical direction,p,q, is reduced, it will help control the spread of the virus.(3)
If the cure rateγ is increased, it will help suppress the virus.


## 6. Numerical Experiments

Numerical simulations are an important tool for the quantitative analysis of models. Next, a series of numerical examples were made to visualize all the above theories.


### 6.1. Stability Analysis

ForR0≤1, E0 is globally asymptotically stable according to Theorem 2. Figure 2 shows how the various types of states in the network evolve in time for R0=0.3119,0.5242,0.6783 and 0.8333 where the corresponding parameter values are shown in Table 3. From the graph, we can see that if R0≤1, the virus in the network eventually tends to die out.Figure 2
Evolution trend chart of system (3) at (a) R0=0.3119, (b) R0=0.5242, (c) R0=0.6783, and (d) at R0=0.8333.
(a)(b)(c)(d)Table 3
The typicalR0 and associated parameter values of system (3).
R0ξβηpqγϵS0L0B00.311 90.660.010.20.10.10.30.60.20.30.40.524 20.260.010.20.10.10.30.50.20.30.40.678 30.160.010.20.10.10.30.50.20.30.40.833 30.10.010.20.10.10.30.60.20.30.4ForR0>1, after the previous analysis and elaboration, Theorem 4 has been confirmed. Figure 3 shows how the various types of states in the network evolve in time for R0=3.9880,6.1771,11.2353, and 121.720 6, where the corresponding parameter values are shown in Table 4. From the graph, we can see that if R0>1, the virus in the network will always be there and tend to stabilizes.Figure 3
Evolution trend chart of system (3) at (a) R0=3.9880, (b) R0=6.1771, (c) R0=11.2353, and (d) R0=121.7206.
(a)(b)(c)(d)Table 4
The typicalR0 and associated parameter values of system (3).
11F0E0R0ξβηpϵγqS0L0B03.988 00.630.50.20.70.60.30.60.50.10.36.177 10.390.560.20.50.60.30.60.50.10.311.235 30.230.60.20.50.60.30.60.50.10.3121.720 60.0230.60.20.50.60.30.680.50.10.3


### 6.2. Influence of Parameters on the Value ofR0

From the previous subsection, we can see that the basic regeneration number determines whether the virus tends to die out in the network or not. So analyzing the effect of parameters in the system on the basic regeneration number can help to contain the spread of the virus. It can be seen that Figures4–6 show that increases in β, p, and q produce an equal proportional increase in R0, respectively. Figure 7 shows a decline in the size of R0 as the parameter γ increases. The parameters are shown in Table 5. These numerical simulations validate the analysis in Section 5.Figure 4
Influence of parameterβ on R0.Figure 5
Influence of parameterp on R0.Figure 6
Influence of parameterq on R0.Figure 7
Influence of parameterγ on R0.Table 5
Simulation parameters.
FigureξβηpqγϵFigure40.8∗0.20.10.10.30.6Figure50.80.20.2∗0.10.30.6Figure60.80.20.20.1∗0.30.6Figure70.80.20.10.10.3∗0.6


### 6.3. Influence of Parameters on the Number of Infected Nodes

When the basic regeneration number is inevitably greater than one, we take steps to reduce the number of infected nodes. Analyzing the impact of parameters on infected nodes helps us to select measures to contain the virus. In this subsection, we analyze the parameters that affect the level of stability of the infected computer, and Figures8–13 show the results of the analysis.Figure 8
Influence of parameterξ on L+B.Figure 9
Influence of parameterβ on L+B.Figure 10
Influence of parameterp on L+B.Figure 11
Influence of parameterq on L+B.Figure 12
Influence of parameterγ on L+B.Figure 13
Influence of parameterη on L+B.


## 6.1. Stability Analysis

ForR0≤1, E0 is globally asymptotically stable according to Theorem 2. Figure 2 shows how the various types of states in the network evolve in time for R0=0.3119,0.5242,0.6783 and 0.8333 where the corresponding parameter values are shown in Table 3. From the graph, we can see that if R0≤1, the virus in the network eventually tends to die out.Figure 2
Evolution trend chart of system (3) at (a) R0=0.3119, (b) R0=0.5242, (c) R0=0.6783, and (d) at R0=0.8333.
(a)(b)(c)(d)Table 3
The typicalR0 and associated parameter values of system (3).
R0ξβηpqγϵS0L0B00.311 90.660.010.20.10.10.30.60.20.30.40.524 20.260.010.20.10.10.30.50.20.30.40.678 30.160.010.20.10.10.30.50.20.30.40.833 30.10.010.20.10.10.30.60.20.30.4ForR0>1, after the previous analysis and elaboration, Theorem 4 has been confirmed. Figure 3 shows how the various types of states in the network evolve in time for R0=3.9880,6.1771,11.2353, and 121.720 6, where the corresponding parameter values are shown in Table 4. From the graph, we can see that if R0>1, the virus in the network will always be there and tend to stabilizes.Figure 3
Evolution trend chart of system (3) at (a) R0=3.9880, (b) R0=6.1771, (c) R0=11.2353, and (d) R0=121.7206.
(a)(b)(c)(d)Table 4
The typicalR0 and associated parameter values of system (3).
11F0E0R0ξβηpϵγqS0L0B03.988 00.630.50.20.70.60.30.60.50.10.36.177 10.390.560.20.50.60.30.60.50.10.311.235 30.230.60.20.50.60.30.60.50.10.3121.720 60.0230.60.20.50.60.30.680.50.10.3


## 6.2. Influence of Parameters on the Value ofR0

From the previous subsection, we can see that the basic regeneration number determines whether the virus tends to die out in the network or not. So analyzing the effect of parameters in the system on the basic regeneration number can help to contain the spread of the virus. It can be seen that Figures4–6 show that increases in β, p, and q produce an equal proportional increase in R0, respectively. Figure 7 shows a decline in the size of R0 as the parameter γ increases. The parameters are shown in Table 5. These numerical simulations validate the analysis in Section 5.Figure 4
Influence of parameterβ on R0.Figure 5
Influence of parameterp on R0.Figure 6
Influence of parameterq on R0.Figure 7
Influence of parameterγ on R0.Table 5
Simulation parameters.
FigureξβηpqγϵFigure40.8∗0.20.10.10.30.6Figure50.80.20.2∗0.10.30.6Figure60.80.20.20.1∗0.30.6Figure70.80.20.10.10.3∗0.6


## 6.3. Influence of Parameters on the Number of Infected Nodes

When the basic regeneration number is inevitably greater than one, we take steps to reduce the number of infected nodes. Analyzing the impact of parameters on infected nodes helps us to select measures to contain the virus. In this subsection, we analyze the parameters that affect the level of stability of the infected computer, and Figures8–13 show the results of the analysis.Figure 8
Influence of parameterξ on L+B.Figure 9
Influence of parameterβ on L+B.Figure 10
Influence of parameterp on L+B.Figure 11
Influence of parameterq on L+B.Figure 12
Influence of parameterγ on L+B.Figure 13
Influence of parameterη on L+B.


## 7. Conclusion

Considering the impact of the vertical transmission of computer viruses on new uninfected nodes, a novel dynamic model of network virus transmission is put forward, and this model is discussed and analyzed. The conclusions drawn from the analysis of the model are as follows:(1)
This article studies an SLBRS model in which the virus infection can spread vertically and horizontally(2)
It is confirmed that this model is asymptotically stable globally(3)
Numerical simulation verifies the theoretical analysis of model balance(4)
Perform system parameter sensitivity analysis onR0(5)
The parameters affecting the stability level of the infected computer are analyzedTo our minds, the model proposed in this article more truly reflects some of the characteristics of network viruses than previous models. We believe that in the later work, the model can be extended on the complex network, and consider some of the most optimal control strategies.



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*Source: 2576685-2022-02-09.xml*