Dynamical Behavior of Hybrid Propagation of Computer Viruses

Zhu, Qingyi; Xiang, Pingfan; Luo, Xuhang; Gan, Chenquan

doi:https://doi.org/10.1155/2022/2576685

Security and Communication Networks

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Volume 2022 | Article ID 2576685 | https://doi.org/10.1155/2022/2576685

Dynamical Behavior of Hybrid Propagation of Computer Viruses

Qingyi Zhu,^1,2Pingfan Xiang,²Xuhang Luo,¹and Chenquan Gan³

Academic Editor: Mamoun Alazab

Received03 Nov 2021

Revised01 Jan 2022

Accepted12 Jan 2022

Published09 Feb 2022

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 number , 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 on system parameters and the parameters affecting the stability level of infected computers are both analyzed.

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.

Based 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 Section 2, 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 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 nodes , latent nodes , breaking-out nodes , and recovering nodes .

Let , , , and 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

Figure 1 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.

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 of nodes becoming nodes is (4)Since the probabilities that the descendants of and nodes are nodes are and , respectively, satisfying , , the probability that the newly accessed computers are nodes is

2.2. Model Expression

Based on the previous description and assumptions, the state transition differential system of the model is

Based on the above theoretical analysis, we know that , and differential equations of system (2) state can be converted intoThe initial states of all kinds of computers in system (3), respectively, satisfy: , , and . The initial value belongs to the positive invariant set:

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:

For one thing, from the above description, the virus-free equilibrium is calculated as . In addition, data analysis of system (3) was performed to determine the unique endemic equilibrium point . Specific results are as follows:

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 equilibrium is local asymptotically stable if , but unstable if .

Proof. We obtained the Jacobian matrix of system (3) at disease-free equilibrium as follows:Accordingly, the characteristic equation is easily obtained by the matrix (7) asClearly, , , and if , which intimates . The correctness of Theorem 1canbe proved according to stability theory in [36].

Theorem 2. If , is globally asymptotically stable relative to .

Proof. Give a Lyapunov function,Obviously, the Lyapunov function is positive definite. Through direct computing, the differential equation of the function is obtained:When , it is calculated that . It is observed that . Furthermore, when and only when and are satisfied. In addition, when tends to be infinite or tends to infinity, one can get 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 , the global stability of in 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 of and gave analysis and proof.

Theorem 3. In the case of , the virose equilibrium is locally asymptotically stable.

Proof. We obtained the Jacobian matrix of system (3) at endemic equilibrium as follows:The characteristic equation of is , wherewe haveBy calculation, , , , and if . On the basis of the Routh–Hurwitz criterion [38], Theorem 3 is proved.
Theorem 1 proves that the virus-free equilibrium is unstable under the condition of . Besides that, as a result of belonging to the boundary of the feasible region , 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 . In other words, there is a positive constant independent of the initial state of system (3), which satisfies

Theorem 4. The virus equilibrium is globally asymptotically stable under the condition of .

Proof. Now we will use a geometric method [41] to illustrate that is globally stable. From the above analysis, we know that system (3) has a compact attractive set , and the only viral equilibrium exists in system (3).
Set , and express the vector field of system (3) by . Find the Jacobian determinant, the Jacobian of the general solution of system (3), that is,By calculation, we obtain the second additive compound Jacobian matrix [42, 43] asNow, set the function aswhereNote that is the positive constant given in Proposition 1. Then, there areWe rewrite the matrix used in the geometric method of global stability [41]:whereThe vectors in are denoted by , where is selected as the norm of . Suppose that the Lozinskii measure about this norm is represented as , we can obtain the following estimates [44]:whereAccordingly, represents the Lozinskii measure of the norm of the vector, and , are the matrix norms about vector norm.
Therefore, we obtainFrom (18) and (19), we can obtainNow rewriting system (3), we haveAdjust the consistent continuous constant in 4.1 so that there is a constant greater than zero in that is independent of the initial value and satisfiesSubstitute (28) into (26), and (29) into (27). From (18) and (19), using formula (27), we have, for ,Consequently, by (23), and (31) and (32), where . For each answer of system (3) satisfying and , there ismeaning that from (23), proving Theorem 4. □

5. Further Discussion

Combined with the analysis in the previous sections, we first analyze the sensitivity of to system parameters and give a visualization diagram. Then, we studied the influence of relevant parameters on and drew a series of visualization graphs.

As described earlier, denotes the infection rate of uninfected computers, and and are part of infected newborns in the latent internal computers and breaking-out internal computers classes, respectively. In addition, 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 , , , and [45] as follows, respectively:From these numerical results, we can draw the following conclusions:(1)If the infection rate is reduced, the spread of the virus can be controlled.(2)If the birth rate of nodes in the vertical direction, , is reduced, it will help control the spread of the virus.(3)If the cure rate 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

For , is globally asymptotically stable according to Theorem 2. Figure 2 shows how the various types of states in the network evolve in time for and where the corresponding parameter values are shown in Table 3. From the graph, we can see that if , the virus in the network eventually tends to die out.

(a)

(b)

(c)

(d)

For , 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 , and 121.720 6, where the corresponding parameter values are shown in Table 4. From the graph, we can see that if , the virus in the network will always be there and tend to stabilizes.

(a)

(b)

(c)

(d)

6.2. Influence of Parameters on the Value of

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 Figures 4–6 show that increases in , , and produce an equal proportional increase in , respectively. Figure 7 shows a decline in the size of as the parameter increases. The parameters are shown in Table 5. These numerical simulations validate the analysis in Section 5.

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 Figures 8–13 show the results of the analysis.

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 on (5)The parameters affecting the stability level of the infected computer are analyzed

To 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.

Abbreviations

SEI:	Susceptible-exposed-infected
SEIQRS:	Susceptible-exposed-infected-quarantined-recovered-susceptible
SEIQV:	Susceptible-exposed-infected-quarantined-vaccinated
SEIR:	Susceptible-exposed-infected-recovered
SEIRS:	Susceptible-exposed-infected-recovered-susceptible
SIR:	Susceptible-infected-recovered
SIRS:	Susceptible-infected-recovered-susceptible
SIS:	Susceptibility-infection-susceptibility
SLBRS:	Susceptible-latent-breaking-recovered-susceptible
SLBS:	Susceptible-latent-breaking-susceptible.

Data Availability

Data sharing is not applicable to this article as no datasets were generated.

Conflicts of Interest

All authors declare that there are no conflicts of interest.

Authors’ Contributions

The authors claim that the research was realized in collaboration with the same responsibility. All authors read and approved the last version of the manuscript.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant no. 61903056), Major Project of Science and Technology Research Program of Chongqing Education Commission of China (Grant no. KJZDM201900601), Chongqing Research Program of Basic Research and Frontier Technology (Grant nos. cstc2019jcyj-msxmX0681, cstc2021jcyj-msxmX0530, and cstc2021jcyj-msxmX0761), Chongqing Municipal Key Laboratory of Institutions of Higher Education (Grant no. cqupt-mct-201901), Chongqing Key Laboratory of Mobile Communications Technology (Grant no. cqupt-mct-202002), and Engineering Research Center of Mobile Communications, Ministry of Education (Grant no. cqupt-mct202006).

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