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Open Access

Peer-reviewed

Research Article

Propagation Effect of a Virus Outbreak on a Network with Limited Anti-Virus Ability

  • Yonghong Xu ,

    xyh8810@63.com

    Affiliation The Key Laboratory of Biotechnology for Medicinal Plants of Jiangsu Province, Jiangsu Normal University, Xuzhou, 221116, China

    ⨯
  • Jianguo Ren

    Affiliation College of Computer, Jiangsu Normal University, Xuzhou, 221116, China

    ⨯

Propagation Effect of a Virus Outbreak on a Network with Limited Anti-Virus Ability

  • Yonghong Xu, 
  • Jianguo Ren
PLOS
x

Abstract

This paper describes a new computer virus spreading model which takes into account the possibility of a virus outbreak on a network with limited anti-virus ability. Then, the model is investigated for the existence of equilibria and their stabilities are proved and illustrated. Moreover, it is found that these two factors are not only relative to the threshold value determining whether the virus becomes extinct or not, but that they are also relative to the virus epidemic levels. Theoretical and experimental results indicate that, in some ways, it would be practically possible to eradicate the virus or suppress its prevalence below a suitable level. Consequently, some suggestions are proposed that may help eradicate or suppress virus propagation over a real computer network.

Citation: Xu Y, Ren J (2016) Propagation Effect of a Virus Outbreak on a Network with Limited Anti-Virus Ability. PLoS ONE 11(10): e0164415. https://doi.org/10.1371/journal.pone.0164415

Editor: Gui-Quan Sun, Shanxi University, CHINA

Received: June 21, 2016; Accepted: September 23, 2016; Published: October 27, 2016

Copyright: © 2016 Xu, Ren. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Data Availability: All relevant data are within the paper.

Funding: This work was supported by the National Natural Science Foundation of China under grant no. 61304117, and the Doctorate Teacher Support Project of Jiangsu Normal University under grant no. 15XLR027.

Competing interests: The authors have declared that no competing interests exist.

1. Introduction

Computer viruses are programs that attempt to parasitize themselves on a host and spread to other computers, mainly through the Internet. They have become a great threat to both computer users and network resources. Consequently, the development of appropriate models to understand the mechanisms governing the spread of computer viruses is becoming of importance. Currently, a lot of computer virus propagation models such as SIR model [1–3], SLB model [4–5], SEIQV model [6] as well as their extensions [7–17], have been proposed. These models have mainly been built upon classical epidemic models [18–28] and investigated the behavior of computer virus spread over computer networks by the analogy between computer and biological viruses [29].

To the best of our knowledge, all of the known computer virus models [1–17] that describe virus propagation neglect the fact that viruses possess a paroxysmal nature in common, as demonstrated when a large number of computer viruses have the possibility of an outbreak absence of aura. Furthermore, antivirus techniques always lag behind virus techniques, and thus during that investable lag from the outbreak of a new virus to the widespread application of the anti-virus software aiming to conquer this virus the computer is susceptible to the attack. As a result, the virus would not be entirely eradicated, but instead be temporarily suppressed and therefore still capable of infecting other computers, which may lead to huge losses. Almost all of the existing models assume, first, that during the time [dt, t+dt], once infected, any susceptible computer is in its latency (i.e., all of the S computers become the E ones), and secondly, that by using the anti-virus software on the affected network, the virus can be completely and immediately eliminated. However, these assumptions are inconsistent to the facts that, in general, most viruses break out suddenly. Zuo et al [30] have shown that there is no any perfect anti-virus software that can detect and clear all kinds of viruses.

This paper pays attention to the spreading behaviors of a computer virus during a virus outbreak on a network with limited anti-virus ability. Accordingly, a new computer virus propagation model is proposed, which incorporates those two kinds of new state transitions. First, we give the threshold value R0, which determines whether the virus disappears completely or not, and find that the dynamic behavior of the proposed model is determined by it. Next, the proposed model admits a virus-free equilibrium which is globally asymptotically stable that results in virus eradication if R0≤1 and admits a virus equilibrium which is globally asymptotically stable whereas R0>1. System parameters are considered to analyze the dynamical behaviors of the model. It is found that (1) the outbreak probability and transmission rate due to the limited anti-virus ability are strongly relative to the threshold value R0, and (2) the outbreak probability and transmission rate are heavily connected with the virus epidemic levels. These results imply that, to some degree, it would be practically possible to eradicate the virus or inhibit its prevalence below a suitable level. Finally, theoretical and experimental studies reveal the effect of system parameters on virus propagation. Consequently, some suggestions are made that may help suppress virus propagation over a computer network.

In the next section, we establish a mathematical model to be discussed. In Section 3, we study the stability of the virus-free and virus equilibrium of the model, respectively. The given parameters are considered to analyze the dynamic behaviors of the model. In Section 4, we use theoretical and experimental research to reveal the effect of parameters on virus propagation to which corresponding suggestions are made. Finally, some conclusions are given in Section 5.

2. Mathematical Model

Based on the computer virus propagation models involving the S, E, I and other compartments in Ref. [31–33] and ignoring infection details, the total number of computers connected the Internet are divided into four compartments:

  1. Susceptible compartments (S):The set of all external uninfected computers that are connected to the network, i.e. susceptible computers.
  2. Exposed compartments (E): The set of all latent computers, i.e. infected computers where all viruses are latent.
  3. Infected compartments (I): The set of all infected computers where all of the viruses are currently breaking out.
  4. Recovered compartments (R): The set of all recovered computers that have run the anti-virus software.

Let S(t), E(t), I(t) and R(t) denote their numbers at time t, respectively. The involved parameters b denotes the rate at which external computers are connected to the network, β denotes the rate at which, when having a connection to one latent computer, one susceptible computer can become latent one, γ denotes the rate at which one latent computer breaks out, α denotes the recovery rate of infected computers and depends on the ability of the anti-virus software, μ denotes the rate at which one computer is removed from the network. Furthermore, since most viruses break out suddenly, with the time period [dt, t+dt], due to a possible connection with infected computers, every susceptible computer is either latent with probability (1-p) βI(t)dt, or breaks out with probability pβI(t)dt, where p>0 is a constant. Since the existing anti-virus ability of any given network is limited, the virus is temporarily suppressed with probability ε, where ε > 0 is a constant. At time t, every infected computer is either latent sequentially with probability ε or recovered with probability α.

From the work above, the transmission between the model can be expressed by the following system of differential equations: (1)

Because the first three equations in (1) are independent of R, we can consider the following reduced model: (2)

With initial conditions S(0) ≥ 0, E(0) ≥ 0, I(0) ≥ 0 and all the parameters are positive constants. It is easily verified that the set (3) is positively invariant for this system.

3. Model Analysis

This section is devoted to understanding the dynamic behaviors of the model (2). First, we will obtain a threshold value R0 for the model and it is defined as the number of virus-free computers that are infected by a single infected computer. It is easy to see that the model (2) always admits a virus-free equilibrium E0 = (S0, 0, 0). Then let (4)

Then,

Hence, next generation matrix for the model is (5)

Again, the spectral radius R0 of the matrix K is the basic reproduction number of the model, i.e.

(6)

3.1 The virus-free equilibrium and its stability

Model (2) always has a unique equilibrium . Then the characteristic equation of the linearization of model (2) near E0 is (7)

Clearly, the eigenvalues are −μ and other roots are decided by the following equations: (8) where (9)

If R0 < 1, then a = μ + γ − pβS0 + ε + α + μ > 0, and (10)

The two roots of Eq (9) both have negative real parts, by the Hurwitz criterion we can obtain the following result:

Theorem 3.1.

When R0 < 1, the virus-free equilibrium E0 is local stability.

Now, it is turn to examine the global stability of virus-free equilibrium by use of the Lyapunov direct method. The following Theorem is obtained:

Theorem 3.2.

When R0 ≤ 1, the virus-free equilibrium E0 is global stability.

Proof. Define (11) then (12)

As all the model parameters are positive, it follows that for R0 < 1 with if and only if I = 0 or R0 = 1. Hence, V is a lyapunov function on Ω. Thus, I → 0 as t → ∞. I = 0 in the first equation of model (2) shows that S → b / μ as t → ∞. Therefore, it follows from the Lasalle’s invariance principle that every solution of the model, under the initial conditions in Ω, approaches E0 as t → ∞.

Example 1. Consider model (2) with parameters b = 5, β = 0.07, μ = 0.6, γ = 0.25, ε = 0.3, p = 0.7 and R0≈0.57<1. The condition of virus-free equilibrium stability is satisfied and hence the virus-free equilibrium E0(8.33,0,0) is asymptotically stable (see Fig 1), i.e. the virus infection can become extinct after some period of time.

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Fig 1. A plot indicates that the viruses can become extinct.

https://doi.org/10.1371/journal.pone.0164415.g001

3.2 The virus equilibrium and its stability

Now, let us investigate the existence and stability of virus equilibrium of model (2). By calculations, the unique equilibrium E* is given by (13)

And the corresponding characteristic equation of the linearization of model (2) is (14) which equals the following equation: (15) where (16) (17) (18)

It is easy to verify that a1a2–a3 > 0 holds, then according to the Routh-Hurwitz criterion, the following theorem is given:

Theorem 3.3.

Virus equilibrium E* is locally asymptotically stable when R0 > 1.

Remark 1. This Theorem expresses that, under the assumption of virus outbreak on a network with limited anti-virus ability, virus epidemic would attain a certain level, and the model admits no the periodic and chaotic dynamic behaviors.

Theorem 3.4.

Virus equilibrium E* is globally asymptotically stable.

Proof. We consider the following Lyapunov function constructed as (19)

Meanwhile, for simplicity, denote

then time derivative of V(t) along the solution of Eq (2) is (20)

By using the following equations (21)

Eq (20) can be written as (22)

Since the arithmetical mean is great than or equal to the geometrical mean, thus, for x > 0 and the equality holds if and only if x = 1; for y,z > 0 and the equality holds if and only if y = z; for x,y,z > 0 and the equality holds if and only if x = y = z. Therefore, V(t)' ≤ 0 for x,y,z > 0 and V(t)' = 0 holds if and only if x = 1 and y = z. Thus, it follows from LaSalle Invariance Principle that the virus equilibrium E* is globally asymptotically stable.

Example 2. Consider model (2) with parameters b = 5, β = 0.15, μ = 0.6, ε = 0.03, p = 0.9, γ = 0.4 where R0 = 1.90>1 satisfies the stable conditions for the virus equilibrium (4.38, 0.34, 3.60), and hence the virus equilibrium is asymptotically stable (see Fig 2), i.e., the virus can maintain its infection to some extent.

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Fig 2. A plot indicates that the viruses become steady.

https://doi.org/10.1371/journal.pone.0164415.g002

Example 3. Consider model (2) with parameters b = 5, β = 0.15, μ = 0.6, p = 0.5, γ = 0.5, the global dynamical behavior of every variables of the model (2) with varying parameterεis shown in Fig 3ε-S(t), Fig 4ε-E(t) and Fig 5ε-I(t).

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Fig 3. Dynamical behaviors of every system variable with parametersε-S(t.).

https://doi.org/10.1371/journal.pone.0164415.g003

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Fig 4. Dynamical behaviors of every system variable with parametersε-E(t).

https://doi.org/10.1371/journal.pone.0164415.g004

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Fig 5. Dynamical behaviors of every system variable with parametersε-I(t).

https://doi.org/10.1371/journal.pone.0164415.g005

Example 4. Consider model (2) with parameters b = 5, β = 0.15, μ = 0.6, ε = 0.2, γ = 0.5, the global dynamical behavior of every variables of the model (2) with varying parameter p is shown in Fig 6 p-S(t), Fig 7 p-E(t) and Fig 8 p-I(t).

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Fig 6. Dynamical behaviors of every system variable with parameters p-S(t).

https://doi.org/10.1371/journal.pone.0164415.g006

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Fig 7. Dynamical behaviors of every system variable with parameters p-E(t).

https://doi.org/10.1371/journal.pone.0164415.g007

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Fig 8. Dynamical behaviors of every system variable with parameters p-I(t).

https://doi.org/10.1371/journal.pone.0164415.g008

4. Further Discussions and Suggestions

Theorem 3.2 and 3.3 tell us, to some degree, it would be possible to eradicate the virus or inhibit its epidemics below a proper level in practice. In order to do this, it is critical to have an overall knowledge of the effects of parameters on the virus propagation. By the proof of Theorem 3.1, we have (23)

Since breakout probability p and transmission rateε play a key role, then differentiating R0 with respect to them, we can obtain: (24)

Hence, we obtain the following Theorem:

Theorem 4.1 Consider model (2), the following claims hold true

  1. R0 is increasing with p, and is decreasing with ε.

Example 5. Consider the effect ofε, p on R0 in model (2) with parameters b = 5, β = 0.075, μ = 0.6, γ = 0.5, Fig 9 shows that R0 is increasing with p and decreasing withε.

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Fig 9. An illustration of effect of p andεon R0.

https://doi.org/10.1371/journal.pone.0164415.g009

Next, we are ready to study the effect of parameters on virus epidemic level I*. A direct calculation yields: (25)

Theorem 4.2 Consider model (2), the following claims hold true

  1. I* is increasing with p and decreasing with ε.

Example 6. Consider the effect ofε, p on I* in model (2) with parameters b = 5, β = 0.15, μ = 0.6, γ = 0.5, Fig 10 shows that I* is increasing with p and decreasing with ε, respectively.

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Fig 10. An illustration of effect of p andεon I*.

https://doi.org/10.1371/journal.pone.0164415.g010

Based on analyses above, the following suggestions may help eliminate or suppress the spread of computer viruses in a real network:

  1. When R0 is well below one, Theorem 4.2 demonstrates that R0 is more sensitive to change in p than inε. This sensitivity informs us that outbreak is more important than passive detection for computer viruses, which leads us to eliminate the virus by dissecting its logical structure [34], analysis of function of every module and their interrelation so as to uncover the source of the virus burst.
  2. When R0 is significantly greater than one, it leads us to suppress virus propagation to an acceptable level (i.e. decreasing the number of infected computers I* by either updating the anti-virus timely or reinstalling the system to dispel the bias of anti-virus software).

5. Conclusions

By taking into consideration the possibility of a virus outbreak on a network with limited anti-virus ability, a new computer virus spreading model is proposed and analyzed. The results are as follows:

  1. The outbreak probability and transmission rate are strongly relative to the threshold value R0, which determines whether the virus may become extinct or not. More specifically, R0 is proportional to the former and inversely proportional to the latter.
  2. The outbreak probability and transmission rate are heavily connected with the virus epidemic level I*. More specifically, I* increases with the former and decreases with the latter.epidemic levels
  3. The virus-free equilibrium is globally stable if R0 ≤ 1 whereas the virus equilibrium is globally stable if R0 > 1.

These results indicate that, in some ways, it would be practically possible to eradicate the virus or suppress its prevalence below an acceptable level. Theoretical and experimental results reveal the effects of the parameters on virus propagation. Consequently, the proposed suggestions may help eradicate or suppress virus propagation over a real computer network.

The main goal of this paper is to better understand the propagation of computer virus qualitatively rather qualitatively. In our view, our work give an insight into the investigation of computer virus spreading by incorporating the objective realistic factors into compartment modeling, which could attract growing attention and interest. Meanwhile, the proposed model is based on the homogeneous network, which can be expanded into heterogeneously node-based model in complex networks, and it is one of our next objectives.

Acknowledgments

The work is supported by the National Natural Science Foundation of China under Grant No. 61304117, and the Doctorate Teacher Support Project of Jiangsu Normal University under Grant no 15XLR027.

Author Contributions

  1. Conceptualization: JGR YHX.
  2. Formal analysis: JGR.
  3. Funding acquisition: YHX JGR.
  4. Investigation: YHX.
  5. Methodology: JGR YHX.
  6. Writing – original draft: YHX JGR.
  7. Writing – review & editing: YHX.

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