Propagation of Computer Virus under Human Intervention: A Dynamical Model

Abstract

This paper examines the propagation behavior of computer virus under human intervention. A dynamical model describing the spread of computer virus, under which a susceptible computer can become recovered directly and an infected computer can become susceptible directly, is proposed. Through a qualitative analysis of this model, it is found that the virus-free equilibrium is globally asymptotically stable when the basic reproduction number 𝑅₀≤1, whereas the viral equilibrium is globally asymptotically stable if 𝑅₀>1. Based on these results and a parameter analysis, some appropriate measures for eradicating the spread of computer virus across the Internet are recommended.

1. Introduction

Due to their striking features such as destruction, polymorphism, and unpredictability [1, 2], computer viruses have come to be one major threat to our work and daily life [3, 4]. With the rapid advance of computer and communication technologies, computer virus programs are becoming increasingly sophisticated so that developing antivirus software is becoming increasingly expensive and time-consuming [5]. Dynamical modeling of the spread process of computer virus is an effective approach to the understanding of behavior of computer viruses because, on this basis, some effective measures can be posed to prevent infection. In the past decade or so, a number of epidemic models (SEIR model [6, 7], SEIRS model [8], SIRS model [9–15], SEIQV model [16], SEIQRS model [17] and SAIR model [4], were simply borrowed to depict the spread of computer virus.

In reality, human intervention plays an important role in slowing down the propagation of computer viruses or preventing the breakout of computer viruses, under which a susceptible computer can become recovered directly, and an infected computer can become susceptible directly. To our knowledge, however, all the previous models did not consider the effect of human intervention.

In this paper, a new computer virus propagation model, which incorporates the above mentioned effects of human intervention, is proposed. The dynamics of this model are investigated. Specifically, the virus-free equilibrium is globally asymptotically stable when the basic reproduction number 𝑅₀≤1, whereas the viral equilibrium is globally asymptotically stable if 𝑅₀>1. Based on these results and a parameter analysis, some effective strategies for eradicating computer viruses are advised.

The subsequent materials of this paper are organized as follows: Section 2 formulates the model; Section 3 shows the global stability of the virus-free equilibrium; Section 4 proves the global stability of the viral equilibrium; Some policies are posed in Section 5 for controlling virus spread; finally, this work is summarized in Section 6.

2. Assumptions and Model Formulation

At any given time, computers all over the world are classified as internal or external depending on whether it is currently accessing to the Internet or not, and all internal computers are further categorized into three classes: (1) susceptible computers, that is, virus-free computers having no immunity; (2) infected computers; (3) recovered computers, that is, virus-free computers having immunity. At time 𝑡, let 𝑆(𝑡), 𝐼(𝑡), and 𝑅(𝑡) denote the concentrations (i.e., percentages) of susceptible, infected, and recovered computers in all internal computers, respectively. Then 𝑆(𝑡)+𝐼(𝑡)+𝑅(𝑡)≡1. Without ambiguity, 𝑆(𝑡), 𝐼(𝑡), and 𝑅(𝑡) will be abbreviated as S, 𝐼, and 𝑅, respectively.

Our model is based on the following assumptions. (A1) All newly accessed computers are virus-free. Furthermore, due to the effect of newly accessed computers, at any time the percentage of susceptible computers increases by 𝛿. (A2)At any time an internal computer is disconnected from the Internet with probability 𝛿. (A3)Due to the effect of previously infected computers, at any time the percentage of infected computers increases by 𝛽𝑆𝐼, where 𝛽 is a positive constant. (A4)Due to the effect of cure, at any time an infected computer becomes recovered with probability 𝛾₁, or becomes susceptible with probability 𝛾₂. (A5)Due to the loss of immunity, at any time a recovered computer becomes susceptible with probability 𝛼₂. (A6)Due to the availability of new vaccine, at any time a susceptible computer becomes recovered with probability 𝛼₁𝐼. This collection of assumptions can be schematically shown in Figure 1, from which one can derive the following model describing the propagation of computer virus: 𝑆̇=𝛿−𝛼₁𝑆𝐼−𝛿𝑆+𝛾₂𝐼−𝛽𝑆𝐼+𝛼₂̇𝑅,𝐼=𝛽𝑆𝐼−𝛾₂𝐼−𝛿𝐼−𝛾₁̇𝐼,𝑅=𝛾₁𝐼+𝛼₁𝑆𝐼−𝛿𝑅−𝛼₂𝑅, (2.1) with initial conditions 𝑆(0)≥0, 𝐼(0)≥0 and 𝑅(0)≥0.

Figure 1 

The transfer diagram of the SIRS model.

The basic  reproduction  number, 𝑅₀, is defined as the average number of susceptible computers that are infected by a single infected computer during its life span. From the above model, one can derive the basic reproduction number 𝑅₀ as 𝑅₀=𝛽𝛾₁+𝛾₂.+𝛿 (2.2)

Because 𝑆+𝐼+𝑅≡1, system (2.1) simplifies to the following planar system: 𝑆̇=𝛿−𝛼₁𝑆𝐼−𝛿𝑆+𝛾₂𝐼−𝛽𝑆𝐼+𝛼₂(),̇1−𝑆−𝐼𝐼=𝛽𝑆𝐼−𝛾₂𝐼−𝛿𝐼−𝛾₁𝐼, (2.3) with initial conditions 𝑆(0)≥0 and 𝐼(0)≥0. Clearly, the feasible region for this system is Ω={(𝑆,𝐼)∶𝑆≥0,𝐼≥0,𝑆+𝐼≤1}, which is positively invariant.

3. The Virus-Free Equilibrium and Its Stability

System (2.3) always has a virus-free equilibrium 𝐸⁰(1,0). Next, let us consider its global stability by means of the Direct Lyapunov Method.

Theorem 3.1. 𝐸⁰ is globally asymptotically stable with respect to Ω if 𝑅₀≤1.

Proof. Let 𝑉(𝑡)=𝐼(𝑡), then 𝑉^′(𝑡)_(2.3)=𝐼̇=𝛽𝑆𝐼−𝛾₂𝐼−𝛿𝐼−𝛾₁𝐼𝛾=𝛽𝐼𝑆−¹+𝛾₂+𝛿𝛽1=𝛽𝐼𝑆−𝑅₀. (3.1) Because 𝑅₀≤1 and 𝑆+𝐼≤1, we have 𝑉′(𝑡)|₍₃₎≤0. Moreover, 𝑉′(𝑡)|₍₃₎=0 if and only if (𝑆,𝐼)=(1,0). Thus, the claimed result follows from the LaSalle Invariance Principle.

Example 3.2. Consider a system of the form (2.3) and with 𝛽=0.3, 𝛿=0.1, 𝛼₁=0.2, 𝛼₂=0.4, 𝛾₁=0.1, and 𝛾₂=0.2. Then 𝑅₀=0.75<1. By Theorem 3.1, the virus-free equilibrium is globally asymptotically stable. Figure 2 demonstrates how 𝑆(𝑡) and 𝐼(𝑡) evolve with time 𝑡 if 𝑆(0)=0.5 and 𝐼(0)=0.4.

Figure 2 

Evolutions of 𝑆(𝑡) and 𝐼(𝑡) for the system with 𝛽=0.3, 𝛿=0.1, 𝛼1=0.2, 𝛼2=0.4, 𝛾1=0.1 and 𝛾2=0.2, provided 𝑆(0)=0.5 and 𝐼(0)=0.4.

4. The Viral Equilibrium and Its Stability

When 𝑅₀>1, it is easy to verify that system (2.3) has a unique viral equilibrium 𝐸^∗(𝑆^∗,𝐼^∗), where 𝑆^∗=𝛿+𝛾₁+𝛾₂𝛽=1𝑅₀,𝐼^∗=𝛿+𝛼₂𝑅₀−1𝛼₁+𝛿+𝛾₁+𝛼₂𝑅₀>0. (4.1)

First, consider the local stability of 𝐸^∗.

Theorem 4.1. 𝐸^∗ is locally asymptotically stable if 𝑅₀>1.

Proof. For the linearized system of system (2.3) at 𝐸^∗, the corresponding Jacobian matrix is 𝐽_𝐸∗=−𝛼₂−𝛿−𝛼₁+𝛽𝐼^∗−𝛼₂+𝛾₁+𝛿+𝛼₁𝑆^∗𝛽𝐼^∗0. (4.2) The characteristic equation of 𝐽_𝐸∗ is 𝜆²+𝑘₁𝜆+𝑘₂=0, (4.3) where 𝑘₁=𝛼₁+𝛽𝐼^∗+𝛼₂𝑘+𝛿>0,₂=𝛼₂+𝛾₁+𝛿+𝛼₁𝑆^∗𝛽𝐼^∗>0. (4.4) It follows from the Hurwitz criterion that the two roots of (4.3) have negative real parts. Hence, the claimed result follows.

We are ready to study global stability of 𝐸^∗. Let Ω′=Ω−𝐸⁰, then we have as following.

Theorem 4.2. 𝐸^∗ is globally asymptotically stable with respect to Ω′ if 𝑅₀>1.

Proof. From system (2.3), we have 𝛿−𝛼₁𝑆^∗𝐼^∗−𝛿𝑆^∗+𝛾₂𝐼^∗−𝛽𝑆^∗𝐼^∗+𝛼₂1−𝑆^∗−𝐼^∗=0. (4.5) Note that 1𝛽𝛼₂−𝛾₂𝑆^∗+𝛼₁1+1=𝛽𝛼₂+𝛾₁+𝛿𝑆^∗+𝛼₁>0. (4.6) Define the Lyapunov function as 𝑉(𝑡)=^𝑆_𝑆∗𝑥−𝑆^∗𝑥𝑑𝑥+(𝑑+1)^𝐼_𝐼∗𝑥−𝐼^∗𝑥𝑑𝑥, (4.7) where 𝑑=(1/𝛽)((𝛼₂−𝛾₂)/𝑆^∗+𝛼₁). Then 𝑉^′(𝑡)_(2.3)=𝑆1−^∗𝑆𝑆̇+(𝑑+1)1−𝐼^∗𝐼𝐼̇=𝑆1−^∗𝑆𝛿−𝛼₁𝑆𝐼−𝛿𝑆+𝛾₂𝐼−𝛽𝑆𝐼+𝛼₂(+()𝐼1−𝑆−𝐼)𝑑+11−^∗𝐼𝛽𝑆𝐼−𝛾₂𝐼−𝛿𝐼−𝛾₁𝐼=𝑆1−^∗𝑆𝛿−𝛼₁𝑆𝐼−𝛿𝑆+𝛾₂𝐼−𝛽𝑆𝐼+𝛼₂(+(𝐼1−𝑆−𝐼)𝑑+1)1−^∗𝐼𝑆1−^∗𝑆=𝑆𝛽𝑆𝐼1−^∗𝑆𝛼₁𝑆^∗𝐼^∗−𝛼₁𝑆𝐼+𝛿𝑆^∗−𝑆+𝛾₂𝐼−𝐼^∗+𝛼₂𝑆^∗−𝑆+𝛼₂𝐼^∗−𝐼+𝛽𝑑𝑆𝐼+𝛽𝑑𝑆^∗𝐼^∗−𝛽(𝑑+1)𝑆𝐼^∗=𝑆1−^∗𝑆𝑆^∗−𝑆𝛼₁𝐼^∗+𝛿+𝛼₂+𝛽𝐼^∗+𝐼−𝐼^∗𝛽𝑑−𝛼₁𝛼𝑆−²−𝛾₂𝛽𝑑−𝛼₁𝑆=−^∗−𝑆²𝑆𝛿+𝛼₂+𝛽𝐼^∗−𝛼₂−𝛾₂𝑆^∗𝛼𝐼+²−𝛾₂𝑆^∗𝐼^∗+𝛼₁𝐼^∗. (4.8) If 𝛼₂≤𝛾₂, because 𝛽𝐼^∗+((𝛼₂−𝛾₂)/𝑆^∗)𝐼^∗=((𝛼₂+𝛾₁+𝛿)/𝑆^∗)𝐼^∗>0, thus 𝛿+𝛼₂+𝛽𝐼^∗−𝛼₂−𝛾₂𝑆^∗𝛼𝐼+²−𝛾₂𝑆^∗𝐼^∗+𝛼₁𝐼^∗>0. (4.9) If 𝛼₂>𝛾₂, from (4.5) we get 𝛿+𝛼₂+𝛽𝐼^∗+𝛼₂−𝛾₂𝑆^∗𝐼^∗+𝛼₁𝐼^∗𝑆^∗𝛼₂−𝛾₂=𝛿+𝛼₂+𝛽𝐼^∗+𝛼₂−𝛾₂𝑆^∗𝐼^∗+𝛼₁𝐼^∗𝑆^∗𝛼₂−𝛾₂+𝛿−𝛼₁𝑆^∗𝐼^∗−𝛿𝑆^∗+𝛾₂𝐼^∗−𝛽𝑆^∗𝐼^∗+𝛼₂1−𝑆^∗−𝐼^∗1𝛼₂−𝛾₂=𝛿+𝛼₂𝛼₂−𝛾₂>1>𝐼. (4.10) Hence, we have 𝛿+𝛼₂+𝛽𝐼^∗−𝛼₂−𝛾₂𝑆^∗𝛼𝐼+²−𝛾₂𝑆^∗𝐼^∗+𝛼₁𝐼^∗>0. (4.11) Furthermore, it is easy to see that 𝑉^′(𝑡)|_(2.3)≤0, and 𝑉^′(𝑡)|_(2.3)=0 if and only if (𝑆,𝐼)=(𝑆^∗,𝐼^∗). Hence, the claimed result follows from the LaSalle invariance principle.

Example 4.3. Consider a system of the form (2.3) and with 𝛽=0.3, 𝛿=0.1, 𝛼₁=0.2, 𝛼₂=0.4, 𝛾₁=0.1, and 𝛾₂=0.05. Then 𝑅₀=1.2>1. It follows from Theorem 4.2 that the viral equilibrium is globally asymptotically stable. Figure 3 displays how 𝑆(𝑡) and 𝐼(𝑡) evolve with time 𝑡 if 𝑆(0)=0.5 and 𝐼(0)=0.4.

Figure 3 

Evolutions of 𝑆(𝑡) and 𝐼(𝑡) for the system with 𝛽=0.3, 𝛿=0.1, 𝛼1=0.2, 𝛼2=0.4, 𝛾1=0.1, and 𝛾2=0.05, provided 𝑆(0)=0.5 and 𝐼(0)=0.4.

5. Discussions

As was indicated in the previous two sections, in order to eradicate computer viruses, one should take actions to keep 𝑅₀ below one.

From (2.2), it is easy to see that 𝑅₀ is increasing with 𝛽, and is decreasing with 𝛾₁,𝛾₂, and 𝛿, respectively. This implies that prevention is more important than cure, and higher disconnecting rate from the Internet contributes to the suppression of virus diffusion.

As a consequence, it is highly recommended that one should regularly update the antivirus software even if their computer is not noticeably infected, and timely disconnect the computer from the Internet whenever this connection is unnecessary. Also, filtering and blocking suspicious messages with firewall is rewarding.

6. Conclusions

By considering the possibility that an infected computer becomes susceptible as well as the possibility that a susceptible computer becomes recovered, a new computer virus propagation model has been proposed. The dynamics of this model has been fully studied. On this basis, some effective measures for controlling the spread of computer viruses across the Internet have been posed.

Acknowledgment

The authors are greatly indebted to the anonymous reviewers for their valuable suggestions. This work is supported by the Natural Science Foundation of China (Grant no. 10771227), the Doctorate Foundation of Educational Ministry of China (Grant no. 20110191110022), and the Research Funds for the Central Universities (Grant nos. CDJXS12180007 and CDJXS10181130).