An Impulse Model for Computer Viruses

Abstract

Computer virus spread model concerning impulsive control strategy is proposed and analyzed. We prove that there exists a globally attractive infection-free periodic solution when the vaccination rate is larger than 𝜃₀. Moreover, we show that the system is uniformly persistent if the vaccination rate is less than 𝜃₁. Some numerical simulations are finally given to illustrate the main results.

1. Introduction

Computer virus is a kind of computer program that can replicate itself and spread from one computer to others. Viruses mainly attack the file system and worms use system vulnerability to search and attack computers. As hardware and software technology develop and computer networks become an essential tool for daily life, the computer virus starts to be a major threat. Consequently, the trial on better understanding of the computer virus propagation dynamics is an important matter for improving the safety and reliability in computer systems and networks. Similar to the biological virus, there are two ways to study this problem: microscopic and macroscopic models. Following a macroscopic approach, since [1, 2] took the first step towards modeling the spread behavior of computer virus, much effort has been done in the area of developing a mathematical model for the computer virus propagation [3–13]. These models provide a reasonable qualitative understanding of the conditions under which viruses spread much faster than others.

In [4], the authors investigated a differential 𝑆𝐼𝑅𝑆 model by making the following assumptions. (H1)The total population of computers is divided into three groups: susceptible, infected, and recovered computers. Let 𝑆,𝐼, and 𝑅 denote the numbers of susceptible, infected and recovered computers, respectively. (H2)New computers are attached to the computer network with constant rate 𝑏. For the sake of antivirus software, some new nodes have temporary immunity with probability 𝑝, some have not with probability 1−𝑝. Hence, new nodes are added into the susceptible class with rate (1−𝑝)𝑏 and into the recovered class with rate 𝑝𝑏. (H3)Computers are disconnected to the computer network with the constant rate 𝜇 and are disconnected from the attack of malicious object with probability 𝛼. (H4)𝑆 computers become 𝐼 with constant rate 𝛽 and with constant time delay 𝜏₁; 𝑅 computers become 𝑆 with constant rate 𝛾 and with constant time delay 𝜏₁; 𝐼 computers become 𝑅 with constant rate 𝛾.

According to the above assumptions, the following model (see Figure 1) is derived: 𝑑𝑆=𝑑𝑡1−𝑝𝑏−𝜇𝑆−𝛽𝑆𝑡−𝜏₁𝐼𝑡−𝜏₁+𝜐𝑅𝑡−𝜏₂,𝑑𝐼𝑑𝑡=𝛽𝑆𝑡−𝜏₁𝐼𝑡−𝜏₁−𝜇+𝛾+𝛼𝐼,𝑑𝑅𝑑𝑡=𝑝𝑏−𝛾𝐼−𝜇𝑅−𝜐𝑅𝑡−𝜏₂. (1.1)

Figure 1 

Original model.

As we know, antivirus software is a kind of computer program which can detect and eliminate known viruses. There are two common methods that an antivirus software application uses to detect viruses: using a list of virus signature definitions and using a heuristic algorithm to find viruses based on common behaviors. It has been observed that it does not always work in detecting a novel computer virus by using the heuristic algorithm. On the other hand, obviously, it is impossible for antivirus software to find new computer viruss signature definitions on the dated list. So, to keep the antivirus soft in high efficiency, it is important to ensure that it is updated. Based on the above facts, we propose an impulsive system to model the process of periodic installing or updating antivirus software on susceptible computers at fixed time for controlling the spread of computer virus.

Based on above facts, we propose the following assumptions.(H5) The antivirus software is installed or updated at time 𝑡=𝑘𝑇(𝑘∈𝑁), where  𝑇 is the period of the impulsive effect.(H6)𝑆 computers are successfully vaccinated from 𝑆 class to 𝑅 class with rate 𝜃 (0<𝜃<1).

According to the above assumptions (H1)–(H6), and for the reason of simplicity we propose the following model with one time delay (see Figure 2): 𝑑𝑆=()𝑑𝑡1−𝑝𝑏−𝜇𝑆−𝛽𝑆𝐼𝑡−𝜏+𝜈𝑅,𝑑𝐼()−𝑑𝑡=𝛽𝑆𝐼𝑡−𝜏𝜇+𝛾+𝛼𝐼,𝑡≠𝑘𝑇,𝑘∈𝑍⁺,𝑑𝑅𝑆𝑡𝑑𝑡=𝑝𝑏+𝛾𝐼−𝜇𝑅−𝜈𝑅,₊=()𝑆(𝑡),𝐼𝑡1−𝜃₊(𝑡)𝑅𝑡=𝐼,𝑡=𝑘𝑇,₊=𝑅(𝑡)+𝜃𝑆(𝑡). (1.2)

Figure 2 

Impulse Model.

The total population size 𝑁(𝑡) can be determined by 𝑁(𝑡)=𝑆(𝑡)+𝐼(𝑡)+𝑅(𝑡) to form the differential equation 𝑁̇(𝑡)=𝑏−𝜇𝑁(𝑡)−𝛼𝐼(𝑡), (1.3) which is derived by adding the equations in system (1.1). Thus the total population size 𝑁 may vary in time. From (1.2), we have 𝑏−𝜇+𝛼𝑁(𝑡)≤𝑁̇(𝑡)≤𝑏−𝜇𝑁(𝑡). (1.4) It follows that 𝑏/(𝜇+𝛼)≤inflim_𝑥→∞𝑁(𝑡)≤lim_𝑥→∞sup𝑁(𝑡)≤𝑏/𝜇.

Before going into any details, we simplify model (1.1) and restrict our attention to the following model: 𝑑𝑆=()(),𝑑𝑡1−𝑝𝑏+𝛽𝑆𝐼𝑡−𝜏+𝜈𝑁−𝑆−𝐼𝑑𝐼()−𝑑𝑡=𝛽𝑆𝐼𝑡−𝜏𝜇+𝛾+𝛼𝐼,𝑡≠𝑘𝑇,𝑘∈𝑍⁺,𝑑𝑁𝑆𝑡𝑑𝑡=𝑏−𝜇𝑁−𝛼𝐼,₊=()𝑆(𝑡),𝐼𝑡1−𝜃₊(𝑡)𝑁𝑡=𝐼,𝑡=𝑘𝑇,₊=𝑁(𝑡). (1.5) The initial conditions for (1.5) are 𝜙₁(𝜉),𝜙₂(𝜉),𝜙₃(𝜉)∈𝐶₊=𝐶[−𝜔,0],𝑅³₊,𝜙₂(𝜉)>0,𝑖=1,2,3. (1.6) From physical considerations, we discuss system (1.5) in the closed set Ω=(𝑆,𝐼,𝑁)∈𝑅³₊𝑏∣0≤𝑆+𝐼≤𝜇𝑏,0≤𝑁≤𝜇, (1.7) where 𝑅³₊ denotes the nonnegative cone of 𝑅³ including its lower-dimensional faces. Note that it is positively invariant with respect to (1.7).

The organization of this paper is as follows. In Section 2, we first state three lemmas which are essential to our proofs and establish sufficient condition for the global attractivity of infection-free periodic solution. The sufficient condition for the permanence of the model is obtained in Section 3. Some numerical simulations are performed in Section 4. In the final section, a brief conclusion is given and some future research directions are also pointed out.

2. Global Attractivity of Infection-Free Periodic Solution

In this section, we prove that the infection-free periodic solution is globally attractive under some conditions. To prove the main results, two lemmas (given in [14]) which are essential to the proofs are stated here.

Lemma 2.1 (see [14], Lemma 1). Consider the following impulsive system: (𝑡)(𝑡)𝑢𝑡𝑢̇=𝑎−𝑏𝑢,𝑡≠𝑘𝜏,₊=(1−𝜃)𝑢(𝑡),𝑡=𝑘𝜏, (2.1) where 𝑎>0, 𝑏>0, 0<𝜃<1. Then there exists a unique positive periodic solution of system (2.1) 𝑢_𝑒(𝑡)=𝑎𝑏+𝑢^∗−𝑎𝑏𝑒^{−𝑏(𝑡−𝑘𝜏)},𝑘𝜏<𝑡≤(𝑘+1)𝜏, (2.2) which is globally asymptotically stable, where 𝑢^∗=𝑎(1−𝜃)(1−𝑒^−𝑏𝜏)/(𝑏(1−(1−𝜃)𝑒^−𝑏𝜏)).

Lemma 2.2 (see [14], Lemma 2). Consider the following linear neutral delay equation: 𝑥̇(𝑡)+𝜆𝑥̇(𝑡−𝜔)+𝑎₂𝑥(𝑡)−𝑎₁𝑥(𝑡−𝜔)=0. (2.3) If |𝜆|<1,𝑎²₁<𝑎²₂ or −𝑎₁=𝑎₂≠0, then increasing 𝜏 does not change the stability of (2.3).

When 𝜆=0, (2.3) becomes 𝑥̇(𝑡)=𝑎₁𝑥(𝑡−𝜔)−𝑎₂𝑥(𝑡). (2.4)

Corollary 2.3. Consider system (2.4) and assume that 𝑎₁,𝑎₂,𝜔>0;𝑥(𝑡)>0 for −𝜔≤𝑡≤0. Then we have the following statements: (i)assume that 𝑎₁<𝑎₂, then lim_𝑥→∞𝑥(𝑡)=0; (ii)assume that 𝑎₁>𝑎₂, then lim_𝑥→∞𝑥(𝑡)=+∞.
From the third and sixth equations of system (1.5), we have lim_𝑡→∞𝑁(𝑡)=𝑏/𝜇. Further, if 𝐼(𝑡)≡0, we have the following limit system 𝑑𝑆=𝑑𝑡1−𝑝𝑏+𝜈𝑏𝜇−𝜇+𝜈𝑆,𝑡≠𝑘𝑇,𝑘∈𝑍⁺,𝑑𝑁𝑆𝑡𝑑𝑡=𝑏−𝜇𝑁,₊=()𝑆(𝑡)𝑁𝑡1−𝜃,𝑡=𝑘𝑇,₊=𝑁(𝑡). (2.5) From the second and fourth equations of system (3.5), we have lim_𝑡→∞𝑁(𝑡)=𝑏/𝜇 and have the following limit systems of (3.5): 𝑑𝑆=𝑑𝑡1−𝑝𝑏+𝜈𝑏𝜇−𝑆𝑡𝜇+𝜈𝑆,𝑡≠𝑘𝑇,₊=(1−𝜃)𝑆(𝑡),𝑡=𝑘𝑇. (2.6) According to Lemma 2.1, we know that the periodic solution of system (3.10) is of the form 𝑆_𝑒(𝑡)=𝜇1−𝑝+𝜈𝑏𝜇+𝑆𝜇+𝜈^∗−𝜇1−𝑝+𝜈𝑏𝜇𝑒𝜇+𝜈^{−(𝜇+𝜈)(𝑡−𝑘𝜏)},𝑘𝜏<𝑡≤(𝑘+1)𝜏, (2.7) and it is globally asymptotically stable, where 𝑆^∗=(((1−𝑝)𝑏+𝜈𝑏/𝜇)(1−𝜃)(𝑒^{(𝜇+𝜈)𝜏}−1)/((𝜇+𝜈)(𝑒^{(𝜇+𝜈)𝜏}−1+𝜃))).

Theorem 2.4. The infection-free periodic solution (𝑆_𝑒(𝑡),0,𝑏/𝜇) of system (1.5) is globally attractive provided that 𝑅₀<1, where 𝑅₀=1−𝑝𝑏+𝜈𝑏/𝜇𝛽𝑒^{(𝜇+𝜈)𝜏}−1𝜇+𝛾+𝛼𝜇+𝜈𝑒^{(𝜇+𝜈)𝜏}.−1+𝜃 (2.8)

Proof. Since 𝑅₀<1, we can choose 𝜀₁>0 sufficiently small such that 𝛽1−𝑝𝑏+𝜈𝑏/𝜇𝑒^{(𝜇+𝜈)𝜏}−1𝜇+𝜈𝑒^{(𝜇+𝜈)𝜏}−1+𝜃+𝜀₁<𝜇+𝛾+𝛼. (2.9) It follows from the third equation of system (1.5) that 𝑁̇(𝑡)≤𝑏−𝜇𝑁(𝑡). (2.10) There exists an integer 𝑘₁>0 such that 𝑁(𝑡)≤𝑏/𝜇,𝑡>𝑘₁𝜏.
From the first equation of system (1.5), we have 𝑆̇(𝑡)≤1−𝑝𝑏+𝜈𝑏𝜇−𝜇+𝜈𝑆. (2.11) For 𝑡>𝑘₁𝜏,𝑘>𝑘₁, we consider the following comparison differential system: 𝑥̇(𝑡)=1−𝑝𝑏+𝜈𝑏𝜇−𝑥(𝑡)𝑥𝑡𝜇+𝜈,𝑡≠𝑘𝜏,₊=(1−𝜃)𝑥(𝑡),𝑡=𝑘𝜏. (2.12) In view of Lemma 2.1, we know that the unique periodic solution of system (2.12) is of the form 𝑆(𝑡)<𝑆_𝑒(𝑡)+𝜀₁≤1−𝑝𝑏+𝜈𝑏/𝜇𝑒^{(𝜇+𝜈)𝜏}−1𝜇+𝜈𝑒^{(𝜇+𝜈)𝜏}−1+𝜃+𝜀₁=̇𝛿,𝑘𝜏<𝑡≤(𝑘+1)𝜏,𝑘>𝑘₂, (2.13) and it is globally asymptotically stable. From (1.5), we have 𝐼̇(𝑡)≤𝛽𝛿𝐼(𝑡−𝜏)−𝜇+𝛾+𝛼𝐼(𝑡),𝑡>𝑘𝜏,𝑘>𝑘₂. (2.14) From (2.9), we have that 𝛽𝛿<𝜇+𝛾+𝛼. According to Corollary 2.3 we have lim_𝑡→∞𝐼(𝑡)=0. (2.15) Therefore, for any 𝜀₁>0 (sufficiently small), there exists an integer 𝑘₃>𝑘₂ such that 𝐼(𝑡)<𝜀 for all 𝑡>𝑘₃𝜏. From the third equation of system (1.5), we have 𝑁(𝑡)>𝑏−𝜇𝑁(𝑡)−𝛼𝜀₁,𝑡>𝑘₃𝜏. (2.16) Consider the comparison equation 𝑧̇(𝑡)=𝑏−𝛼𝜀₁−𝜇𝑧(𝑡),for𝑡>𝑘₃𝜏. (2.17) It is easy to see that lim_𝑡→∞𝑧(𝑡)=(𝑏−𝛼𝜀₁)/𝜇. It follows by the comparison theorem that there exists an integer 𝑘₄>𝑘₃ such that 𝑁(𝑡)≥𝑏−𝛼𝜀₁𝜇−𝜀₁,∀𝑡>𝑘₄𝜏. (2.18) Since 𝜀₁ is arbitrarily small, from lim_𝑡→∞sup𝑁(𝑡)≤𝑏/𝜇 and (2.18) we have lim_𝑡→∞𝑁(𝑡)=𝑏𝜇. (2.19) It follows from (2.15) and (2.19) that there exists 𝑘₅>𝑘₄ such that 𝐼(𝑡)<𝜀₁(𝑡)>𝑏,𝑁𝜇−𝜀₁,for𝑡>𝑘₅𝜏. (2.20) Hence, from the first equation of system (1.5) we have that 𝑆̇(𝑡)≥1−𝑝𝑏−𝜇𝑆(𝑡)−𝛽𝑆(𝑡)𝜀₁+𝜈𝑏𝜇−𝑠𝜀₁(𝑡)=𝑏−𝑆1−𝑝𝑏+𝜈𝜇−2𝜀₁𝜈−𝜇+𝛽𝜀₁+𝜈𝑆(𝑡),for𝑡>𝑘₅𝜏. (2.21) Consider the following comparison impulsive differential equations for 𝑡>𝑘₅𝜏 and 𝑘>𝑘₅, (𝑡)=𝑏𝑢̇1−𝑝𝑏+𝜈𝜇−2𝜀₁𝜈−𝜇+𝛽𝜀₁𝑢(𝑡)𝑢𝑡+𝜈,𝑡≠𝑘𝜏,₊=(1−𝜃)𝑢(𝑡),𝑡=𝑘𝜏. (2.22) In view of Lemma 2.1, we periodic solution of system 𝑢_𝑒(𝑡)=Γ+𝑢^∗−Γ𝑒^{−(𝜇+𝛽𝜀1+𝜈)(𝑡−𝑘𝜏)},𝑘𝜏<𝑡≤(𝑘+1)𝜏, (2.23) which is globally asymptotically stable, where Γ=1−𝑝𝑏+𝜈𝑏/𝜇−2𝜀₁𝜈𝜇+𝛽𝜀₁+𝜈,𝑢^∗()=Γ1−𝜃1−𝑒^{−(𝜇+𝛽𝜀1+𝜈)𝜏}1−(1−𝜃)𝑒^{−(𝜇+𝛽𝜀1+𝜈)𝜏}. (2.24) According to the comparison theorem for impulsive differential equation, there exists an integer 𝑘₆>𝑘₅ such that 𝑆(𝑡)>𝑢_𝑒(𝑡)−𝜀₁,𝑘𝜏<𝑡≤(𝑘+1)𝜏,𝑘>𝑘₆. (2.25) Because 𝜀₁ is arbitrarily small, it follows from (2.25) that 𝑆_𝑒(𝑡)=1−𝑝𝑏+𝜈𝑏/𝜇𝑒^{(𝜇+𝜈)𝜏}−1𝜇+𝜈𝑒^{(𝜇+𝜈)𝜏}()−1+𝜃,𝑘𝜏<𝑡≤𝑘+1𝜏, (2.26) is globally attractive, that is, lim_𝑥→∞𝑆(𝑡)=𝑆_𝑒(𝑡). (2.27) It follows from (2.15), (2.19), (2.27), and the restriction 𝑁(𝑡)=𝑆(𝑡)+𝐼(𝑡)+𝑅(𝑡) that lim_𝑥→∞𝑅(𝑡)=𝑏/𝜇−𝑆_𝑒(𝑡). Hence, the infection-free periodic solution (𝑆_𝑒(𝑡),0,𝑏/𝜇) of system (1.5) is globally attractive. The proof is completed.

Corollary 2.5. The infection-free periodic solution (𝑆_𝑒(𝑡),0,𝑏/𝜇) of system (1.5) is globally attractive, if 𝜃>𝜃₀ where 𝜃₀=(𝑒^{(𝜇+𝜈)𝜏}−1)(((1−𝑝+𝜈/𝜇)/((𝜇+𝛾)(𝜇+𝛾+𝛼)))−1).

Theorem 2.4 determines the global attractivity of (1.5) in Ω for the case 𝑅₀<1. Its realistic implication is that the infected computers vanish so the computer virus removed from the network. Corollary 2.5 implies that the computer virus will disappear if the vaccination rate is larger than 𝜃₀.

3. Permanence

In this section, we say the computer virus is local if the infectious population persists above a certain positive level for sufficiently large time. The locality viruses can be well captured and studied through the notion of permanence.

Definition 3.1. System (1.5) is said to be uniformly persistent if there is an 𝜑>0 (independent of the initial data) such that every solution  (𝑆(𝑡),𝐼(𝑡),𝑅(𝑡),𝑁(𝑡)) with initial conditions (1.7) of system (1.5) satisfies lim_𝑡→∞inf𝑆(𝑡)≥𝜑,lim_𝑡→∞inf𝐼(𝑡)≥𝜑,lim_𝑡→∞inf𝑅(𝑡)≥𝜑. (3.1)

Definition 3.2. System (1.5) is said to be permanent if there exists a compact region Ω₀∈intΩ such that every solution of system (1.5) with initial data (1.7) will eventually enter and remain in region Ω₀. Denote 𝑅₁=𝛽1−𝑝𝑏𝑒^𝜇𝜏−1𝜇+𝛾+𝛼(𝑒^𝜇𝜏).−1+𝜃 (3.2)

Theorem 3.3. Suppose that 𝑅₁>1. Then there is a positive constant 𝑚_𝐼 such that each positive solution (𝑆(𝑡),𝐼(𝑡),𝑅(𝑡)) of system (1.5) satisfies 𝐼(𝑡)≥𝑚_𝐼, for 𝑡 large enough.

Proof. Now, we will prove there exist 𝑚_𝐼>0 and a sufficiently large 𝑡_𝑝 such that 𝐼(𝑡)≥𝑚_𝐼 holds for all 𝑡>𝑡_𝑝. Suppose that 𝐼(𝑡)<𝑚^∗_𝐼 for all 𝑡>𝑡₀. From the first equation of (1.5), we have 𝑆̇(𝑡)>1−𝑝𝑏−𝜇+𝛽𝑚^∗_𝐼𝑆. (3.3)
Consider the following comparison system: 𝑧₂̇(𝑡)=1−𝑝𝑏−𝜇+𝛽𝑚^∗_𝐼𝑧₂(𝑡)𝑧,𝑡≠𝑘𝜏,₂𝑡⁺=(1−𝜃)𝑧₂(𝑡),𝑡=𝑘𝜏𝑆̇(𝑡)>1−𝑝𝑏−𝜇+𝛽𝑚^∗_𝐼𝑆. (3.4) By Lemma 2.1, we know that, there exists 𝑡₁ such that 𝑆(𝑡)≥𝑧₂(𝑡)>𝑧^∗₂(𝑡)𝑏𝑒−𝜀≥1−𝑝^{(𝜇+𝛽𝑚∗𝐼)𝜏}−1𝜇+𝛽𝑚^∗_𝐼𝑒^{(𝜇+𝛽𝑚∗𝐼)𝜏}−1+𝜃−𝜀=̇𝑆>0,for𝑡>𝑡₁. (3.5) It follows from the second equation of (1.5) that 𝐼(̇𝑡)>𝛽𝑆𝐼(𝑡−𝜏)−(𝜇+𝛾+𝛼)𝐼(𝑡). Consider the comparison system 𝑧₃̇(𝑡)=𝛽𝑆𝐼(𝑡−𝜏)−(𝜇+𝛾+𝛼)𝑧₃(𝑡). Noting that 𝑅₁>1 and 𝜀 is sufficiently small, we have 𝛽𝑆>(𝜇+𝛾+𝛼).
Corollary 2.3 implies that 𝑡→∞,  𝐼(𝑡)>𝑧₃(𝑡)→∞. This contradicts 𝐼(𝑡)≤𝑏. Hence, we can claim that, for any 𝑡₀>0, it is impossible that 𝐼(𝑡)<𝑚^∗_𝐼∀𝑡≥𝑡₀. (3.6) By the claim, we are left to consider two cases. First, 𝐼(𝑡)≤𝑚^∗_𝐼 for 𝑡 large enough. Second, 𝐼(𝑡) oscillates about 𝑚^∗_𝐼 for 𝑡 large enough. Obviously, there is nothing to prove for the first case. For the second case, we can choose 𝑡₂>𝑡₁ and 𝜉>0 satisfy 𝐼𝑡₂=𝐼𝑡₂+𝜉=𝑚^∗_𝐼,𝐼(𝑡)<𝑚^∗_𝐼,for𝑡₂<𝑡<𝑡₂+𝜉 (3.7)𝐼(𝑡) is uniformly continuous since the positive solutions to (1.5) are ultimately bounded and 𝐼(𝑡) is not effected by impulses.
Therefore, it is certain that there exists 𝜂 (0<𝜂<𝜏, and 𝜂 is independent of the choice of 𝑡₂) such that 𝐼(𝑡)≥𝑚^∗_𝐼2,for𝑡₂≤𝑡≤𝑡₂+𝜂. (3.8) In this case, we shall consider the following three possible cases in term of the sizes of 𝜂, 𝜉 and 𝜏.  Case 1. If 𝜉≤𝜂<𝜏, then it is obvious that 𝐼(𝑡)≥𝑚^∗_𝐼/2, for 𝑡∈[𝑡₂,𝑡₂+𝜉].  Case 2. If 𝜂<𝜉≤𝜏, then from the second equation of system (1.5), we obtain 𝐼(̇𝑡)>−(𝜇+𝛾+𝛼)𝐼(𝑡). Since 𝐼(𝑡₂)=𝑚^∗_𝐼, it is obvious that 𝐼(𝑡)>𝑚^∗_𝐼𝑒^{−(𝛾+𝛼+𝜇)𝜏}=̇𝑚_𝐼^∗∗, for 𝑡∈[𝑡₂,𝑡₂+𝜉].  Case 3. If 𝜂<𝜏≤𝜉, it is easy to obtain that 𝐼(𝑡)>𝑚_𝐼^∗∗ for 𝑡∈[𝑡₂,𝑡₂+𝜏]. Then, proceeding exactly the proof for above claim, we have that 𝐼(𝑡)>𝑚_𝐼^∗∗ for 𝑡∈[𝑡₂+𝜏,𝑡₂+𝜉].
Owing to the randomicity of 𝑡₂, we obtain that there exists 𝑚_𝐼=̇min{𝑚^∗_𝐼/2,𝑚_𝐼^∗∗}, such that 𝐼(𝑡)>𝑚_𝐼 holds for all 𝑡>𝑡_𝑝. The proof of Theorem 3.3 is completed.

Theorem 3.4. Suppose 𝑅₁>1. Then system (1.5) is permanent.

Proof. Let (𝑆(𝑡),𝐼(𝑡),𝑁(𝑡)) be any solution to system (1.5). First, from the first equation of system (1.5), we have 𝑆(̇𝑡)>(1−𝑝)𝑏−(𝜇+𝛽𝑏)𝑆. Consider the following comparison system: 𝑧₁̇(𝑡)=1−𝑝𝑏−𝜇+𝛽𝑏𝑧₁(𝑡)𝑧,𝑡≠𝑘𝜏,₁𝑡⁺=(1−𝜃)𝑧₁(𝑡),𝑡=𝑘𝜏. (3.9)
By Lemmas 2.1 and 2.2, we know that for any sufficiently small 𝜀>0, there exists 𝑡₁ (𝑡₁ is sufficiently large) such that 𝑆(𝑡)≥𝑧₂(𝑡)>𝑧^∗₂(𝑡)𝑏𝑒−𝜀≥1−𝑝^{(𝜇+𝛽𝑚∗𝐼)𝜏}−1𝜇+𝛽𝑚^∗_𝐼𝑒^{(𝜇+𝛽𝑚∗𝐼)𝜏}−1+𝜃−𝜀=̇𝑚_𝑆>0,𝑘𝜏<𝑡≤(𝑘+1)𝜏. (3.10) From the third equation of (1.5), we have 𝑁̇(𝑡)>𝑛−𝜇𝑁(𝑡)−𝛼𝑁(𝑡). It is easy to see that 𝑁(𝑡)>𝑏/(𝜇+𝛼)−𝜀=̇𝑚_𝑁. Let Ω₀={(𝑆,𝐼,𝑁)∈𝑅³₊∣𝑚_𝜉≤𝑆,𝑚_𝐼≤𝐼,𝑚_𝑁≤𝑁≤𝑏/𝜇,𝑆+1≤𝑏/𝜇}. By Theorem 3.3 and above discussions, we know that the set Ω₀ is a global attractor in Ω, and of course, every solution of system (1.5) with initial conditions (1.7) will eventually enter and remain in region Ω₀. Therefore, system (1.5) is permanent. The proof is completed.

Corollary 3.5. It follows from Theorem 3.4 that the system (1.5) is uniformly persistent provided that 𝜃<𝜃₁, where 𝜃₁=(𝑒^𝜇𝜏−1)((𝛽(1−𝑝)𝑏/((𝜇+𝛼)(𝜇+𝛼+𝛾)))−1).

4. Numerical Simulations

In this section, we perform some numerical simulations to show the geometric impression of our results. To demonstrate the global attractivity of infection-free periodic solution to system (1.5), we take the following parameter values: 𝑏=1,𝜇=0.5,𝛾=0.3,𝛼=0.02,𝛽=0.3,𝜈=0.7,𝜃=0.4,  and 𝜏=1. In this case, we have 𝑅₀=0.5894<1. In Figures 3(a), 3(b), and 3(c) display respectively the susceptible, infected and recovered population of system (1.5) with initial conditions: 𝑆(0)=3,𝐼(0)=4 and 𝑅(0)=5. Figure 3(d) shows their corresponding phase-portrait.

(a)

(b)

(c)

(d)

Figure 3 

Global attractivity of infection-free periodic solution.

To demonstrate the permanence of system (1.5) we take following set parameter values: 𝑏=10,𝜇=0.3,𝛾=0.3,𝛼=0.02,𝛽=0.3,𝜃=0.2,𝜈=0.7,  and 𝜏=1. In this case, we have 𝑅₁=2.2195>1. In Figures 4(a), 4(b), and 4(c) display, respectively, the susceptible, infected and recovered population of system (1.5) with initial conditions: 𝑆(0)=3,𝐼(0)=4, and 𝑅(0)=5. Figure 4(d) shows their corresponding phase portrait.

(a)

(b)

(c)

(d)

Figure 4 

permanence of system.

5. Conclusion

We have analyzed the delayed 𝑆𝐼𝑅𝑆 model with pulse vaccination and varying total population size. We have shown that 𝑅₁>1 or 𝜃<𝜃₁ implies that the disease will be endemic, whereas 𝑅₀<1 or 𝜃>𝜃₀ implies that the disease will fade out. We have also established sufficient condition for the permanence of the model. Our results indicate that a short interpulse time or a large pulse vaccination rate will lead to eradication of the computer virus.

In this paper, we have only discussed two cases: (i) 𝑅₀<1 (or 𝜃>𝜃₀) and (ii) 𝑅₁>1  (or 𝜃<𝜃₁). But for closed interval [𝑅₀, 𝑅₁] (or [𝜃₁,𝜃₀]), the dynamical behavior of model (3) have not been studied, and the threshold parameter for the reproducing number (or the pulse vaccination rate) between the extinction of the computer viruses and the uniform persistence of the viruses have not been obtained. These issues would be left to our future consideration.

Acknowledgment

This paper is supported by the National Natural Science Foundation of China (no. 61170320), the Natural Science Foundation of Guangdong Province (no. S2011040002981) and the Scientific research Foundation of Guangdong Medical College (no. KY1048).