Optimal control strategy for a novel computer virus propagation model on scale-free networks

doi:10.1016/j.physa.2016.01.028

Physica A: Statistical Mechanics and its Applications

Volume 451, 1 June 2016, Pages 251-265

https://doi.org/10.1016/j.physa.2016.01.028 Get rights and content

Highlights

•
Propose a novel SLBOS computer virus propagation model on scale-free networks.
•
Obtain the spreading threshold and global stability criterions.
•
Investigate existence of an optimal control for the control problem.
•
Some numerical simulations are given to illustrate the main results.

Abstract

This paper aims to study the combined impact of reinstalling system and network topology on the spread of computer viruses over the Internet. Based on scale-free network, this paper proposes a novel computer viruses propagation model—SLBOSmodel. A systematic analysis of this new model shows that the virus-free equilibrium is globally asymptotically stable when its spreading threshold is less than one; nevertheless, it is proved that the viral equilibrium is permanent if the spreading threshold is greater than one. Then, the impacts of different model parameters on spreading threshold are analyzed. Next, an optimally controlled SLBOS epidemic model on complex networks is also studied. We prove that there is an optimal control existing for the control problem. Some numerical simulations are finally given to illustrate the main results.

Introduction

Computer virus is a kind of computer program that can replicate itself and spread from one computer to others [1]. Computer viruses have caused enormous financial losses in the past few decades. With the popularization of the Internet and wireless networks, the epidemic capability of computer viruses has been overwhelmingly magnified. Indeed, computer viruses have turned out to be a major threat to our work and life [2]. The dynamical modeling, as it is known, is a vital approach to studying the way computer virus spreads on the Internet. Since Kephart and White [3], who followed the idea suggested by Cohen [4] and Murray [5], presented the first propagation model of computer virus, numerous relevant efforts in this field have been done.

In the past decade, the work on computer virus epidemiology has been mainly focused on the following two topics:

(i)
Viruses spreading on fully-connected networks that are established based on the assumption that every computer on the network is equally likely to be accessed by any other computer across the network. Some classical models range from conventional models, such as the Susceptible–Infected–Susceptible (SIS) models [3], [6], Susceptible–Infected–Removed (SIR) models [7], [8], Susceptible–Infectious–Recovered–Susceptible (SIRS) models [9], [10], [11], Susceptible–Infected–External–Susceptible (SIES) models [12], Susceptible–Exposed–Infectious–Recovered (SEIR) models [13], [14], Susceptible–Exposed–Infectious–Recovered–Susceptible (SEIRS) models [15], Susceptible–Exposed–Infectious–Quarantined–Recovered–Susceptible (SEIQRS) models [16], Susceptible–Latent–Breaking–Susceptible (SLBS) models [17], [18], [19], [20], [21], [22], Susceptible–Infected–Countermeasure–Susceptible (SICS) models [23], [24], and some other models [25], [26], [27], [28], to unconventional models such as delayed models [29], [30], [31], [32], [33], [34], [35], [36] and stochastic models [22], [37].
(ii)
Viruses spreading on complex networks, which was stimulated by the discovery that the Internet follows a power-law degree distribution [38], [39], [40]. These pioneering work has aroused intense interest in the impact of network topology on virus spreading. As a result, multifarious network-based virus epidemic models, ranging from Susceptible–Infected (SI) models [41], [42], [43] and SIS models [44], [45], [46], [47] to SIR models [48], [49], [50], [51], [52] and SLBS models [53] have been inspected.

To better understand the combined impact of both reinstalling system and network topology on virus spreading, in this paper we propose a novel SLBOS model by assuming that the underlying network is scale-free, i.e. its degree distribution follows a power law distribution. A comprehensive study of the model is conducted. Specifically, the spreading threshold $R_{0}$ is calculated, the virus-free equilibrium is shown to be globally asymptotically stable if $R_{0} < 1$ , and the viral equilibrium is proved to be permanent if $R_{0} > 1$ . To better control computer virus propagation, an optimally controlled SLBOS epidemic model on complex networks is also proposed.

The layout of this paper is as follows: Section 2 deals with the relevant mathematical framework (notations, hypotheses, and model formulation). Section 3 determines the spreading threshold and equilibria for this model. In Sections 4 Stability of the virus-free equilibrium, 5 Permanence of the virose equilibrium, we examine the global stability of the virus-free equilibrium and the permanence of the viral equilibrium, respectively. The effects of system parameters on virus spreading are analyzed in Section 6. In Section 7, the analysis of optimization problems is presented. Some numerical simulations are performed in Section 8. Finally, Section 9 outlines this work.

Section snippets

Description of the new model

We shall use a graph $G = (V, E)$ to represent the Internet topology, where nodes and edges stand for computers and communication links among computers, respectively. Its node degrees asymptotically comply with a power-law distribution, $P (k) \sim k^{- τ}$ , where $P$ ( $k$ ) means the probability that a node chosen randomly from the Internet is of degree $k$ [39], [40]. At any time a node has two states: within system, and without system. The nodes within system have three states: virus-free nodes, infected nodes

Spreading threshold and equilibria

Let $R_{0} = \frac{β_{1} δ (γ + μ) + β_{2} δ α}{(δ + μ) (γ + μ) (α + μ)} \frac{〈 k^{2} 〉}{〈 k 〉},$ where $〈 k^{2} 〉$ stands for the second origin moment of the node degree, $〈 k^{2} 〉 ≔ \sum_{k} k^{2} P (k)$ . Then, we have

Theorem 1

Consider system (4). The following assertions hold.

(1)
There is always a virus-free equilibrium, $E_{0} = {(S_{k}^{0}, 0, 0)}^{T}$ , where $S_{k}^{0} = \frac{δ}{δ + μ}, 1 \leq k \leq Λ$ .
(2)
There is no virose equilibrium if $R_{0} \leq 1$ .
(3)
There is a unique virose equilibrium, $E_{*} = {(S^{*}, L^{*}, B^{*})}^{T}$ , if $R_{0} > 1$ ,
where $S_{k}^{*} = \frac{δ (γ + μ) (α + μ)}{(δ + μ) (γ + μ) (α + μ) + (δ + μ) (γ + μ + α) k Θ_{*}},$ $L_{k}^{*} = \frac{k Θ_{*} δ (γ + μ)}{(δ + μ) (γ + μ) (α + μ) + (δ + μ) (γ + μ + α) k Θ_{*}},$ $B_{k}^{*} = \frac{k Θ_{*} δ α}{(δ + μ) (γ + μ) (α + μ) + (δ + μ) (}$

Stability of the virus-free equilibrium

This section examines the stability of the virus-free equilibrium $E_{0}$ . For convenience, let $Ω = {x = (x_{1}, x_{2}, \dots, x_{3 Λ}) | x_{i} \geq 0 for all 1 \leq i \leq 3 Λ, x_{i} + x_{i + Λ} + x_{i + 2 Λ} \leq 1 for all 1 \leq i \leq Λ} .$ Let $x (t) = {(S (t), L (t), B (t))}^{T}$ and rewrite system (4) in matrix–vector notation as $x (t) = A x (t) + H (x (t)),$ with initial condition $x (0) \in Ω$ , where $A = (\begin{matrix} A_{11} & A_{12} & A_{13} \\ A_{21} & A_{22} & A_{23} \\ A_{31} & A_{32} & A_{33} \end{matrix}),$ $A_{11} = (- δ - μ) E_{Λ},$ $A_{12} = (\begin{matrix} - (1 \times 1) P (1) S_{k}^{0} \frac{β_{1}}{〈 k 〉} & - (1 \times 2) P (2) S_{k}^{0} \frac{β_{1}}{〈 k 〉} & \dots & - (1 \times Λ) P (Λ) S_{k}^{0} \frac{β_{1}}{〈 k 〉} \\ - (2 \times 1) P (1) S_{k}^{0} \frac{β_{1}}{〈 k 〉} & - (2 \times 2) P (2) S_{k}^{0} \frac{β_{1}}{〈 k 〉} & \dots & - (2 \times Λ) P (Λ) S_{k}^{0} \frac{β_{1}}{〈 k 〉} \\ \dots & \dots & \dots & \dots \\ - (Λ \times 1) P (1) S_{k}^{0} \frac{β_{1}}{〈 k 〉} & - (Λ \times 2) P (2) S_{k}^{0} \frac{β_{1}}{〈 k 〉} & \dots & - (Λ \times Λ \end{matrix}$

Permanence of the virose equilibrium

To better understand the dynamical properties of system (4), we give

Theorem 4

Consider system (4) and suppose $R_{0} > 1$ . If $L (0) + B (0) > 0$ , then, for all $1 \leq k \leq Λ$ , we have $lim_{t \to \infty} inf {L_{k} (t) + B_{k} (t)} > 0 .$

Proof

According to assertion (2) of Lemma 4, there exists $k_{0} (1 \leq k_{0} \leq Λ)$ such that $lim_{t \to \infty} inf {L_{k_{0}} (t) + B_{k_{0}} (t)} > 0 .$ Let $β_{0} = min {β_{1}, β_{2}}$ . Then $lim_{t \to \infty} inf Θ (L (t), B (t)) = lim_{t \to \infty} inf \frac{\sum_{k} k P (k) [β_{1} L_{k} (t) + β_{2} B_{k} (t)]}{〈 k 〉} \geq β_{0} \frac{k_{0} P (k_{0})}{〈 k 〉} lim_{t \to \infty} inf (L_{k_{0}} (t) + B_{k_{k}} (t)) > 0 .$ As ${lim}_{t \to \infty} inf {S_{k_{0}} + (t) L_{k_{0}} (t) + B_{k_{0}} (t)} = \frac{δ}{δ + μ}$ , let $σ = {lim}_{t \to \infty} inf Θ (L (t), B (t))$ . Then $σ > 0$ . There exists $τ > 0$

Further discussions

In order to eradicate virus infections, measures ensuring $R_{0} < 1$ should be taken. To have a close look at the impact of different parameters on $R_{0}$ , let us do the following calculations. $\frac{\partial R_{0}}{\partial β_{1}} = \frac{δ (γ + μ)}{(δ + μ) (γ + μ) (α + μ)} \frac{〈 k^{2} 〉}{〈 k 〉} > 0,$ $\frac{\partial R_{0}}{\partial β_{2}} = \frac{δ α}{(δ + μ) (γ + μ) (α + μ)} \frac{〈 k^{2} 〉}{〈 k 〉} > 0,$ $\frac{\partial R_{0}}{\partial γ} = - \frac{β_{2} α δ}{(δ + μ) (α + μ) {(γ + μ)}^{2}} \frac{〈 k^{2} 〉}{〈 k 〉} < 0,$ $\frac{\partial R_{0}}{\partial δ} = \frac{[β_{1} (γ + μ) + β_{2} α] μ}{(γ + μ) (α + μ) {(δ + μ)}^{2}} \frac{〈 k^{2} 〉}{〈 k 〉} > 0,$ $\frac{\partial R_{0}}{\partial μ} = - \frac{β_{1} δ (δ + α + 2 μ)}{{[(α + μ) (δ + μ)]}^{2}} \frac{〈 k^{2} 〉}{〈 k 〉} < 0 .$

From these computational results, we can draw the following conclusions:

(a)
Reducing the infection rate, $β_{1}$ and $β_{2}$ , could

The optimal control problem

Optimal control techniques are widely used in applying optimal strategies to control computer viruses [36], [55], [56], [57]. To address the challenges of obtaining an optimal computer worm control strategy, we exploit optimal control theory in Ref. [58].

In the system (4), we have three state variables $S_{k} (t)$ , $L_{k} (t)$ and $B_{k} (t)$ . For the optimal control problem, we consider the control variable $u_{k} (t) \in U$ to be the percentage of infected computers become recovered computers under the effects of

Numerical simulations

In this section, some numerical simulations are given to show the geometric impression of our results. To demonstrate the global stability of infection-free solution of system (4), we take the following set of parameter values: $β_{1} = 0.004$ , $β_{2} = 0.009$ , $α = 0.1$ , $γ = 0.2$ , $δ = 0.15$ , $μ = 0.3$ , which runs on a scale-free network with $Λ = 1000$ and $τ = 2$ . In this case, we have $R_{0} = 0.6457 < 1$ , see Fig. 2.

To demonstrate the permanence of system (4), we take the following set of parameter values: $β_{1} = 0.4$ , $β_{2} = 0.3$ , $α = 0.2$ , $γ = 0.2$

Conclusions

To better understand the combined impact of reinstalling system and network topology on virus spreading, a new model capturing the epidemics of computer viruses on scale-free networks has been proposed. The spreading threshold for the model has been determined. The global asymptotic stability of the virus-free equilibrium has been shown when the threshold is below one, whereas the permanence of the virose equilibrium has been proved if the threshold is above one. The effect of some system

Acknowledgments

The authors are indebted to the anonymous reviewers and the editor for their valuable suggestions that have greatly improved the quality of this paper. This work is supported by Natural Science Foundation of China (#11347150, #61304117). This work is supported by Natural Science Foundation of Guangdong Province, China (#2014A030310239).

References (59)

F. Cohen
Computer viruses: Theory and experiments
Comput. Secur.
(1987)
L. Billings et al.
A unified prediction of computer virus spread in connected networks
Phys. Lett. A
(2002)
J. Ren et al.
A novel computer virus model and its dynamics
Nonliear Anal. RWA
(2012)
Q. Zhu et al.
Modeling and analysis of the spread of computer virus
Commun. Nonlinear Sci. Numer. Simul.
(2012)
C. Gan et al.
A propagation model of computer virus with nonlinear vaccination probability
Commun. Nonlinear Sci. Numer. Simul.
(2014)
C. Gan et al.
An epidemic model of computer viruses with vaccination and generalized nonlinear incidence rate
Appl. Math. Comput.
(2013)
H. Yuan et al.
Network virus-epidemic model with the point-to-group information propagation
Appl. Math. Comput.
(2008)
H. Yuan et al.
Towards controlling virus propagation in information systems with pointto- group information sharing
Decis. Support Syst.
(2009)
B.K. Mishra et al.
Dynamic model of worms with vertical transmission in computer network
Appl. Math. Comput.
(2011)
B.K. Mishra et al.
SEIQRS model for the transmission of malicious objects in computer network
Appl. Math. Model.
(2010)

A. d’Onofrio

A note on the global behavior of the network-based SIS epidemic model

Nonlinear Anal. Real World Appl.

(2008)

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Optimal control strategy for a novel computer virus propagation model on scale-free networks

Highlights

Abstract

Introduction

Section snippets

Description of the new model

Spreading threshold and equilibria

Stability of the virus-free equilibrium

Permanence of the virose equilibrium

Further discussions

The optimal control problem

Numerical simulations

Conclusions

Acknowledgments

Comput. Secur.

Phys. Lett. A

Nonliear Anal. RWA

Commun. Nonlinear Sci. Numer. Simul.

Commun. Nonlinear Sci. Numer. Simul.

Appl. Math. Comput.

Appl. Math. Comput.

Decis. Support Syst.

Appl. Math. Comput.

Appl. Math. Model.

Appl. Math. Comput.

Commun. Nonlinear Sci. Numer. Simul.

Discrete Dyn. Nat. Soc.

Nonlinear Anal. Real World Appl.

Discrete Dyn. Nat. Soc.

Physica A

Appl. Math. Comput.

Comput. Secur.

Appl. Math. Model.

Math. Comput. Modelling

Appl. Math. Comput.

Appl. Math. Comput.

Appl. Math. Comput.

Chaos Solitons Fractals

Appl. Math. Comput.

Physica A

Nonlinear Anal. Real World Appl.

Comput. Statist. Data Anal.

Nonlinear Anal. Real World Appl.