On the Optimal Dynamic Control Strategy of Disruptive Computer Virus

Bi, Jichao; Yang, Xiaofan; Wu, Yingbo; Xiong, Qingyu; Wen, Junhao; Tang, Yuan Yan

doi:https://doi.org/10.1155/2017/8390784

Discrete Dynamics in Nature and Society

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Epidemic Processes on Complex Networks

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Volume 2017 | Article ID 8390784 | https://doi.org/10.1155/2017/8390784

On the Optimal Dynamic Control Strategy of Disruptive Computer Virus

Jichao Bi,¹Xiaofan Yang,¹Yingbo Wu,¹Qingyu Xiong,¹Junhao Wen,¹and Yuan Yan Tang²

Academic Editor: Yong Deng

Received14 Dec 2016

Revised13 Feb 2017

Accepted23 Feb 2017

Published19 Mar 2017

Abstract

Disruptive computer viruses have inflicted huge economic losses. This paper addresses the development of a cost-effective dynamic control strategy of disruptive viruses. First, the development problem is modeled as an optimal control problem. Second, a criterion for the existence of an optimal control is given. Third, the optimality system is derived. Next, some examples of the optimal dynamic control strategy are presented. Finally, the performance of actual dynamic control strategies is evaluated.

1. Introduction

The proliferation of computer networks has brought huge benefits to human society. Meanwhile, it offers a shortcut to spread computer viruses, inflicting large economic losses [1]. Consequently, containing the prevalence of digital viruses has been one of the major concerns in the field of cybersecurity. The spreading dynamics of computer virus has been widely adopted as the standard method for assessing the viral prevalence [2]. Since the seminal work by Kephart and White [3, 4], a multitude of computer virus-spreading models, ranging from the population-level models [5–12] and the network-level models [13–17] to the node-level models [18–22], have been proposed.

One of the central tasks in cybersecurity is to develop control strategies of computer virus so that, subject to limited budgets, the losses caused by computer infections are minimized [23]. In recent years, the optimal design problem of virus control strategies has been modeled as static optimization problems [24–28]. The optimal static control strategies, however, only apply to the small-timescale situations where the network state keeps unchanged. In the realistic situations where the network state is varying over time, the optimal design problem of virus control strategies can be modeled as dynamic optimal control problems [29–33]. The optimal dynamic control strategies outperform their static counterparts, because the former not only are more cost-effective but apply to different timescales.

A disruptive computer virus is defined as a computer virus whose life period consists of two consecutive phases: the latent phase and the disruptive phase. In the latent phase, a disruptive virus staying in a host does not perform any disruptive operations. Rather, the virus tries to infect as many hosts as possible by sending its copies to them. In the disruptive phase, a disruptive virus staying in a host performs a variety of operations that disrupt the host, such as distorting data, deleting data or files, and destroying the operating system. To assess the prevalence of disruptive viruses, a number of virus-spreading models, which are referred to as the Susceptible-Latent-Bursting-Susceptible (SLBS) models, have been suggested [34–38]. The main distinction between the SLBS models and the traditional SEIS models lies in that the latent hosts in the former possess strong infecting capability, whereas the exposed individuals in the latter possess no infecting capability at all. Recently, the basic SLBS models have been extended towards different directions [39–43]. At the population-level, Chen et al. [44] developed an optimal dynamic control strategy of disruptive viruses.

All of the above-mentioned SLBS models are population-level; that is, they are based on the assumption that every infected host in the population is equally likely to infect any other susceptible host. These models have two striking defects: (a) the personalized features of different hosts cannot be taken into consideration and (b) the impact of the structure of the virus-propagating network on the viral prevalence cannot be revealed by studying the models. To overcome these defects, Yang et al. [45] presented a node-level SLBS model. In our opinion, optimal dynamic control strategies of disruptive viruses should be developed at the node-level, so as to achieve the best cost-efficiency.

This paper is intended to develop at the node-level an optimal dynamic control strategy of disruptive computer viruses. First, the development problem is modeled as an optimal control problem. Second, a criterion for the existence of an optimal control for the optimal control problem is given. Third, the optimality system for the optimal control problem is presented. Next, some exemplar optimal dynamic control strategies are given. Finally, the difference between the cost-efficiency of an arbitrary control strategy and that of the optimal dynamic strategy is estimated.

The subsequent materials of this work are organized as follows. Section 2 presents the preliminary knowledge on optimal control theory. Sections 3 and 4 formulate and study the optimal control problem, respectively. Some numerical examples are given in Section 5. Section 6 estimates the aforementioned difference. Finally, Section 7 closes this work.

2. Fundamental Knowledge

For fundamental knowledge on optimal control theory, see [46].

Consider the following optimal control problem.

Lemma 1. Problem has an optimal control if the following five conditions hold simultaneously. (C₁) is closed and convex.(C₂) There is such that the adjunctive dynamical system is solvable.(C₃) is bounded by a linear function in .(C₄) is convex on .(C₅) for some vector norm , , and .

3. Formulation of the Optimal Control Problem

Consider a population of hosts (nodes) labelled . As with the traditional SLBS models, assume that at any time every node in the population is in one of three possible states: susceptible, latent, and disruptive. Susceptible nodes are those that are not infected with any disruptive computer virus. Latent nodes are those that are infected with some disruptive viruses and all of them are in the latent phase. Disruptive nodes are those that are infected with some disruptive viruses and some of them are in the disruptive phase. Let , 1, and 2 denote that at time node is susceptible, latent, and disruptive, respectively. LetAs (), the vector probabilistically captures the state of the population at time .

Suppose a dynamic control strategy will be carried out during the time frame . Let us impose a set of statistical hypotheses as follows.(H₁)A susceptible node is infected by a latent node at rate . Let .(H₂)A susceptible node is infected by a disruptive node at rate . Let .(H₃)Due to the outburst of latent viruses, a latent node becomes disruptive at rate . Let .(H₄)Due to the action of new patches, at time a latent node becomes susceptible at a controllable rate and . Hereafter, the symbol stands for the set of all Lebesgue square integrable functions defined on the interval . Moreover, the cost needed to achieve the rate at the infinitesimal time interval is , , and . This accords with the intuition that the cost increases with .(H₅)Due to the action of new patches, at time a disruptive node becomes susceptible at a controllable rate and . Moreover, the cost needed to achieve the rate at the infinitesimal time interval is , . This conforms to the intuition that the cost increases with .

Figure 1 shows hypotheses (H₁)–(H₅) schematically.

Let denote a very small time interval. Hypotheses (H₁)–(H₅) imply the following relations.As a result, we haveBy the total probability formula, we getTransposing the terms and from the right to the left and dividing both sides by , we get Letting , we get the following dynamical model.where , . We refer to the model as the controlled SLBS model, where the control, stands for a dynamic control strategy of disruptive computer viruses. The admissible set of controls is

Model (7) can be written in matrix notation as

Given a dynamic control strategy . The total loss can be measured by , and the total cost can be gauged by . As a result, the performance of a dynamic control strategy can be measured by Hence, developing an optimal dynamic control strategy of disruptive viruses can be modeled as solving the following optimal control problem.

A solution to the optimal control problem stands for an optimal dynamic control strategy of disruptive viruses. For convenience, let

4. A Theoretical Study of the Optimal Control Problem

In this section, we shall study the optimal control problem presented in the previous section.

4.1. Existence of an Optimal Control

As a solution to the optimal control problem stands for an optimal dynamic control strategy of disruptive viruses, it is critical to show that there is such an optimal control. For that purpose, let us show that the five conditions in Lemma 1 hold true simultaneously.

Lemma 2. The admissible set is closed.

Proof. Let be a limit point of , a sequence of points in such that The completeness of implies . Hence, the claim follows from the observation that

Lemma 3. The admissible set is convex.

Proof. Letand . As is a real vector space, we get So, the claim follows from the observation that

Lemma 4. There is such that model (7) is solvable.

Proof. Substituting into model (7), we getAs is continuously differentiable, the claim follows from the Continuation Theorem for Differential Systems [47].

Lemma 5. is bounded by a linear function in .

Proof. The claim follows from the observation that, for ,

Lemma 6. is convex on if .

Proof. The Hessian of with respect to , is always positive semidefinite. This implies the convexity of .

Lemma 7. , where stands for the -norm of vectors.

Proof. We have

We are ready to present the main result of this subsection.

Theorem 8. Problem has an optimal control if .

Proof. Lemmas 2–7 show that the five conditions in Lemma 1 are all met. Hence, the existence of an optimal control follows from Lemma 1.

4.2. The Optimality System

As the optimality system for the optimal control problem offers a method for numerically solving the problem, it is critical to determine the optimality system. For that purpose, consider the corresponding Hamiltonian where is the adjoint.

Theorem 9. Suppose is an optimal control for problem with ; is the solution to the controlled SLBS model with . Then, there exists with such that where and .

Proof. According to the Pontryagin Minimum Principle [26], there exists such that Thus, the first equations in the claim follow by direct calculations. As the terminal cost is unspecified and the final state is free, the transversality condition holds. By using the optimality condition we get (a) either or or and (b) either or or . So, the last equations in the claim follow.
By combining the above discussions, we get the optimality system for problem with as follows. where , , , .
By applying the forward-backward Euler scheme to the optimality system, we can obtain the numerical solution to the optimal control problem , that is, an optimal dynamic control strategy of disruptive viruses.

5. Numerical Examples

This section gives some examples of the optimal dynamic control strategy of disruptive computer viruses. Given a dynamic control strategy . Define the average control (AC) function, the average cumulative loss (ACL) function, the average cumulative cost (ACC) function, and the average cumulative performance (ACP) function as follows. These functions form an evaluation criterion of dynamic control strategies of disruptive viruses.

5.1. Scale-Free Network

Scale-free networks are a large class of networks having widespread applications. For our purpose, generate a scale-free network with nodes using the Barabasi-Albert method [48].

Example 10. Consider an optimal control problem on the virus-spreading network , where the parameters and the initial conditions are set as follows. (a), , , , , and .(b) and , .(c) and , .(d) and , .For the optimal dynamic control strategy to the optimal control problem and some static control strategies, the AC functions, the ACL functions, the ACC functions, and the ACP function are shown in Figure 2.

(a)

(b)

(c)

(d)

5.2. Small-World Network

Small-world networks are another large class of networks having widespread applications. For our purpose, generate a small-world network with nodes using the Watts-Strogatz method [49].

Example 11. Consider an optimal control problem on the virus-spreading network , where the parameters and the initial conditions are set as follows. (a), , , , , and .(b) and , .(c) and , .(d) and , .For the optimal dynamic control strategy to the optimal control problem and some static control strategies, the AC functions, the ACL functions, the ACC functions, and the ACP function are shown in Figure 3.

(a)

(b)

(c)

(d)

5.3. Realistic Network

Consider a network with nodes cut out from the database of Stanford University [50].

Example 12. Consider an optimal control problem on the virus-spreading network , where the parameters and the initial conditions are set as follows. (a), , , , , and .(b) and , .(c) and , .(d) and , .For the optimal dynamic control strategy to the optimal control problem and some static control strategies, the AC functions, the ACL functions, the ACC functions, and the ACP function are shown in Figure 4.

(a)

(b)

(c)

(d)

6. Performance Evaluation

The previous discussions manifest that if the parameters in the optimal control problem are all available, then an optimal dynamic control strategy can be obtained by numerically solving the optimality system. In realistic scenarios, however, some of these parameters might be unavailable. In such situations, it is necessary to estimate the performance of an actual dynamical control strategy in comparison with that of the optimal dynamical control strategy. Now let us present such an estimation.

Theorem 13. Consider the optimal control problem . Let be the optimal dynamic control strategy, an arbitrary dynamic control strategy. Then,where

Proof. Let denote the -norm. Let denote the solution to the SLBS model with control and the solution to the SLBS model with control . As we get So,As we get Thus, As , we get Applying the Gronwall inequality [47], we get Hence, we deduce that

Although this estimation is rough, it takes the first step towards the accurate performance evaluation of actual dynamic control strategies of disruptive computer viruses.

7. Conclusions and Remarks

This paper has studied the problem of containing disruptive computer viruses in a cost-effective way. The problem has been modeled as an optimal control problem. A criterion for the existence of an optimal control has been given, and the optimality system has been derived. Some examples of the optimal dynamic control strategy have been presented. Finally, the performance of an actual control strategy of disruptive viruses has been estimated.

Towards this direction, there are a number of problems that are worth studying. First, the bandwidth resources consumed in the virus control process should be measured and incorporated in the cost. Second, the optimal dynamic control problem should be investigated under sophisticated epidemic models such as the impulsive epidemic models [51, 52], the stochastic epidemic models [53–55], and the epidemic models on time-varying networks [56–58]. Last, it is rewarding to apply the methodology developed in this paper to the optimal dynamic control of rumor spreading [59–61].

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This work is supported by Natural Science Foundation of China (Grant nos. 61572006, 71301177), National Sci-Tech Support Plan (Grant no. 2015BAF05B03), Basic and Advanced Research Program of Chongqing (Grant no. cstc2013jcyjA1658), and Fundamental Research Funds for the Central Universities (Grant no. 106112014CDJZR008823).

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Copyright © 2017 Jichao Bi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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