Discrete Dynamics in Nature and Society
Volume 2016 (2016), Article ID 5649584, 21 pages
http://dx.doi.org/10.1155/2016/5649584

Research Article

Dynamical Analysis of a Computer Virus Model with Delays

Juan Liu,¹ Carlo Bianca,² and Luca Guerrini³

¹Department of Mathematics and Physics, Bengbu University, Bengbu 233030, China
²Laboratoire de Physique Statistique, Ecole Normale Supérieure, PSL Research University, Université Paris Diderot Sorbonne Paris-Cité, Sorbonne Universités, UPMC Univ Paris 06, CNRS, 24 rue Lhomond, 75005 Paris, France
³Department of Management, Polytechnic University of Marche, 60121 Ancona, Italy

Received 5 August 2016; Accepted 29 September 2016

Academic Editor: Vincenzo Scalzo

Copyright © 2016 Juan Liu 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.

Abstract.

An SIQR computer virus model with two delays is investigated in the present paper. The linear stability conditions are obtained by using characteristic root method and the developed asymptotic analysis shows the onset of a Hopf bifurcation occurs when the delay parameter reaches a critical value. Moreover the direction of the Hopf bifurcation and stability of the bifurcating period solutions are investigated by using the normal form theory and the center manifold theorem. Finally, numerical investigations are carried out to show the feasibility of the theoretical results.

1. Introduction

The advances of information technology and the wide-spread popularity of computer networks have increased the interest in the computer viruses. In the past decades, many epidemic models such as SIR model [1, 2], SIRS model [3, 4], SEIR model [5–7], SEIRS model [8] and SEIQRS model [9] characterizing the spread of computer viruses in networks were investigated by many scholars. It is worthwhile to note that the dynamical models above neglect the time delay in the spreading process of computer viruses. To our knowledge, there have been some computer virus models with time delay proposed to depict the spread of a computer virus. In [10, 11], Muroya et al. studied the global stability of a delayed SIRS computer virus propagation model, respectively. In [12], Feng et al. investigated the Hopf bifurcation of a delayed SIRS viral infection model in computer networks by regarding the delay due to temporary immune period of the recovered computers as a bifurcation parameter. In [13], Dong et al. analyzed the Hopf bifurcation of a delayed SEIR computer virus model with multistate antivirus by regarding the time delay due to the period that the computers use antivirus software to clean the viruses as the bifurcation parameter. In [14], Zhang and Yang studied the Hopf bifurcation of the following SIQR computer virus model with time delay:

where , , , and denote the numbers of nodes in states susceptible, infectious, quarantined, and recovered at time , respectively. , , , , , , , and are the parameters of system (1) and is the time delay due to the latent period of the computer virus. Zhang and Yang obtained the sufficient conditions for the local stability and existence of local Hopf bifurcation by regarding the delay as a bifurcation parameter and investigated the properties of the Hopf bifurcation by using the normal form method and center manifold theory. As stated in [13], it usually takes a period to clean the viruses in the computers infected by viruses for antivirus software. Therefore, it is reasonable to incorporate the time delay due to the period that antivirus software uses to clean viruses in the infectious and the quarantined computers into system (1). Bearing all above in mind, this paper deals with the analysis of the Hopf bifurcation of the following system with two delays:

where is the time delay due to the latent period of the computer virus and is the time delay due to the period that the antivirus software uses to clean the computer viruses in the infectious and the quarantined nodes.

This paper mainly investigates the effect of the two delays on system (2). The remainder of this paper is organized as follows. The local stability of the positive equilibrium and existence of local Hopf bifurcation are analyzed in Section 2 by choosing different combination of the two delays as a bifurcation parameter. Direction of the Hopf bifurcation, stability, and period of the bifurcating periodic solutions on the center manifold are determined in Section 3. Some numerical simulations are presented to illustrate the validity of the main results in Section 4. Finally a critical analysis and further research directions are contained in Section 5.

2. Local Stability of the Positive Equilibrium and Existence of Local Hopf Bifurcation

According to the analysis in [14] we know that if , system (2) has a unique positive equilibrium , where

The characteristic equation of system (2) at is

from which we obtain

where

with

Case 1 (). When , (5) becomes

Thus, if the condition (9) holds, then the positive equilibrium of system (2) is locally asymptotically stable when .

Case 2 (, ). When , , (5) becomes

where

Let be the root of (10); we get

from which one can obtain

where

Let ; then (13) becomes

Obviously, if all the coefficients of system (2) are given, then we can obtain all the roots of (15) by Matlab software package easily. Thus, we make the following assumption.
(15) has at least one positive root.
If the condition holds, then there exists a positive root for (15) such that (10) has a pair of purely imaginary roots . For ,

where

Substituting into the left side of (10) and taking the derivative with respect to , one can obtain

Thus,

where and .
Thus, if the condition holds, then . According to the Hopf bifurcation theorem in [15], we have the following results.

Theorem 1. If the conditions - hold, then the positive equilibrium is asymptotically stable for and system (2) undergoes a Hopf bifurcation at the positive equilibrium when .

Case 3 (, ). When , , (5) becomes

where

Multiplying by , (20) becomes

Let () be the root of (22); then

It follows that

where

Then, we can get

with

Let ; (26) becomes

Similar as in Case 2, we make the following assumption.
(28) has at least one positive root.

If condition holds, there exists a positive root for (22) such that (22) has a pair of purely imaginary root . For ,

Differentiating the two sides of (22) regarding , we obtain

where

where

Therefore, if condition () holds, then Re. Thus, we have the following results according to the Hopf bifurcation theorem in [15].

Theorem 2. If conditions - hold, then the positive equilibrium is asymptotically stable for and system (2) undergoes a Hopf bifurcation at the positive equilibrium when .

Case 4 (, and ). Let be the root of (5); then we obtain

where

Then, we can obtain the following equation with respect to :

where

In order to give the main results in this paper, we make the following assumption.
(35) has at least one positive root.
If the condition holds, then there exists a positive root for (35) such that (5) has a pair of purely imaginary root . For ,

Substituting into (5) and differentiating both sides of it with respect to , then

where

We therefore derive that

where

Therefore, if condition holds, then . Thus, we have the following results according to the Hopf bifurcation theorem in [15].

Theorem 3. For system (2), if conditions - hold and , then the positive equilibrium is asymptotically stable for and system (2) undergoes a Hopf bifurcation at the positive equilibrium when and a family of periodic solutions bifurcate from near .

Case 5 (, and ). Multiplying by , (5) becomes

Let be the root of (42); then we get

where

Then, we get

Thus, we can obtain the following function with respect to :

Suppose that (46) has at least one positive root.
If condition holds, there exists a positive root of (46) such that (46) has a pair of purely imaginary roots . For ,

Taking the derivative with respect to in (42), we can obtain

where

Define

Obviously, if the condition , then . Thus, we have the following results according to the Hopf bifurcation theorem in [15].

Theorem 4. For system (2), if the conditions - hold and , then the positive equilibrium is asymptotically stable for and system (2) undergoes a Hopf bifurcation at the positive equilibrium when and a family of periodic solutions bifurcate from near .

3. Properties of the Hopf Bifurcation

In this section, we investigate the direction of the Hopf bifurcation and the stability of the Hopf bifurcation of system (2) when and by using the normal form theory and the center manifold theorem in [15]. Throughout this section, we assume that .

Let , , , , and , and normalize the time delay by . Then system (1) can be transformed into as

where

By the Riesz representation theorem, there is a matrix function with bounded variation components , such that

In fact, we choose

For , we define

Then system (51) is equivalent to the following operator equation:

Next, we define the adjoint operator of

and a bilinear inner product

where .

Let be the eigenvectors of corresponding to and be the eigenvectors of corresponding to . Then, we obtain

From (58), we can obtain

such that , .

Following the algorithms given in [15] and using similar computation process in [16], we can get the coefficients which determine the properties of the Hopf bifurcation:

with

where and can be determined by the following equations, respectively,

with

Then, we can get the following coefficients:

in which determines the direction of the Hopf bifurcation; determines the stability of the bifurcation periodic solutions and determines the period of the bifurcation periodic solutions. From the conclusion of [15], we have the main results in this section.

Theorem 5. For system (2), determines the direction of the Hopf bifurcation: if , the Hopf bifurcation is supercritical (subcritical). determines the stability of the bifurcating periodic solution: if the bifurcating periodic solutions are stable (unstable). determines the period of the bifurcating periodic solution: if , the period of the bifurcating periodic solutions increases (decreases).

4. Numerical Investigations

In this section, a numerical example is given to support the theoretical results in Sections 2 and 3. We choose the same values of the parameters as in [14] and let , , , , , , , , and . Then, we get the following system:

Then, we can get and the unique positive equilibrium of system (66) and condition holds.

For , , we obtain , . By Theorem 1, we can conclude that the positive equilibrium is locally asymptotically stable when . In this case, the propagation of computer viruses can be predicted and controlled easily. This property can be illustrated by Figures 1–3. However, if we choose , the positive equilibrium becomes unstable and a Hopf bifurcation occurs, which can be illustrated by Figures 4–6. When a Hopf bifurcation occurs, the state of computer viruses propagation changes from an equilibrium point to a limit cycle and it means that the propagation of computer viruses is out of control. Similarly, we have , when , . The corresponding waveforms and phase plots are shown as in Figures 7–12.

Figure 1: The trajectory of , , , and when .

Figure 2: The phase plot of , , and when .

Figure 3: The phase plot of , , and when .

Figure 4: The trajectory of , , , and when .

Figure 5: The phase plot of , , and when .

Figure 6: The phase plot of , , and when .

Figure 7: The trajectory of , , , and when .

Figure 8: The phase plot of , , and when .

Figure 9: The phase plot of , , and when .

Figure 10: The trajectory of , , , and when .

Figure 11: The phase plot of , , and when .

Figure 12: The phase plot of , , and when .

For , and . We can obtain and then we obtain . Figures 13–15 show that the positive equilibrium of system (66) is asymptotically stable when and Figures 16–18 show that there is a Hopf bifurcation occurs at the positive equilibrium of system (66) and a family of periodic solutions bifurcate from when . Similarly, we have and when , . The corresponding waveforms and plots are shown in Figures 19–24. In addition, we can obtain , , . Thus, we can conclude that the Hopf bifurcation is supercritical, the bifurcating periodic solutions are stable, and the period of the periodic solutions increases according to Theorem 5.

Figure 13: The trajectory of , , , and when and .

Figure 14: The phase plot of , , and when and .

Figure 15: The phase plot of , , and when and .

Figure 16: The trajectory of , , , and when and .

Figure 17: The phase plot of , , and when and .

Figure 18: The phase plot of , , and when and .

Figure 19: The trajectory of , , , and when and .

Figure 20: The phase plot of , , and when and .

Figure 21: The phase plot of , , and when and .

Figure 22: The trajectory of , , , and when and .

Figure 23: The phase plot of , , and when and .

Figure 24: The phase plot of , , and when and .

5. Critical Analysis and Research Perspectives

In this paper, an SIQR computer virus model with two delays is investigated. Compared with the system considered in [14], the system in this paper is more general because it accounts for not only the time delay due to the latent period of the computer virus but also the time delay due to the period that the antivirus software uses to clean the computer viruses in the infectious and the quarantined nodes. The sufficient conditions for the stability of the positive equilibrium and existence of the Hopf bifurcation for the possible combinations of two delays are obtained. When the conditions are satisfied, then there exists a critical value of the delay below which system (2) is locally asymptotically stable and above which system (2) is unstable. The direction and the stability of the bifurcating periodic solutions are determined by applying the normal theory and the center manifold theorem. Numerical simulations show that the computer viruses may be controlled by shortening the time delay due to the latent period of the computer viruses and the time delay due to the period that the antivirus software uses to clean the computer viruses in the infectious and the quarantined nodes.

However, it should be pointed out that we suppose that the recovered computers have a permanent immunization period and they can no longer be infected. This is not consistent with real situation. In order to overcome this limitation and considering that the recovered computers may be infected again after a temporary immunity period, it is definitely an interesting work to investigate the following more general SIQRS model with multiple delays:

where is the transition rate from the recovered computers to the susceptible computers. is the temporary immunity period after which a recovered computer may be infected again. We leave the analysis of the more complicated bifurcations of system (67) as the future work.

Further research directions include the possibility of linking the results obtained with the model proposed in the present paper with the results coming from the networks theory; see the recent review paper [17]. Specifically, the interest focuses on the possibility to gain a deep understanding of the impact of the network topology on the viral prevalence; see [18, 19]. In this context, the model proposed in this paper, which is a coarse-grained compartment-level model, can be considered as a network where all nodes are classified into a few compartments according to their states and whose degree distribution takes into account the rule of interactions proposed in the present paper.

Competing Interests

The authors declare that there are no competing interests regarding the publication of this paper.

Acknowledgments

This work was supported by Natural Science Foundation of the Higher Education Institutions of Anhui Province (KJ2015A144).

References

J. R. C. Piqueira and V. O. Araujo, “A modified epidemiological model for computer viruses,” Applied Mathematics and Computation, vol. 213, no. 2, pp. 355–360, 2009.

J. G. Ren, X. F. Yang, Q. Y. Zhu, L.-X. Yang, and C. M. Zhang, “A novel computer virus model and its dynamics,” Nonlinear Analysis: Real World Applications, vol. 13, no. 1, pp. 376–384, 2012.

C. Gan, X. Yang, W. Liu, Q. Zhu, and X. Zhang, “Propagation of computer virus under human intervention: a dynamical model,” Discrete Dynamics in Nature and Society, vol. 2012, Article ID 106950, 8 pages, 2012.

C. Gan, X. Yang, and Q. Zhu, “Propagation of computer virus under the influences of infected external computers and removable storage media: modeling and analysis,” Nonlinear Dynamics, vol. 78, no. 2, pp. 1349–1356, 2014.

H. Yuan and G. Q. Chen, “Network virus-epidemic model with the point-to-group information propagation,” Applied Mathematics and Computation, vol. 206, no. 1, pp. 357–367, 2008.

H. Yuan, G. Chen, J. Wu, and H. Xiong, “Towards controlling virus propagation in information systems with point-to-group information sharing,” Decision Support Systems, vol. 48, no. 1, pp. 57–68, 2009.

M. Peng, X. He, J. Huang, and T. Dong, “Modeling computer virus and its dynamics,” Mathematical Problems in Engineering, vol. 2013, Article ID 842614, 5 pages, 2013.

B. K. Mishra and S. K. Pandey, “Dynamic model of worms with vertical transmission in computer network,” Applied Mathematics and Computation, vol. 217, no. 21, pp. 8438–8446, 2011.

B. K. Mishra and N. Jha, “SEIQRS model for the transmission of malicious objects in computer network,” Applied Mathematical Modelling, vol. 34, no. 3, pp. 710–715, 2010.

Y. Muroya, Y. Enatsu, and H. X. Li, “Global stability of a delayed SIRS computer virus propagation model,” International Journal of Computer Mathematics, vol. 91, no. 3, pp. 347–367, 2014.

J. Ren, X. Yang, L.-X. Yang, Y. Xu, and F. Yang, “A delayed computer virus propagation model and its dynamics,” Chaos, Solitons & Fractals, vol. 45, no. 1, pp. 74–79, 2012.

L. P. Feng, X. F. Liao, H. Q. Li, and Q. Han, “Hopf bifurcation analysis of a delayed viral infection model in computer networks,” Mathematical and Computer Modelling, vol. 56, no. 7-8, pp. 167–179, 2012.

T. Dong, X. Liao, and H. Li, “Stability and Hopf bifurcation in a computer virus model with multistate antivirus,” Abstract and Applied Analysis, vol. 2012, Article ID 841987, 16 pages, 2012.

Z. Z. Zhang and H. Z. Yang, “Hopf bifurcation of an SIQR computer virus model with time delay,” Discrete Dynamics in Nature and Society, vol. 2015, Article ID 101874, 8 pages, 2015.

B. D. Hassard, N. D. Kazarinoff, and Y. H. Wan, Theory and applications of Hopf bifurcation, vol. 41, Cambridge University Press, Cambridge, Uk, 1981.

M. Ferrara, L. Guerrini, and G. M. Bisci, “Center manifold reduction and perturbation method in a delayed model with a mound-shaped Cobb-Douglas production function,” Abstract and Applied Analysis, vol. 2013, Article ID 738460, 6 pages, 2013.

R. Pastor-Satorras, C. Castellano, P. Van Mieghem, and A. Vespignani, “Epidemic processes in complex networks,” Reviews of Modern Physics, vol. 87, no. 3, pp. 925–979, 2015.

L.-X. Yang, M. Draief, and X. Yang, “The impact of the network topology on the viral prevalence: a node-based approach,” PLoS ONE, vol. 10, no. 7, article e0134507, 2015.

L. X. Yang, M. Draief, and X. Yang, “Heterogeneous virus propagation in networks: a theoretical study,” Mathematical Methods in the Applied Sciences, 2016.