Qualitative analysis for a delayed epidemic model with latent and breakingout over the Internet
 Zizhen Zhang^{1}Email author and
 Yougang Wang^{1}
https://doi.org/10.1186/s1366201710749
© The Author(s) 2017
Received: 9 October 2016
Accepted: 22 December 2016
Published: 27 January 2017
Abstract
We generalize a delayed computer virus model, known as the SLBQRS model in a computer network, by introducing the time delay due to the period that the antivirus software uses to clean viruses in the breaking out computers and the quarantined computers. By choosing the delay as the parameter, we prove the existence of a Hopf bifurcation as the delay crosses a critical value. Moreover, we study properties of the Hopf bifurcation by applying the center manifold theorem and the normal form theory. Finally, we carry out numerical simulations to support the obtained theoretical conclusions.
Keywords
1 Introduction
With the rapid development of the communication technology and network applications, the Internet has become an important platform for people sharing news and ideas [1]. Meanwhile, the Internet has become a powerful mechanism for propagating malicious computer virus programs such as worms, Trojans horses, and so on. What is more serious, enormous financial losses and social panic have also been caused by these computer viruses [2]. Therefore, it is urgent to understand the behavior of computer viruses and to pose effective measures of controlling their spread across the Internet.
The subsequent materials of this paper are organized as follows. In Section 2, we prove the local stability of the viral equilibrium and the existence of a Hopf bifurcation. Section 3 is devoted to investigations in direction of the Hopf bifurcation and stability of the bifurcating periodic solutions. Then, we validate the obtained results by using simulations in Section 4. We summarize this work in Section 5.
2 Stability of viral equilibrium and existence of Hopf bifurcation
The discussion of the distribution of positive real roots of equation (13) is similar to that in [20]. Obviously, if \(e_{0}<0\), then equation (13) has at least one positive root. In the following, we discuss the distribution of the roots of equation (13) as \(e_{0}\geq 0\).
Lemma 1
 (i)
If \(\triangle_{0}<0\), then equation (13) has no positive real roots;
 (ii)
If \(\triangle_{0}\geq 0\), \(c_{2}\geq 0\), and \(c_{0}>0\), then equation (13) has no positive real roots;
 (iii)
If (i) and (ii) are not satisfied, then equation (13) has positive real roots if and only if there exists at least one \(v^{*}\in \{v_{1}, v_{2}, v_{3}, v_{4}\}\) such that \(v^{*}>0\) and \(h(v^{*})\leq 0\).
Lemma 2
 (i)
If \(\triangle_{2}<0\) and \(\triangle_{3}<0\), then equation (13) has no positive real roots;
 (ii)
If (i) is not satisfied, then equation (13) has positive real roots if and only if there exists at least one \(v^{*}\in \{v_{1}, v_{2}, v_{3}, v_{4}\}\) such that \(v^{*}>0\) and \(h(v^{*})\leq 0\).
Lemma 3
Suppose that \(e_{0}\geq 0\), \(c_{1}\neq 0\), and \(s_{*}< c_{2}\). Then equation (13) has positive real roots if and only if \(\frac{c_{1}^{2}}{4(c_{2}s_{*})^{2}}+\frac{s_{*}}{2}=0\) and \(\bar{v}>0\), \(h(\bar{v})\leq 0\).
 \((H_{2})\) :

(a) \(e_{0}<0\); (b) \(e_{0}\geq 0\), \(c_{1}=0\), and \(c_{2}<0\) or \(c_{0}>0\), and there exists at least one \(v^{*}\in \{v_{1}, v_{2}, v _{3}, v_{4}\}\) such that \(v^{*}>0\) and \(h(v^{*})\leq 0\); (c) \(e_{0}\geq 0\), \(c_{1}\neq 0\), \(s_{*}>c_{2}\), \(\triangle_{2}\geq 0\), or \(\triangle_{3}\geq 0\), and there exists at least one \(v^{*}\in \{v _{1}, v_{2}, v_{3}, v_{4}\}\) such that \(v^{*}>0\) and \(h(v^{*})\leq 0\); (d) \(e_{0}\geq 0\), \(c_{1}\neq 0\), \(s_{*}< c_{2}\), \(\frac{c_{1}^{2}}{4(c _{2}s_{*})^{2}}+\frac{s_{*}}{2}=0\), and \(\bar{v}>0\), \(h(\bar{v}) \leq 0\).
 \((H_{3})\) :

\(h^{\prime }(\omega_{0}^{2})\neq 0\)
Theorem 1
For system (2), if conditions \((H_{1})\)\((H_{3})\) hold, then the viral equilibrium \(E_{*}(S_{*}, L_{*}, B_{*}, Q_{*}, R_{*})\) is asymptotically stable for \(\tau \in [0, \tau_{0})\). A Hopf bifurcation occurs at the viral equilibrium \(E_{*}(S_{*}, L_{*}, B _{*}, Q_{*}, R_{*})\) when \(\tau =\tau_{0}\), and a family of periodic solutions bifurcate from the viral equilibrium \(E_{*}(S_{*}, L_{*}, B _{*}, Q_{*}, R_{*})\) near \(\tau =\tau_{0}\).
3 Stability of the bifurcating periodic solutions
Then, it is easy to see that \(A(0)\) and \(A^{*}(0)\) are adjoint operators and that \(\pm i\omega_{0}\) are the eigenvalues of \(A(0)\) and also the eigenvalues of \(A^{*}(0)\).
Theorem 2
 (i)
The Hopf bifurcation at the viral equilibrium \(E_{*}(S_{*}, L_{*}, B_{*}, Q_{*}, R_{*})\) is supercritical if \(\mu_{2}>0\) and subcritical if \(\mu_{2}<0\);
 (ii)
The bifurcating periodic solutions are stable if \(\beta_{2}<0\) and unstable if \(\beta_{2}>0\);
 (iii)
The period of the bifurcating periodic solutions increases if \(T_{2}>0\) and decreases if \(T_{2}<0\).
4 Numerical simulations
Further, we obtain \(\lambda^{\prime }(\tau_{0})=0.53862.2362i\) and \(C_{1}(0)=0.94667.6058i\) by some complex computations. Then, we can get \(\beta_{2}=1.8932<0\), \(\mu_{2}=1.7575>0\), and \(T_{2}=0.3931>0\). According to Theorem 2, we can conclude that the Hopf bifurcation is supercritical, the bifurcating periodic solutions are stable, and the period of the bifurcating periodic solutions increases. Since the bifurcating periodic solutions are stable, we know that the five kinds of computers in system (32) may coexist in an oscillatory mode from the viewpoint of biology. In this case, it is difficult to control the propagation of computer virus described in system (32).
5 Conclusions
A delayed SLBQRS computer virus model is proposed based on the model studied in [17]. Compared with the model studied in [17], the model in this paper incorporates the time delay due to the period that the antivirus software uses to clean the viruses in the breaking out computers and the quarantined computers. So, our delayed SLBQRS computer virus model is more general.
We mainly investigate the effect of the time delay on the model. We prove that there exists a critical value \(\tau_{0}\) of the time delay below which the model is stable and above which the model will lose its stability. Practically speaking, propagation of the computer viruses can be controlled easily when the model is stable. However, it will be out of control when the model is unstable. That is, the time delay in system (2) can affect propagation of the computer viruses. Furthermore, properties of the Hopf bifurcation at the critical value \(\tau_{0}\) are also investigated.
It should be pointed out that the delayed model in this paper only considers the time delay due to the period that the antivirus software uses to clean the viruses. There are also some other types of delay in system (2) such as latent delay and delay due to a temporary immunity period of the recovered computers. We will investigate dynamics of system (2) with multiple delays in the near future.
Declarations
Acknowledgements
The authors would like to thank the editor and anonymous referees for their constructive suggestions on improving the presentation of the paper. This work was supported by Natural Science Foundation of the Higher Education Institutions of Anhui Province (No. KJ2015A144) and Anhui Provincial Natural Science Foundation (Nos. 1608085QF151, 1608085QF145).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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