Bifurcation Analysis for an SEIRS-V Model with Delays on the Transmission of Worms in a Wireless Sensor Network

Hindawi Mathematical Problems in Engineering Volume 2017, Article ID 9898726, 15 pages https://doi.org/10.1155/2017/9898726 Research Article Bifurcation Analysis for an SEIRS-V Model with Delays on the Transmission of Worms in a Wireless Sensor Network Zizhen Zhang and Yougang Wang School of Management Science and Engineering, Anhui University of Finance and Economics, Bengbu 233030, China Correspondence should be addressed to Zizhen Zhang; Received 10 February 2016; Accepted 23 May 2016; Published 18 January 2017 Academic Editor: Vladimir Turetsky Copyright © 2017 Z. Zhang and Y. Wang. 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. Hopf bifurcation for an SEIRS-V model with delays on the transmission of worms in a wireless sensor network is investigated. We focus on existence of the Hopf bifurcation by regarding the diverse delay as a bifurcation parameter. The results show that propagation of worms in the wireless sensor network can be controlled when the delay is suitably small under some certain conditions. Then, we study properties of the Hopf bifurcation by using the normal form theory and center manifold theorem. Finally, we give a numerical example to support the theoretical results. 1. Introduction In recent years, wireless sensor networks have received extensive attention due to their vast potential in many application environments. However, security of wireless sensor networks still remains one of the most critical challenges because sensor nodes are often placed in a hostile or dangerous environment [1]. Many epidemiological models [2–6] have been proposed to study and predict the spread of viruses in wireless networks motivated by the pioneering work of Murray [7] and Kephart and White [8, 9]. In [10], Mishra and Keshri proposed the following SEIRS-V model to describe the propagation of worms in a wireless sensor network: 𝑑𝑆 (𝑡) = 𝐴 − 𝛽𝑆 (𝑡) 𝐼 (𝑡) − (𝜁 + 𝑝) 𝑆 (𝑡) + 𝛿𝑅 (𝑡) 𝑑𝑡 + 𝜂𝑉 (𝑡) , 𝑑𝐸 (𝑡) = 𝛽𝑆 (𝑡) 𝐼 (𝑡) − (𝜁 + 𝛼) 𝐸 (𝑡) , 𝑑𝑡 𝑑𝐼 (𝑡) = 𝛼𝐸 (𝑡) − (𝜁 + 𝜀 + 𝛾) 𝐼 (𝑡) , 𝑑𝑡 𝑑𝑅 (𝑡) = 𝛾𝐼 (𝑡) − (𝜁 + 𝛿) 𝑅 (𝑡) , 𝑑𝑡 𝑑𝑉 (𝑡) = 𝑝𝑆 (𝑡) − (𝜁 + 𝜂) 𝑉 (𝑡) , 𝑑𝑡 (1) where 𝑆(𝑡), 𝐸(𝑡), 𝐼(𝑡), 𝑅(𝑡), and 𝑉(𝑡) denote the number of susceptible, exposed (infected, but not infectious), infectious, recovered, and vaccinated sensor nodes at time 𝑡, respectively. 𝐴, 𝑝, 𝛼, 𝛽, 𝛾, 𝛿, 𝜂, 𝜀, and 𝜁 are the positive parameters of system (1) and for the specific meanings of them one can refer to [10]. Considering the time delays in system (1), Zhang and Si [11] proposed the following delayed SEIRS-V system: 𝑑𝑆 (𝑡) = 𝐴 − 𝛽𝑆 (𝑡) 𝐼 (𝑡) − (𝜁 + 𝑝) 𝑆 (𝑡) + 𝛿𝑅 (𝑡 − 𝜏2 ) 𝑑𝑡 + 𝜂𝑉 (𝑡 − 𝜏2 ) , 𝑑𝐸 (𝑡) = 𝛽𝑆 (𝑡) 𝐼 (𝑡) − (𝜁 + 𝛼) 𝐸 (𝑡) , 𝑑𝑡 2 Mathematical Problems in Engineering 𝑑𝐼 (𝑡) = 𝛼𝐸 (𝑡) − (𝜁 + 𝜀) 𝐼 (𝑡) − 𝛾𝐼 (𝑡 − 𝜏1 ) , 𝑑𝑡 bifurcation of a ring of five neurons with delays. In [14], Bianca et al. investigated Hopf bifurcation of an economic growth model with two delays. Considering that there is a latent period of worms in the exposed nodes in system (3), we study the following system with delays: 𝑑𝑅 (𝑡) = 𝛾𝐼 (𝑡 − 𝜏1 ) − 𝜁𝑅 (𝑡) − 𝛿𝑅 (𝑡 − 𝜏2 ) , 𝑑𝑡 𝑑𝑉 (𝑡) = 𝑝𝑆 (𝑡) − 𝜁𝑉 (𝑡) − 𝜂V (𝑡 − 𝜏2 ) , 𝑑𝑡 (2) where 𝜏1 ≥ 0 is the time delay due to the period that antivirus software uses to clean worms in the infected nodes; 𝜏2 ≥ 0 is the time delay due to the temporary immunity period of the recovered and the vaccinated nodes. For the convenience of analysis, Zhang and Si [11] let 𝜏1 = 𝜏2 and considered the following system: 𝑑𝑆 (𝑡) = 𝐴 − 𝛽𝑆 (𝑡) 𝐼 (𝑡) − (𝜁 + 𝑝) 𝑆 (𝑡) + 𝛿𝑅 (𝑡 − 𝜏) 𝑑𝑡 + 𝜂𝑉 (𝑡 − 𝜏) , 𝑑𝐸 (𝑡) = 𝛽𝑆 (𝑡) 𝐼 (𝑡) − (𝜁 + 𝛼) 𝐸 (𝑡) , 𝑑𝑡 𝑑𝐼 (𝑡) = 𝛼𝐸 (𝑡) − (𝜁 + 𝜀) 𝐼 (𝑡) − 𝛾𝐼 (𝑡 − 𝜏) , 𝑑𝑡 (3) 𝑑𝑅 (𝑡) = 𝛾𝐼 (𝑡 − 𝜏) − 𝜁𝑅 (𝑡) − 𝛿𝑅 (𝑡 − 𝜏) , 𝑑𝑡 𝑑𝑉 (𝑡) = 𝑝𝑆 (𝑡) − 𝜁𝑉 (𝑡) − 𝜂𝑉 (𝑡 − 𝜏) . 𝑑𝑡 Zhang and Si [11] investigated existence and properties of the Hopf bifurcation of system (3). It should be pointed out that one of the significant features of worms in networks is its latent characteristic. Therefore, there exists a certain period before the exposed nodes develop themselves into the infectious ones. In addition, as far as we know, there have been some papers that deal with research of Hopf bifurcation of dynamical systems with multiple delays in recent years [12–15]. In [12], Xu et al. studied Hopf 𝑑𝑆 (𝑡) = 𝐴 − 𝛽𝑆 (𝑡) 𝐼 (𝑡) − (𝜁 + 𝑝) 𝑆 (𝑡) + 𝛿𝑅 (𝑡 − 𝜏2 ) 𝑑𝑡 + 𝜂𝑉 (𝑡 − 𝜏2 ) , 𝑑𝐸 (𝑡) = 𝛽𝑆 (𝑡) 𝐼 (𝑡) − 𝛼𝐸 (𝑡 − 𝜏1 ) − 𝜁𝐸 (𝑡) , 𝑑𝑡 𝑑𝐼 (𝑡) = 𝛼𝐸 (𝑡 − 𝜏1 ) − (𝜁 + 𝜀) 𝐼 (𝑡) − 𝛾𝐼 (𝑡 − 𝜏2 ) , 𝑑𝑡 (4) 𝑑𝑅 (𝑡) = 𝛾𝐼 (𝑡 − 𝜏2 ) − 𝜁𝑅 (𝑡) − 𝛿𝑅 (𝑡 − 𝜏2 ) , 𝑑𝑡 𝑑𝑉 (𝑡) = 𝑝𝑆 (𝑡) − 𝜁𝑉 (𝑡) − 𝜂𝑉 (𝑡 − 𝜏2 ) , 𝑑𝑡 where 𝜏1 is the time delay due to the latent period of worms in the exposed nodes and 𝜏2 is the time delay due to the period that the antivirus software uses to clean worms in the infected nodes and that due to the temporary immunity period of the recovered and the vaccinated nodes. The structure of this paper is as follows. In Section 2, we obtain sufficient conditions for local stability of the positive equilibrium and existence of a Hopf bifurcation of system (4). In Section 3, we deal with the properties of the Hopf bifurcation by using the normal form theory and center manifold theorem. Some numerical simulations are carried out in Section 4 with the aim of verifying the obtained analytic results. Finally, conclusions and future work are summarized. 2. Hopf Bifurcation Analysis By a direct computation, we know that if 𝑅0 = (𝛼𝛽𝐴(𝜁 + 𝜂) + 𝑝𝜂(𝜁 + 𝛼)(𝜁 + 𝜀 + 𝛾))/(𝜁 + 𝑝)(𝜁 + 𝛼)(𝜁 + 𝜂)(𝜁 + 𝜀 + 𝛾) > 1, then system (4) has a unique positive equilibrium 𝐷∗ (𝑆∗ , 𝐸∗ , 𝐼∗ , 𝑅∗ , 𝑉∗ ) in which 𝑆∗ = (𝜁 + 𝛼) (𝜁 + 𝜀 + 𝛾) , 𝛼𝛽 𝐸∗ = 𝜁+𝜀+𝛾 𝐼∗ , 𝛼 𝑅∗ = 𝛾 𝐼 , 𝜁+𝛿 ∗ 𝑉∗ = 𝑝 (𝜁 + 𝛼) (𝜁 + 𝜀 + 𝛾) , 𝛼𝛽 (𝜁 + 𝜂) 𝐼∗ = 𝛼𝛽𝐴 (𝜁 + 𝛿) (𝜁 + 𝜂) + 𝑝𝜂 (𝜁 + 𝛼) (𝜁 + 𝛿) (𝜁 + 𝜀 + 𝛾) − (𝜁 + 𝑝) (𝜁 + 𝛼) (𝜁 + 𝛿) (𝜁 + 𝜂) (𝜁 + 𝜀 + 𝛾) . 𝛽 (𝜁 + 𝛼) (𝜁 + 𝛿) (𝜁 + 𝜂) (𝜁 + 𝜀 + 𝛾) − 𝛼𝛽𝛿𝛾 (𝜁 + 𝜂) (5) Mathematical Problems in Engineering 3 𝐶3 = 𝑏33 (𝑎11 + 𝑎22 + 𝑎44 + 𝑎55 ) + 𝑎44 𝑏55 + 𝑎55 𝑏44 The characteristic equation of system (4) at 𝐷∗ is + (𝑎11 + 𝑎22 + 𝑎33 ) (𝑏44 + 𝑏55 ) , 𝜆5 + 𝐴 4 𝜆4 + 𝐴 3 𝜆3 + 𝐴 2 𝜆2 + 𝐴 1 𝜆 + 𝐴 0 + (𝐵4 𝜆4 + 𝐵3 𝜆3 + 𝐵2 𝜆2 + 𝐵1 𝜆 + 𝐵0 ) 𝑒−𝜆𝜏1 𝐶4 = − (𝑏22 + 𝑏44 + 𝑏55 ) , + (𝐶4 𝜆4 + 𝐶3 𝜆3 + 𝐶2 𝜆2 + 𝐶1 𝜆 + 𝐶0 ) 𝑒−𝜆𝜏2 𝐷0 = −𝑎11 𝑎22 (𝑎33 𝑏44 𝑏55 + 𝑎44 𝑏33 𝑏55 + 𝑎55 𝑏33 𝑏44 ) , + (𝐷3 𝜆3 + 𝐷2 𝜆2 + 𝐷1 𝜆 + 𝐷0 ) 𝑒−2𝜆𝜏2 𝐷1 = 𝑏33 𝑏44 (𝑎11 𝑎22 + 𝑎11 𝑎55 + 𝑎22 𝑎55 ) + (𝐸3 𝜆3 + 𝐸2 𝜆2 + 𝐸1 𝜆 + 𝐸0 ) 𝑒−𝜆(𝜏1 +𝜏2 ) + (𝐹2 𝜆2 + 𝐹1 𝜆 + 𝐹0 ) 𝑒−𝜆(𝜏1 +2𝜏2 ) + (𝐺2 𝜆2 + 𝐺1 𝜆 + 𝐺0 ) 𝑒−3𝜆𝜏2 + (𝐻1 𝜆 + 𝐻0 ) 𝑒−𝜆(𝜏1 +3𝜏2 ) = 0, where 𝐴 0 = −𝑎11 𝑎22 𝑎33 𝑎44 𝑎55 , 𝐴 1 = 𝑎11 𝑎22 𝑎33 𝑎44 + 𝑎11 𝑎22 𝑎55 (𝑎33 + 𝑎44 ) + 𝑎33 𝑎44 𝑎55 (𝑎11 + 𝑎22 ) , 𝐴 2 = −𝑎55 (𝑎11 𝑎22 + 𝑎33 𝑎44 + (𝑎11 + 𝑎22 ) (𝑎33 + 𝑎44 )) − (𝑎11 𝑎22 (𝑎33 + 𝑎44 ) + 𝑎33 𝑎44 (𝑎11 + 𝑎22 )) , 𝐴 3 = 𝑎11 𝑎22 + 𝑎33 𝑎44 + (𝑎11 + 𝑎22 ) (𝑎33 + 𝑎44 ) + 𝑎55 (𝑎11 + 𝑎22 + 𝑎33 + 𝑎44 ) , 𝐴 4 = − (𝑎11 + 𝑎22 + 𝑎33 + 𝑎44 + 𝑎55 ) , 𝐵0 = 𝑎44 𝑎55 𝑏32 (𝑎11 𝑎23 − 𝑎13 𝑎21 ) − 𝑎11 𝑎 (...truncated)