Global stability of a delayed SIRS epidemic model with saturation incidence and temporary immunity

Xu, Rui; Ma, Zhien; Wang, Zhiping

doi:10.1016/j.camwa.2010.03.009

Cited by 56 publications

(34 citation statements)

References 16 publications

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“…Qing and Wen introduced the Kermack-McKendrick model, which is also called SIR model [3]. Then many mathematical models [4][5][6][7][8][9] inspired by the SIR model have been employed to constrain the propagation of Internet worms. Some research achievements [10][11][12] showed that the spread dynamic system of malware would be unstable and bifurcation and chaos would appear.…”

Section: Introductionmentioning

confidence: 99%

An Epidemic Model of Computer Worms with Time Delay and Variable Infection Rate

Yao

Yang

et al. 2018

Security and Communication Networks

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With rapid development of Internet, network security issues become increasingly serious. Temporary patches have been put on the infectious hosts, which may lose efficacy on occasions. This leads to a time delay when vaccinated hosts change to susceptible hosts. On the other hand, the worm infection is usually a nonlinear process. Considering the actual situation, a variable infection rate is introduced to describe the spread process of worms. According to above aspects, we propose a time-delayed worm propagation model with variable infection rate. Then the existence condition and the stability of the positive equilibrium are derived. Due to the existence of time delay, the worm propagation system may be unstable and out of control. Moreover, the threshold 0 of Hopf bifurcation is obtained. The worm propagation system is stable if time delay is less than 0 . When time delay is over 0 , the system will be unstable. In addition, numerical experiments have been performed, which can match the conclusions we deduce. The numerical experiments also show that there exists a threshold in the parameter , which implies that we should choose appropriate infection rate ( ) to constrain worm prevalence. Finally, simulation experiments are carried out to prove the validity of our conclusions.

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Section: Introductionmentioning

confidence: 99%

An Epidemic Model of Computer Worms with Time Delay and Variable Infection Rate

Yao

Yang

et al. 2018

Security and Communication Networks

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“…The basic idea was to abstract malware propagation into a probabilistic graph, and described the statistical dependence of malware propagation in arbitrary topologies using a spatial-temporal random process. Based on the ordinary differential equation and the SIR model [19][20][21][22], Wang in [23,24] derived the threshold for a piece of malware to propagate in WSNs, where all the nodes were supposed to be stationary. Recently, [25] considered a modeling framework which mathematically characterizes the process of malware propagation in MWSNs based on the theory of reaction-diffusion equation.…”

Section: Introductionmentioning

confidence: 99%

Stability and bifurcation analysis in a delayed reaction–diffusion malware propagation model

Zhu

Zhao

Wang

2015

Computers & Mathematics with Applications

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a b s t r a c tMobile wireless sensor networks (MWSNs) have become an area of intense research activity due to technical advances in sensors, wireless communications and networking, and signal processing. Many applications, including environment monitoring, battlefield surveillance, and urban search and rescue especially in hazardous situations, are envisaged. However, MWSNs may be vulnerable to malicious interference because of the large-scale characteristics. When a contaminated node communications with its neighbors, multiple copies of the malware are transmitted to its neighbors, which may destroy, block regular communications, or even damage the integrity of regular data packets. Modeling spatial distribution of malware propagation over time is the first step to predict the trend of malware propagation in MWSNs. We propose a novel wireless malware propagation model with the discrete time delay based on reaction-diffusion equations in mobile wireless sensor networks, and study its dynamic behaviors. By analyzing the stability and Hopf bifurcation of the equilibrium of our model, we search for the sufficient conditions, which leads to the malware propagation disappears or continues. Furthermore, we demonstrate that oscillations in this model occur through the destabilization of the stationary solution at a Hopf bifurcation point. And formulas for determining the stability of the bifurcating periodic oscillations are derived by applying the normal form method and center manifold theorem. Finally, we conduct extensive simulations on large-scale MWSNs to evaluate the proposed model. Numerical evidence shows that the spatial-temporal dynamic characteristics of malware propagation in MWSNs are closely related to the packet transmission rate, the communication rang and the mobile behavior of nodes.

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“…For instance, it can change the stability of equilibrium and thus lead to periodic solutions by Hopf bifurcation [6,[10][11][12]14,17,18,21,[27][28][29][30].…”

Section: Introductionmentioning

confidence: 99%

Global Hopf bifurcation analysis of an susceptible-infective-removed epidemic model incorporating media coverage with time delay

Zhao

2016

Journal of Biological Dynamics

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Global stability of a delayed SIRS epidemic model with saturation incidence and temporary immunity

Cited by 56 publications

References 16 publications

An Epidemic Model of Computer Worms with Time Delay and Variable Infection Rate

An Epidemic Model of Computer Worms with Time Delay and Variable Infection Rate

Stability and bifurcation analysis in a delayed reaction–diffusion malware propagation model

Global Hopf bifurcation analysis of an susceptible-infective-removed epidemic model incorporating media coverage with time delay

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