State feedback impulsive control of computer worm and virus with saturated incidence

Meng Zhang; Kaiyuan Liu; Lansun Chen; Zeyu Li

doi:10.3934/mbe.2018067

Article Contents

2018, Volume 15, Issue 6: 1465-1478. Doi: 10.3934/mbe.2018067

This issue Previous Article Enhancement of chemotherapy using oncolytic virotherapy: Mathematical and optimal control analysis Next Article Coexistence of a cross-diffusive West Nile virus model in a heterogenous environment

State feedback impulsive control of computer worm and virus with saturated incidence

1.
School of Science, Beijing University of Civil Engineering and Architecture, Beijing 100044, China
2.
Anshan Normal University, Anshan 114007, China
3.
Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100080, China
4.
Canvard College, Beijing Technology and Business University, Beijing 101118, China

* Corresponding author: Meng Zhang
* Corresponding author: Meng Zhang

Received: May 21, 2018

Accepted: July 23, 2018

Published: December 2018

Lansun Chen is supported by NSFC of China (No.11671346, No.61751317), and Meng Zhang is supported by NSFC of China (No.11701026).

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Abstract

A state feedback impulsive model is set up to discuss the spreading and control of the computer worm and virus. Considering the transmission features, saturated infectious is adopted to describe the spreading in the model, and all the treatment measures, such as patching operating system and updating antivirus software, are assumed to take effect instantly. Then the model is analyzed with a novel method, and the existence and stability of order-1 limit cycle are discussed. Finally, the numerical simulation is listed to verify the result of the paper.

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Mathematics Subject Classification: Primary: 34H15, 34K21; Secondary: 39A23.

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Figure 1. Successor function $F(A) = c-a$

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Figure 2. Trajectory of uncontrolled system. The parameter values: $K = 0.06, \beta = 0.09, \alpha = 8.2, \mu = 0.01$

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Figure 3. region $G$

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Figure 4. Case of $N_A$ coinciding with $A$

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Figure 5. Case of $0<\sigma_1<\sigma_1^*$

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Figure 6. Case of $\sigma_1^*<\sigma_1<1$

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Figure 7. The successor function is monotonically decreasing

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Figure 8. ${S_1},{S_2},\cdots ,{S_{k+1}},{S_{k + 2}},\cdots $ are the subsequent points of ${S_0},{S_1},\cdots ,{S_k},{S_{k + 1}},\cdots $ respectively

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Figure 9. Establish coordinate system $(s,n)$ on point $A$

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Figure 10. Subplot (a) is the trajectory of system (1) and (b) and (c) are time series of $S$ and $I$ respectively

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