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December 2018, 15(6): 1465-1478. doi: 10.3934/mbe.2018067

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

Received  May 22, 2018 Accepted  July 24, 2018 Published  September 2018

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

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.

Citation: Meng Zhang, Kaiyuan Liu, Lansun Chen, Zeyu Li. State feedback impulsive control of computer worm and virus with saturated incidence. Mathematical Biosciences & Engineering, 2018, 15 (6) : 1465-1478. doi: 10.3934/mbe.2018067
References:
 [1] Y. Cai, Y. Kang and W. Wang, Global stability of the steady states of an epidemic model incorporating intervention strategies, Mathematical Biosciences and Engineering, 14 (2017), 1071-1089. doi: 10.3934/mbe.2017056. [2] L. Chen, X. Liang and Y. Pei, The periodic solutions of the impulsive state feedbck dynamical system, Commun. Math. Biol. Neurosci, 14 (2018). [3] A. M. Elaiw, Global properties of a class of virus infection models with multitarget cells, Nonlinear Dynamics, 69 (2012), 423-435. doi: 10.1007/s11071-011-0275-0. [4] A. M. Elaiw, R. M. Abukwaik and E. O. Alzahrani, Global properties of a cell mediated immunity in HIV infection model with two classes of target cells and distributed delays, International Journal of Biomathematics, 7 (2014), 1450055, 25 pp. doi: 10.1142/S1793524514500557. [5] T. Fukunaga and W. Iwasaki, Inactivity periods and postural change speed can explain atypical postural change patterns of Caenorhabditis elegans mutants, BMC Bioinformatics, 18 (2017), p46. doi: 10.1186/s12859-016-1408-8. [6] H. Guo, L. Chen and X. Song, Qualitative analysis of impulsive state feedback control to an algae-fish system with bistable property, Appl. Math. Comput., 271 (2015), 905-922. doi: 10.1016/j.amc.2015.09.046. [7] J. D. Hernández Guillén, A. Martín del Rey and L. Hernández Encinas, Study of the stability of a SEIRS model for computer worm propagation, Physica A: Statistical Mechanics and its Applications, 479 (2017), 411-421. doi: 10.1016/j.physa.2017.03.023. [8] J. Jia, Z. Jin and L. Chang, Structural calculations and propagation modeling of growing networks based on continuous degree, Mathematical Biosciences and Engineering, 14 (2017), 1215-1232. doi: 10.3934/mbe.2017062. [9] E. Kirmani and C. S. Hood, Analysis of a scanning model of worm propagation, Journal in Computer Virology, 6 (2010), 31-42. doi: 10.1007/s11416-008-0111-3. [10] S. Lefschetz, Contribution to the Theory of Nonlinear Oscillations, Vol. I, Princeton University Press, Princeton, 1950. [11] Z. Li, L. Chen and Z. Liu, Periodic solution of a chemostat model with variable yield and impulsive state feedback control, Appl. Math. Model., 36 (2012), 1255-1266. doi: 10.1016/j.apm.2011.07.069. [12] X. Liang, Y. Pei and Y. Lv, Modeling the state dependent impulse control for computer virus propagation under media coverage, Physica A: Statistical Mechanics and its Applications, 491 (2018), 516-527. doi: 10.1016/j.physa.2017.09.058. [13] B. Liu, Y. Tian and B. Kang, Dynamics on a Holling II predator-prey model with statedependent impulsive control, Int. J. Biomath., 5 (2012), 93-110. doi: 10.1142/S1793524512600066. [14] S. Max, A matter of time: On the transitory nature of cyberweapons, Journal of Strategic Studies, 41 (2018), 6-32. [15] G. Pang and L. Chen, Periodic solution of the system with impulsive state feedback control, Nonlinear Dynam., 78 (2014), 743-753. doi: 10.1007/s11071-014-1473-3. [16] D. A. Ray, C. B. Ward and B. Munteanu, etc, Investigating the impact of real-world factors on internet worm propagation, Lecture Notes in Computer Science, 4812 (2007), 10-24. doi: 10.1007/978-3-540-77086-2_2. [17] M. Sun, Y. Liu and S. Liu, etc, A novel method for analyzing the stability of periodic solution of impulsive state feedback model, Appl. Math. Comput., 273 (2016), 425-434. doi: 10.1016/j.amc.2015.09.093. [18] Y. Tian, K. Sun and L. Chen, Modelling and qualitative analysis of a predator-prey system with state-dependent impulsive effects, Math. Comput. Simulat., 82 (2011), 318-331. doi: 10.1016/j.matcom.2011.08.003. [19] A. Wang, Y. Xiao and H. Zhu, Dynamics of a filippov epidemic model with limited hospital beds, Mathematical Biosciences and Engineering, 15 (2018), 739-764. doi: 10.3934/mbe.2018033. [20] C. Wei and L. Chen, Periodic Solution of Prey-Predator Model with Beddington-DeAngelis Functional Response and Impulsive State Feedback Control, Journal of Applied Mathematics, 2012 (2012), Art. ID 607105, 17 pp. doi: 10.1155/2012/607105. [21] C. Wei and L. Chen, Homoclinic bifurcation of prey-predator model with impulsive state feedback control, Appl. Math. Comput., 237 (2014), 282-292. doi: 10.1016/j.amc.2014.03.124. [22] X. Xiao, P. Fu and G. Hu, etc, SAIDR: A new dynamic model for SMS-based worm propagation in mobile networks, IEEE Access, 5 (2017), 9935-9943. doi: 10.1109/ACCESS.2017.2700011. [23] W. Xu, L. Chen and S. Chen, etc, An impulsive state feedback control model for releasing white-headed langurs in captive to the wild, Commun. Nonlinear Sci., 34 (2016), 199-209. doi: 10.1016/j.cnsns.2015.10.015. [24] Y. Ye, Limit Cycle Theory, Shanghai Science and Technology Press, Shanghai, 1984. (in Chinese) [25] S. Yuan, X. Ji and H. Zhu, Asymptotic behavior of a delayed stochastic logistic model with impulsive perturbations, Mathematical Biosciences and Engineering, 14 (2017), 1477-1498. doi: 10.3934/mbe.2017077. [26] Y. Zhang, Y. Li, Q. Zhang and A. Li, Behavior of a stochastic SIR epidemic model with saturated incidence and vaccination rules, Physica A, 501 (2018), 178-187. doi: 10.1016/j.physa.2018.02.191. [27] M. Zhang, G. Song and L. Chen, A state feedback impulse model for computer worm control, Nonlinear Dynam., 85 (2016), 1561-1569. doi: 10.1007/s11071-016-2779-0. [28] Y. Zhang, Y. Zheng, F. Zhao and X. Liu, Dynamical analysis in a stochastic bioeconomic model with stage-structuring, Nonlinear Dynamics, 84 (2016), 1113-1121. doi: 10.1007/s11071-015-2556-5. [29] M. Zhu, X. Guo and Z. Lin, The risk index for an SIR epidemic model and spatial spreading of the infectious disease, Mathematical Biosciences and Engineering, 14 (2017), 1565-1583. doi: 10.3934/mbe.2017081. [30] C. C. Zou, D. Towsley and W. Gong, On the performance of Internet worm scanning strategies, Performance Evaluation, 63 (2006), 700-723. doi: 10.1016/j.peva.2005.07.032. [31] C. C. Zou, D. Towsley and W. Gong, On the performance of Internet worm scanning strategies, Performance Evaluation, 63 (2006), 700-723. doi: 10.1016/j.peva.2005.07.032.

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References:
 [1] Y. Cai, Y. Kang and W. Wang, Global stability of the steady states of an epidemic model incorporating intervention strategies, Mathematical Biosciences and Engineering, 14 (2017), 1071-1089. doi: 10.3934/mbe.2017056. [2] L. Chen, X. Liang and Y. Pei, The periodic solutions of the impulsive state feedbck dynamical system, Commun. Math. Biol. Neurosci, 14 (2018). [3] A. M. Elaiw, Global properties of a class of virus infection models with multitarget cells, Nonlinear Dynamics, 69 (2012), 423-435. doi: 10.1007/s11071-011-0275-0. [4] A. M. Elaiw, R. M. Abukwaik and E. O. Alzahrani, Global properties of a cell mediated immunity in HIV infection model with two classes of target cells and distributed delays, International Journal of Biomathematics, 7 (2014), 1450055, 25 pp. doi: 10.1142/S1793524514500557. [5] T. Fukunaga and W. Iwasaki, Inactivity periods and postural change speed can explain atypical postural change patterns of Caenorhabditis elegans mutants, BMC Bioinformatics, 18 (2017), p46. doi: 10.1186/s12859-016-1408-8. [6] H. Guo, L. Chen and X. Song, Qualitative analysis of impulsive state feedback control to an algae-fish system with bistable property, Appl. Math. Comput., 271 (2015), 905-922. doi: 10.1016/j.amc.2015.09.046. [7] J. D. Hernández Guillén, A. Martín del Rey and L. Hernández Encinas, Study of the stability of a SEIRS model for computer worm propagation, Physica A: Statistical Mechanics and its Applications, 479 (2017), 411-421. doi: 10.1016/j.physa.2017.03.023. [8] J. Jia, Z. Jin and L. Chang, Structural calculations and propagation modeling of growing networks based on continuous degree, Mathematical Biosciences and Engineering, 14 (2017), 1215-1232. doi: 10.3934/mbe.2017062. [9] E. Kirmani and C. S. Hood, Analysis of a scanning model of worm propagation, Journal in Computer Virology, 6 (2010), 31-42. doi: 10.1007/s11416-008-0111-3. [10] S. Lefschetz, Contribution to the Theory of Nonlinear Oscillations, Vol. I, Princeton University Press, Princeton, 1950. [11] Z. Li, L. Chen and Z. Liu, Periodic solution of a chemostat model with variable yield and impulsive state feedback control, Appl. Math. Model., 36 (2012), 1255-1266. doi: 10.1016/j.apm.2011.07.069. [12] X. Liang, Y. Pei and Y. Lv, Modeling the state dependent impulse control for computer virus propagation under media coverage, Physica A: Statistical Mechanics and its Applications, 491 (2018), 516-527. doi: 10.1016/j.physa.2017.09.058. [13] B. Liu, Y. Tian and B. Kang, Dynamics on a Holling II predator-prey model with statedependent impulsive control, Int. J. Biomath., 5 (2012), 93-110. doi: 10.1142/S1793524512600066. [14] S. Max, A matter of time: On the transitory nature of cyberweapons, Journal of Strategic Studies, 41 (2018), 6-32. [15] G. Pang and L. Chen, Periodic solution of the system with impulsive state feedback control, Nonlinear Dynam., 78 (2014), 743-753. doi: 10.1007/s11071-014-1473-3. [16] D. A. Ray, C. B. Ward and B. Munteanu, etc, Investigating the impact of real-world factors on internet worm propagation, Lecture Notes in Computer Science, 4812 (2007), 10-24. doi: 10.1007/978-3-540-77086-2_2. [17] M. Sun, Y. Liu and S. Liu, etc, A novel method for analyzing the stability of periodic solution of impulsive state feedback model, Appl. Math. Comput., 273 (2016), 425-434. doi: 10.1016/j.amc.2015.09.093. [18] Y. Tian, K. Sun and L. Chen, Modelling and qualitative analysis of a predator-prey system with state-dependent impulsive effects, Math. Comput. Simulat., 82 (2011), 318-331. doi: 10.1016/j.matcom.2011.08.003. [19] A. Wang, Y. Xiao and H. Zhu, Dynamics of a filippov epidemic model with limited hospital beds, Mathematical Biosciences and Engineering, 15 (2018), 739-764. doi: 10.3934/mbe.2018033. [20] C. Wei and L. Chen, Periodic Solution of Prey-Predator Model with Beddington-DeAngelis Functional Response and Impulsive State Feedback Control, Journal of Applied Mathematics, 2012 (2012), Art. ID 607105, 17 pp. doi: 10.1155/2012/607105. [21] C. Wei and L. Chen, Homoclinic bifurcation of prey-predator model with impulsive state feedback control, Appl. Math. Comput., 237 (2014), 282-292. doi: 10.1016/j.amc.2014.03.124. [22] X. Xiao, P. Fu and G. Hu, etc, SAIDR: A new dynamic model for SMS-based worm propagation in mobile networks, IEEE Access, 5 (2017), 9935-9943. doi: 10.1109/ACCESS.2017.2700011. [23] W. Xu, L. Chen and S. Chen, etc, An impulsive state feedback control model for releasing white-headed langurs in captive to the wild, Commun. Nonlinear Sci., 34 (2016), 199-209. doi: 10.1016/j.cnsns.2015.10.015. [24] Y. Ye, Limit Cycle Theory, Shanghai Science and Technology Press, Shanghai, 1984. (in Chinese) [25] S. Yuan, X. Ji and H. Zhu, Asymptotic behavior of a delayed stochastic logistic model with impulsive perturbations, Mathematical Biosciences and Engineering, 14 (2017), 1477-1498. doi: 10.3934/mbe.2017077. [26] Y. Zhang, Y. Li, Q. Zhang and A. Li, Behavior of a stochastic SIR epidemic model with saturated incidence and vaccination rules, Physica A, 501 (2018), 178-187. doi: 10.1016/j.physa.2018.02.191. [27] M. Zhang, G. Song and L. Chen, A state feedback impulse model for computer worm control, Nonlinear Dynam., 85 (2016), 1561-1569. doi: 10.1007/s11071-016-2779-0. [28] Y. Zhang, Y. Zheng, F. Zhao and X. Liu, Dynamical analysis in a stochastic bioeconomic model with stage-structuring, Nonlinear Dynamics, 84 (2016), 1113-1121. doi: 10.1007/s11071-015-2556-5. [29] M. Zhu, X. Guo and Z. Lin, The risk index for an SIR epidemic model and spatial spreading of the infectious disease, Mathematical Biosciences and Engineering, 14 (2017), 1565-1583. doi: 10.3934/mbe.2017081. [30] C. C. Zou, D. Towsley and W. Gong, On the performance of Internet worm scanning strategies, Performance Evaluation, 63 (2006), 700-723. doi: 10.1016/j.peva.2005.07.032. [31] C. C. Zou, D. Towsley and W. Gong, On the performance of Internet worm scanning strategies, Performance Evaluation, 63 (2006), 700-723. doi: 10.1016/j.peva.2005.07.032.
Successor function $F(A) = c-a$
Trajectory of uncontrolled system. The parameter values: $K = 0.06, \beta = 0.09, \alpha = 8.2, \mu = 0.01$
region $G$
Case of $N_A$ coinciding with $A$
Case of $0<\sigma_1<\sigma_1^*$
Case of $\sigma_1^*<\sigma_1<1$
The successor function is monotonically decreasing
${S_1},{S_2},\cdots ,{S_{k+1}},{S_{k + 2}},\cdots$ are the subsequent points of ${S_0},{S_1},\cdots ,{S_k},{S_{k + 1}},\cdots$ respectively
Establish coordinate system $(s,n)$ on point $A$
Subplot (a) is the trajectory of system (1) and (b) and (c) are time series of $S$ and $I$ respectively
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