# American Institute of Mathematical Sciences

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

show all references

##### 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
 [1] Piotr Zgliczyński. Steady state bifurcations for the Kuramoto-Sivashinsky equation: A computer assisted proof. Journal of Computational Dynamics, 2015, 2 (1) : 95-142. doi: 10.3934/jcd.2015.2.95 [2] Ben Niu, Weihua Jiang. Dynamics of a limit cycle oscillator with extended delay feedback. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1439-1458. doi: 10.3934/dcdsb.2013.18.1439 [3] A. Aschwanden, A. Schulze-Halberg, D. Stoffer. Stable periodic solutions for delay equations with positive feedback - a computer-assisted proof. Discrete & Continuous Dynamical Systems - A, 2006, 14 (4) : 721-736. doi: 10.3934/dcds.2006.14.721 [4] Benjamin Steinberg, Yuqing Wang, Huaxiong Huang, Robert M. Miura. Spatial Buffering Mechanism: Mathematical Model and Computer Simulations. Mathematical Biosciences & Engineering, 2005, 2 (4) : 675-702. doi: 10.3934/mbe.2005.2.675 [5] Elena Celledoni, Markus Eslitzbichler, Alexander Schmeding. Shape analysis on Lie groups with applications in computer animation. Journal of Geometric Mechanics, 2016, 8 (3) : 273-304. doi: 10.3934/jgm.2016008 [6] Maxime Breden, Jean-Philippe Lessard. Polynomial interpolation and a priori bootstrap for computer-assisted proofs in nonlinear ODEs. Discrete & Continuous Dynamical Systems - B, 2018, 23 (7) : 2825-2858. doi: 10.3934/dcdsb.2018164 [7] Kazuhisa Ichikawa. Synergistic effect of blocking cancer cell invasion revealed by computer simulations. Mathematical Biosciences & Engineering, 2015, 12 (6) : 1189-1202. doi: 10.3934/mbe.2015.12.1189 [8] Thomas Wanner. Computer-assisted equilibrium validation for the diblock copolymer model. Discrete & Continuous Dynamical Systems - A, 2017, 37 (2) : 1075-1107. doi: 10.3934/dcds.2017045 [9] István Balázs, Jan Bouwe van den Berg, Julien Courtois, János Dudás, Jean-Philippe Lessard, Anett Vörös-Kiss, JF Williams, Xi Yuan Yin. Computer-assisted proofs for radially symmetric solutions of PDEs. Journal of Computational Dynamics, 2018, 5 (1&2) : 61-80. doi: 10.3934/jcd.2018003 [10] Yoichi Enatsu, Yukihiko Nakata. Stability and bifurcation analysis of epidemic models with saturated incidence rates: An application to a nonmonotone incidence rate. Mathematical Biosciences & Engineering, 2014, 11 (4) : 785-805. doi: 10.3934/mbe.2014.11.785 [11] Hyeuknam Kwon, Yoon Mo Jung, Jaeseok Park, Jin Keun Seo. A new computer-aided method for detecting brain metastases on contrast-enhanced MR images. Inverse Problems & Imaging, 2014, 8 (2) : 491-505. doi: 10.3934/ipi.2014.8.491 [12] Cheng-Ta Yeh, Yi-Kuei Lin. Component allocation cost minimization for a multistate computer network subject to a reliability threshold using tabu search. Journal of Industrial & Management Optimization, 2016, 12 (1) : 141-167. doi: 10.3934/jimo.2016.12.141 [13] Fang Qin, Ying Jiang, Ping Gu. Three-dimensional computer simulation of twill woven fabric by using polynomial mathematical model. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 1167-1178. doi: 10.3934/dcdss.2019080 [14] Valery A. Gaiko. The geometry of limit cycle bifurcations in polynomial dynamical systems. Conference Publications, 2011, 2011 (Special) : 447-456. doi: 10.3934/proc.2011.2011.447 [15] Fabio S. Priuli. State constrained patchy feedback stabilization. Mathematical Control & Related Fields, 2015, 5 (1) : 141-163. doi: 10.3934/mcrf.2015.5.141 [16] Richard L Buckalew. Cell cycle clustering and quorum sensing in a response / signaling mediated feedback model. Discrete & Continuous Dynamical Systems - B, 2014, 19 (4) : 867-881. doi: 10.3934/dcdsb.2014.19.867 [17] Yoshiaki Muroya, Teresa Faria. Attractivity of saturated equilibria for Lotka-Volterra systems with infinite delays and feedback controls. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-12. doi: 10.3934/dcdsb.2018302 [18] Magdalena Caubergh, Freddy Dumortier, Robert Roussarie. Alien limit cycles in rigid unfoldings of a Hamiltonian 2-saddle cycle. Communications on Pure & Applied Analysis, 2007, 6 (1) : 1-21. doi: 10.3934/cpaa.2007.6.1 [19] Jihua Yang, Liqin Zhao. Limit cycle bifurcations for piecewise smooth integrable differential systems. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2417-2425. doi: 10.3934/dcdsb.2017123 [20] Fangfang Jiang, Junping Shi, Qing-guo Wang, Jitao Sun. On the existence and uniqueness of a limit cycle for a Liénard system with a discontinuity line. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2509-2526. doi: 10.3934/cpaa.2016047

2017 Impact Factor: 1.23