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Optimal Impulse Control of SIR Epidemics Over Scale-Free Networks

  • Vladislav Taynitskiy
  • Elena Gubar
  • Quanyan ZhuEmail author
Chapter
Part of the EAI/Springer Innovations in Communication and Computing book series (EAISICC)

Abstract

Recent wide spreading of Ransomware has created new challenges for cybersecurity over large-scale networks. The densely connected networks can exacerbate the spreading and makes the containment and control of the malware more challenging. In this work, we propose an impulse optimal control framework for epidemics over networks. The hybrid nature of discrete-time control policy of continuous-time epidemic dynamics together with the network structure poses a challenging optimal control problem. We leverage the Pontryagin’s minimum principle for impulsive systems to obtain an optimal structure of the controller and use numerical experiments to corroborate our results.

Notes

Acknowledgements

The work of the second author was supported by the research grant “Optimal Behavior in Conflict-Controlled Systems” (17-11-01079) from Russian Science Foundation.

References

  1. 1.
    Agur, Z., Cojocaru, L., Mazor, G., Anderson, R.M., Danon, Y.L.: Pulse mass measles vaccination across age cohorts. Proc. Natl. Acad. Sci. USA 90, 11698–11702 (1993)CrossRefGoogle Scholar
  2. 2.
    Barabási, A.L., Albert, R.: Emergence of scaling in random networks. Science 286(5439), 509–512 (1999)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Blaquière, A.: Impulsive optimal control with finite or infinite time horizon. J. Optim. Theory Appl. 46, 431–439 (1985)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Chahim, M., Harti, R., Kort, P.: A tutorial on the deterministic impulse control maximum principle: necessary and sufficient optimality conditions. Eur. J. Oper. Res. 219, 18–26 (2012)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Dykhta, V.A., Samsonyuk, O.N.: A maximum principle for smooth optimal impulsive control problems with multipoint state constraints. Comput. Math. Math. Phys. 49, 942–957 (2009)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Fu, X., Small, M., Walker, D.M., Zhang, H.: Epidemic dynamics on scale-free networks with piecewise linear infectivity and immunization. Phys. Rev. E. 77(3), 036113 (2008)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Gubar, E., Zhu, Q.: Optimal control of influenza epidemic model with virus mutations. In: Proceedings 12th Biannual European Control Conference, pp. 3125–3130. IEEE Control Systems Society, New York (2013)Google Scholar
  8. 8.
    Gubar, E., Kumacheva, S., Zhitkova, E., Porokhnyavaya, O.: Impact of propagation information in the model of tax audit. In: Recent Advances in Game Theory and Applications. Static and Dynamic Game Theory: Foundations and Applications, Switzerland, pp. 91–110 (2015)Google Scholar
  9. 9.
    Gubar, E., Zhu, Q., Taynitskiy, V.: Optimal control of multi-strain epidemic processes in complex networks. In: Game Theory for Networks. GameNets 2017. Lecture Notes of the Institute for Computer Sciences, Social Informatics and Telecommunications Engineering, vol. 212, pp. 108–117. Springer, Cham (2017)Google Scholar
  10. 10.
    Kharraz, A., Robertson, W., Balzarotti, D., Bilge, L., Kirda, E.: Cutting the gordian knot: a look under the hood of ransomware attacks. In: International Conference on Detection of Intrusions and Malware, and Vulnerability Assessment, pp. 3–24. Springer, Berlin (2015)Google Scholar
  11. 11.
    Luo, X., Liao, Q.: Ransomware: a new cyber hijacking threat to enterprises. In: Handbook of Research on Information Security and Assurance, pp. 1–6. IGI Global, Hershey (2009)Google Scholar
  12. 12.
    Pastor-Satorras, R., Vespignani A.: Epidemic spreading in scale-free networks. Phys. Rev. Lett. 86(14), 3200 (2001)CrossRefGoogle Scholar
  13. 13.
    Sethi, S.P., Thompson, G.L.: Optimal Control Theory: Applications to Management Science and Economics. Springer, Berlin (2006)zbMATHGoogle Scholar
  14. 14.
    Taynitskiy, V.A., Gubar, E.A., Zhitkova, E.M.: Optimization of protection of computer networks against malicious software. In: Proceedings of International Conference Stability and Oscillations of Nonlinear Control Systems (Pyatnitskiy’s Conference) (2016)Google Scholar
  15. 15.
    Taynitskiy, V., Gubar, E., Zhu Q.: Optimal impulse control of bi-virus SIR epidemics with application to heterogeneous internet of things. In: Constructive Nonsmooth Analysis and Related Topics. Abstracts of the International Conference. Dedicated to the Memory of Professor V.F. Demyanov, pp. 113–116 (2017)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Vladislav Taynitskiy
    • 1
  • Elena Gubar
    • 1
  • Quanyan Zhu
    • 2
    Email author
  1. 1.St. Petersburg State UniversityFaculty of Applied Mathematics and Control ProcessesSaint-PetersburgRussia
  2. 2.Department of Electrical and Computer EngineeringTandon School of Engineering, New York UniversityBrooklynUSA

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