Advertisement

Persistence, Periodicity and Privacy for Positive Systems in Epidemiology and Elsewhere

  • Oliver MasonEmail author
  • Aisling McGlinchey
  • Fabian Wirth
Chapter
Part of the Lecture Notes in Control and Information Sciences book series (LNCIS, volume 471)

Abstract

We first recall and describe some recently published results giving sufficient conditions for persistence and the existence of periodic solutions for switched SIS epidemiological models. We extend the result on the existence of persistent switching signals in two ways. We establish uniform strong persistence where previous work only guaranteed weak persistence; we replace the hypothesis that there exists an unstable matrix in the convex hull of the linearized systems with the weaker assumption that the JLE is positive. In the final section of the chapter, the issue of data privacy for positive systems is addressed.

Keywords

Switched systems SIS models Persistence Joint Lyapunov exponent Differential privacy 

Notes

Acknowledgements

This work was supported, in part, by Science Foundation Ireland grant 13/RC/2094 and co-funded under the European Regional Development Fund through the Southern & Eastern Regional Operational Programme to Lero—the Irish Software Research Centre (http://www.lero.ie)

References

  1. 1.
    van den Driessche, P., Watmough, J.: Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Biosci. 180, 29–48 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Fall, A., Iggidr, A., Sallet, G., Tewa, J.: Epidemiological models and Lyapunov functions. Math. Model. Nat. Phenom. 2, 62–68 (2007)Google Scholar
  3. 3.
    Ait-Rami, M., Bokharaie, V.S., Mason, O., Wirth, F.: Stability criteria for SIS epidemiological models under switching policies. Discret. Contin. Dyn. Syst. Ser. B 19(9), 2865–2887 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    J. Le Ny, Privacy-Preserving Nonlinear Observer Design Using Contraction Analysis. In: Proceedings IEEE 54th Annual Conference on Decision and Control (CDC) (2015)Google Scholar
  5. 5.
    Smith, H.: Monotone Dynamical Systems. American Mathematical Society (1995)Google Scholar
  6. 6.
    Smith, H., Thieme, H.: Dynamical Systems and Population Persistence. American Mathematical Society (2011)Google Scholar
  7. 7.
    Fainshil, L., Margaliot, M., Chigansky, P.: On the stability of positive linear switched systems under arbitrary switching laws. IEEE Trans. Autom. Control 54(4), 897–899 (2009)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Gurvits, L., Shorten, R., Mason, O.: On the stability of switched positive linear systems. IEEE Trans. Autom. Control 52, 1099–1103 (2007)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Mason, O., Wirth, F.: Extremal norms for positive linear inclusions. Linear Algebra Appl. 444, 100–113 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Dwork, C.: Differential Privacy. In: Proceedings of the International Colloquium on Automata, Languages and Programming, pp. 1–12. Springer (2006)Google Scholar
  11. 11.
    Holohan, N., Leith, D., Mason, O.: Differential privacy in metric spaces: numerical categorical and functional data under the one roof. Inform. Sci. 305, 256–268 (2015)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Le Ny, J., Pappas, G.J.: Differentially private filtering. IEEE Trans. Autom. Control 59(2), 341–354 (2014)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Hardin, H., van Schuppen, J.H.: Observers for linear positive systems. Linear Algebra Appl. 425, 571–607 (2007)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Oliver Mason
    • 1
    Email author
  • Aisling McGlinchey
    • 2
  • Fabian Wirth
    • 3
  1. 1.Department of Mathematics & Statistics/Hamilton InstituteMaynooth University and Lero, The Irish Software Research CentreKildareIreland
  2. 2.Department of Mathematics & StatisticsMaynooth University and Lero, The Irish Software Research CentreKildareIreland
  3. 3.Faculty of Computer Science and MathematicsUniversity of PassauPassauGermany

Personalised recommendations