Annali di Matematica Pura ed Applicata (1923 -)

, Volume 197, Issue 6, pp 1729–1737 | Cite as

Uniform boundedness of the attractor in \(H^2\) of a non-autonomous epidemiological system

  • María Anguiano


In this paper, we prove the uniform boundedness of the pullback attractor of a non-autonomous SIR (susceptible, infected, recovered) model from epidemiology considered in Anguiano and Kloeden (Commun Pure Appl Anal 13(1):157–173, 2014). We prove two uniform bounds of this pullback attractor, firstly in the norm \(H_0^1\) and later, under appropriate additional assumptions, in the norm \(H^2\).


SIR epidemic model with diffusion Invariant sets Uniform boundedness in \(H^2\) 

Mathematics Subject Classification

35B41 37B55 



María Anguiano has been supported by Junta de Andalucía (Spain), Proyecto de Excelencia P12-FQM-2466


  1. 1.
    Anderson, R.M., May, R.M.: Infectious Diseases of Humans, Dynamics and Control. Oxford University Press, Oxford (1992)Google Scholar
  2. 2.
    Anguiano, M., Kloeden, P.E.: Asymptotic behavior of the nonautonomous SIR equations with diffusion. Commun. Pure Appl. Anal. 13(1), 157–173 (2014)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Anguiano, M.: $H^2$-boundedness of the pullback attractor for the non-autonomous SIR equations with diffusion. Nonlinear Ana. Theory Methods Appl. 113, 180–189 (2015)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Brauer, F., van den Driessche, P., Wu, J. (eds.): Mathematical Epidemiology, Springer Lecture Notes in Mathematics, 1945. Springer, Heidelberg (2008)Google Scholar
  5. 5.
    Crauel, H., Debussche, A., Flandoli, F.: Random attractors. J. Dyn. Differ. Equ. 9, 307–341 (1997)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Kloeden, P.E., Pötzsche, C., Rasmussen, M.: Discrete-time nonautonomous dynamical systems. In: Stability and Bifurcation Theory for Non-autonomous Differential Equations, Lectures Notes in Mathematics, 2065. Springer, Berlin, Heidelberg (2013)Google Scholar
  7. 7.
    Keeling, M.J., Rohani, P., Grenfell, B.T.: Seasonally forced disease dynamics explored as switching between attractors. Physica D 148, 317–335 (2001)CrossRefGoogle Scholar
  8. 8.
    Kermack, W.O., McKendrick, A.G.: Contributions to the mathematical theory of epidemics (part I). Proc. R. Soc. Lond. Ser. A 115, 700–721 (1927)CrossRefGoogle Scholar
  9. 9.
    Robinson, J.C.: Infinite-Dimensional Dynamical Systems. Cambridge University Press, Cambridge (2001)CrossRefGoogle Scholar
  10. 10.
    Stone, L., Olinky, R., Huppert, A.: Seasonal dynamics of recurrent epidemics. Nature 446, 533–536 (2007)CrossRefGoogle Scholar
  11. 11.
    Tan, W., Ji, Y.: On the pullback attractor for the non-autonomous SIR equations with diffusion. J. Math. Anal. Appl. 449(2), 1850–1862 (2017)MathSciNetCrossRefGoogle Scholar

Copyright information

© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Departamento de Análisis Matemático, Facultad de MatemáticasUniversidad de SevillaSevilleSpain

Personalised recommendations