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Dynamics of Zika Virus Epidemic in Random Environment

  • Yusuke Asai
  • Xiaoying HanEmail author
  • Peter E. Kloeden
Chapter
Part of the Studies in Systems, Decision and Control book series (SSDC, volume 200)

Abstract

A mathematical model for Zika virus dynamics under randomly varying environmental conditions is developed, in which the birth and loss rates for mosquitoes, and environmental influence are modeled as random processes. The resulting system of random ordinary differential equations are studied by the theory of random dynamical systems and dynamical analysis. First the existence, uniqueness, positiveness and boundedness of solutions are discussed. Then the long term dynamics in terms of existence and geometric structures of random attractors and forward omega limit sets are investigated. Moreover, sufficient conditions under which the prevalence of Zika virus among human beings decreases monotonically to zero, as well as conditions under which an epidemic occurs are established.

References

  1. 1.
    Arnold, L.: Random Dynamical Systems. Springer-Verlag, Berlin (1998)Google Scholar
  2. 2.
    Caraballo, T., Han, X.: Applied Nonautonomous and Random Dynamical Systems. Springer-Verlag, Cham, BCAM SpringerBrief (2016)Google Scholar
  3. 3.
    Caraballo, T., Kloeden, P.E., Schmalfuss, B.: Exponentially stable stationary solutions for stochastic evolution equations and their perturbation. Appl. Math. Optim. 50, 183–207 (2004)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Caraballo, T., Łukaszewicz, G., Real, J.: Pullback attractors for asymptotically compact nonautonomous dynamical systems. Nonlinear Analysis TMA 6, 484–498 (2006)CrossRefGoogle Scholar
  5. 5.
    Crauel, H., Kloeden, P.E.: Nonautonomous and random attractors. Jahresber. Dtsch.En Math.-Ver.Igung 117, 173–206 (2015)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Fauci, A.S., Morens, D.M.: Zika Virus in the Americas - yet another arbovirus threat. N. Engl. J. Med. 374, 601–604 (2016)CrossRefGoogle Scholar
  7. 7.
    Flandoli, F., Schmalfuss, B.: Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative noise. Stochast. Stochast. Rep. 59, 21–45 (1996)CrossRefGoogle Scholar
  8. 8.
    Hale, J.K.: Ordinary Differential Equations. Robert E. Krieger Publishing Co., N.Y. (1980)Google Scholar
  9. 9.
    Han, X., Kloeden, P.E.: Random Ordinary Differential Equations and their Numerical Solution. Springer Nature, Singapore (2017)Google Scholar
  10. 10.
    Gao, D., Lou, Y., He, D., Porco, T.C., Kuang, Y., Chowell, G., Ruan, S.: Prevention and control of Zika as a mosquito-borne and sexually transmitted disease: a mathematical modeling analysis, Scientific Reports 6. Article number 28070, (2016)Google Scholar
  11. 11.
    Kloeden, P.E., Rasmussen, M.: Nonautonomous Dynamical Systems. American Mathematical Society, Providence, RI (2011)Google Scholar
  12. 12.
    Kloeden, P.E., Lorenz, T.: Construction of nonautonomous forward attractors. Proc. Amer. Mat. Soc. 144, 259–268 (2016)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Hygiene, Graduate School of MedicineHokkaido UniversitySapporoJapan
  2. 2.Department of Mathematics and Statistics221 Parker Hall Auburn UniversityAuburnUSA

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