Applications of Mathematics

, Volume 57, Issue 6, pp 601–616 | Cite as

Existence of positive periodic solutions of an SEIR model with periodic coefficients

Article

Abstract

An SEIR model with periodic coefficients in epidemiology is considered. The global existence of periodic solutions with strictly positive components for this model is established by using the method of coincidence degree. Furthermore, a sufficient condition for the global stability of this model is obtained. An example based on the transmission of respiratory syncytial virus (RSV) is included.

Keywords

epidemic model coincidence degree Fredholm mapping 

MSC 2010

34C25 54H25 92D30 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    M.R. Al-Ajam, A.R. Bizri, J. Mokhbat, J. Weedon, L. Lutwick: Mucormycosis in the Eastern Mediterranean: a seasonal disease. Epidemiol. Infect. 134 (2006), 341–346.CrossRefGoogle Scholar
  2. [2]
    R.M. Anderson, R.M. May: Population biology of infectious diseases, Part 1. Nature 280 (1979), 361.CrossRefGoogle Scholar
  3. [3]
    R.M. Anderson, R.M. May: Infectious Diseases of Humans, Dynamics and Control. Oxford University, Oxford, 1991.Google Scholar
  4. [4]
    A. J. Arenas, G. Gonzalez, L. Jódar: Existence of periodic solutions in a model of respiratory syncytial virus RSV. J. Math. Anal. Appl. 344 (2008), 969–980.MathSciNetMATHCrossRefGoogle Scholar
  5. [5]
    O. Diekmann, J.A. P. Heesterbeek, J.A. J. Metz: On the definition and the computation of the basic reproduction ratio R 0 in models for infectious diseases in heterogeneous populations. J. Math. Biol. 28 (1990), 365–382.MathSciNetMATHCrossRefGoogle Scholar
  6. [6]
    O. Diekmann, J.A. P. Heesterbeek: Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation. John Wiley & Sons, Chichester, 2000.Google Scholar
  7. [7]
    P. van den Driessche, J. Watmough: Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Biosci. 180 (2002), 29–48.MathSciNetMATHCrossRefGoogle Scholar
  8. [8]
    D. J.D. Earn, J. Dushoff, S.A. Levin: Ecology and evolution of the flu. Trends in Ecology and Evolution 17 (2002), 334–340.CrossRefGoogle Scholar
  9. [9]
    M. Fan, K. Wang: Periodicity in a delayed ratio-dependent predator-prey system. J. Math. Anal. Appl. 262 (2001), 179–190.MathSciNetMATHCrossRefGoogle Scholar
  10. [10]
    R.E. Gaines, J. L. Mawhin: Coincidence Degree, and Nonlinear Differential Equations. Springer, Berlin, 1977.MATHGoogle Scholar
  11. [11]
    J.K. Hale: Ordinary Differential Equations. Wiley-Interscience, New York, 1969.MATHGoogle Scholar
  12. [12]
    G. Herzog, R. Redheffer: Nonautonomous SEIRS and Thron models for epidemiology and cell biology. Nonlinear Anal., Real World Appl. 5 (2004), 33–44.MathSciNetMATHCrossRefGoogle Scholar
  13. [13]
    H.W. Hethcote: The mathematics of infectious diseases. SIAM Review 42 (2000), 599–653.MathSciNetMATHCrossRefGoogle Scholar
  14. [14]
    L. Jódar, R. J. Villanueva, A. Arenas: Modeling the spread of seasonal epidemiological diseases: Theory and applications. Math. Comput. Modelling 48 (2008), 548–557.MathSciNetMATHCrossRefGoogle Scholar
  15. [15]
    Y. Li, Y. Kuang: Periodic solutions of periodic delay Lotka-Volterra equations and Systems. J. Math. Anal. Appl. 255 (2001), 260–280.MathSciNetMATHCrossRefGoogle Scholar
  16. [16]
    M.Y. Li, J. S. Muldowney: Global stability for the SEIR model in epidemiology. Math. Biosci. 125 (1995), 155–164.MathSciNetMATHCrossRefGoogle Scholar
  17. [17]
    W. London, J.A. Yorke: Recurrent outbreaks of measles, chickenpox and mumps. 1. Seasonal variation in contact rates. Am. J. Epidemiol. 98 (1973), 453–468.Google Scholar
  18. [18]
    J. Ma, Z. Ma: Epidemic threshold conditions for seasonally forced SEIR models. Math. Biosci. Eng. 3 (2006), 161–172.MathSciNetMATHCrossRefGoogle Scholar
  19. [19]
    M. Nuño, Z. Feng, M. Martcheva, C.C. Carlos: Dynamics of two-strain influenza with isolation and partial cross-immunity. SIAM J. Appl. Math. 65 (2005), 964–982.MathSciNetMATHCrossRefGoogle Scholar
  20. [20]
    Z. Teng: On the periodic solutions of periodic multi-species competitive systems with delays. Appl. Math. Comput. 127 (2002), 235–247.MathSciNetMATHCrossRefGoogle Scholar
  21. [21]
    Z. Teng, L. Chen: Permanence and extinction of periodic predator-prey systems in a patchy environment with delay. Nonlinear Anal., Real World Appl. 4 (2003), 335–364.MathSciNetMATHCrossRefGoogle Scholar
  22. [22]
    A. Weber, M. Weber, P. Milligan: Modeling epidemics caused by respiratory syncytial virus (RSV). Math. Biosci. 172 (2001), 95–113.MathSciNetMATHCrossRefGoogle Scholar
  23. [23]
    X. Zhang, L. Chen: The periodic solution of a class of epidemic models. Comput. Math. Appl. 38 (1999), 61–71.MATHCrossRefGoogle Scholar
  24. [24]
    T. Zhang, J. Liu, Z. Teng: Stability of Hopf bifurcation of a delayed SIRS epidemic model with stage structure. Nonlinear Anal., Real World Appl. 11 (2010), 293–306.MathSciNetMATHCrossRefGoogle Scholar
  25. [25]
    J. Zhang, Z. Ma: Global dynamics of an SEIR epidemic model with saturating contact rate. Math. Biosci. 185 (2003), 15–32.MathSciNetMATHCrossRefGoogle Scholar
  26. [26]
    T. Zhang, Z. Teng: On a nonautonomous SEIRS model in epidemiology. Bull. Math. Biol. 69 (2007), 2537–2559.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic 2012

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsXi’an Jiaotong UniversityXi’anP.R.China
  2. 2.School of ScienceXi’an Polytechnic UniversityXi’anP.R. China
  3. 3.College of Mathematics and System SciencesXinjiang UniversityUrumqiP.R.China

Personalised recommendations