Journal of Applied Mathematics and Computing

, Volume 46, Issue 1–2, pp 305–319 | Cite as

Threshold dynamics for a nonautonomous schistosomiasis model in a periodic environment

Original Research


In this paper, we investigate a nonautonomous schistosomiasis model in a periodic environment. We obtain a threshold value between the extinction and the uniform persistence. Our main results show that the disease persists if the threshold value is larger than unity. We also prove that there exists a positive periodic solution. Numerical simulations which support our theoretical analysis are also given.


Nonautonomous schistosomiasis model Periodic systems Uniform permanence Extinction 

Mathematics Subject Classification

34D23 92D30 



The research has been supported by the Natural Science Foundation of China (No. 11261004), China Postdoctoral Science Foundation funded project (No. 2012M510039) and the National Key Technologies R & D Program of China (2009BAI78B02), the Natural Science Foundation of Jiangxi Province (20122BAB211010) and the Postgraduate Innovation Fund of Jiangxi Province (YC2012-5121).


  1. 1.
    Edward, T.C., Gesham, M., Lawrence, M.: Modelling within host parasite dynamics of schistosomiasis. Comput. Math. Methods Med. 11(3), 255–280 (2010) CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Zhou, X.N., Wang, L.Y., Chen, M.G., Wu, X.H., Jiang, Q.W., Chen, X.Y., Zheng, J., Utzinger, J.: The public health significance and control of schistosomiasis in China – then and now. Acta Trop. 96, 97–105 (2005) CrossRefGoogle Scholar
  3. 3.
    Zhou, X.N., Guo, J.G., Wu, X.H., Jiang, Q.W., Zheng, J., Dang, H., Wang, X.H., Xu, J., Zhu, H.Q., Wu, G.L., Li, Y.S., Xu, X.J., Chen, H.G., Wang, T.P., Zhu, Y.C., Qiu, D.C., Dong, X.Q., Zhao, G.M., Zhang, S.J., Zhao, N.Q., Xia, G., Wang, L.Y., Zhang, S.Q., Lin, D.D., Chen, M.G., Hao, Y.: Epidemiology of schistosomiasis in the People’s Republic of China, 2004. Emerg. Infect. Dis. 13, 1470–1476 (2007) CrossRefGoogle Scholar
  4. 4.
    MacDonald, G.: The dynamics of helminth infections with special reference to schistosomes. Trans. R. Soc. Trop. Med. Hyg. 59, 489–506 (1965) CrossRefGoogle Scholar
  5. 5.
    Barbour, A.D.: Modeling the transmission of schistosomiasis: an introductory view. Am. J. Trop. Med. Hyg. 55, 135–143 (1996) Google Scholar
  6. 6.
    Gao, S., Liu, Y., Luo, Y.: Control problems of a mathematical model for schistosomiasis transmission dynamics. Nonlinear Dyn. 63, 503–512 (2011) CrossRefMathSciNetGoogle Scholar
  7. 7.
    Anderson, R.M., May, R.M.: Regulation and stability of host-parasite population interactions: I. Regulatory processes. J. Anim. Ecol. 47, 219–247 (1978) CrossRefGoogle Scholar
  8. 8.
    Anderson, R.M., May, R.M.: Regulation and stability of host-parasite interactions: II. Destabilizing processes. J. Anim. Ecol. 47, 249–267 (1978) CrossRefGoogle Scholar
  9. 9.
    Anderson, R.M., May, R.M.: Prevalence of schistosome infections within molluscan populations: observed patterns and theoretical predictions. Parasitology 79, 63–94 (1979) CrossRefGoogle Scholar
  10. 10.
    Castillo-Chavez, C., Thieme, H.R.: Asymptotically autonomous epidemic models. In: Arino, O., Kimmel, M. (eds.) Proceedings of the Third International Conference on Mathematical Population Dynamics, p. 33 (1995) Google Scholar
  11. 11.
    Zhang, T.L., Teng, Z.D.: On a nonautonomous SEIRS model in epidemiology. Bull. Math. Biol. 69, 2537–2559 (2007) CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Anderson, R.M., May, R.M.: Infectious Disease of Humans: Dynamics and Control. Oxford University Press, Oxford (1991) Google Scholar
  13. 13.
    Yukihiko, N., Toshikazu, K.: Global dynamics of a class of SEIRS epidemic models in a periodic environment. J. Math. Anal. Appl. 363, 230–237 (2010) CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Hale, J.K.: Asymptotic Behavior of Dissipative Systems. Math. Surveys Monogr., vol. 25. Amer. Math. Soc., Providence (1988) MATHGoogle Scholar
  15. 15.
    Zhao, X.Q.: Dynamical Systems in Population Biology. CMS Books Math./Ouvrages Math. SMC, vol. 16. Springer, New York (2003) CrossRefMATHGoogle Scholar
  16. 16.
    Wang, W.D., Zhao, X.Q.: Threshold dynamics for compartmental epidemic models in periodic environments. J. Dyn. Differ. Equ. 20(3), 699–717 (2008) CrossRefMATHGoogle Scholar
  17. 17.
    Yang, Y.P., Xiao, Y.N.: Threshold dynamics for an HIV model in periodic environments. J. Math. Anal. Appl. 361, 59–68 (2010) CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Jin, Y., Wang, W.D.: The effect of population dispersal on the spread of a disease. J. Math. Anal. Appl. 308, 343–364 (2005) CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Mukandavire, Z., Chiyaka, C., Garira, W., Musuka, G.: Mathematical analysis of a sex-structured HIV/AIDS model with a discrete time delay. Nonlinear Anal., Theory Methods Appl. 71, 1082–1093 (2009) CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Gao, S.J., Liu, Y.J., Nieto, J.J., Andrade, H.: Seasonality and mixed vaccination strategy in an epidemic model with vertical transmission. Math. Comput. Simul. 81, 1855–1868 (2011) CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Bacaer, N., Dads, E.: On the biological interpretation of a definition for the parameter R0 in periodic population models. J. Math. Biol. 65, 601–621 (2012) CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Heesterbeek, J.A.: A brief history of R0 and a recipe for its calculation. Acta Biotheor. 50, 189–204 (2002) CrossRefGoogle Scholar
  23. 23.
    Inaba, H.: On a new perspective of the basic reproduction number in heterogeneous environments. J. Math. Biol. 65, 309–348 (2012) CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Korean Society for Computational and Applied Mathematics 2014

Authors and Affiliations

  1. 1.Key Laboratory of Jiangxi Province for Numerical Simulation and Emulation TechniquesGannan Normal UniversityGanzhouP.R. China

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