Chinese Annals of Mathematics, Series B

, Volume 31, Issue 4, pp 433–446 | Cite as

A mathematical model with delays for schistosomiasis japonicum transmission

Article

Abstract

A dynamic model of schistosoma japonicum transmission is presented that incorporates effects of the prepatent periods of the different stages of schistosoma into Barbour’s model. The model consists of four delay differential equations. Stability of the disease free equilibrium and the existence of an endemic equilibrium for this model are stated in terms of a key threshold parameter. The study of dynamics for the model shows that the endemic equilibrium is globally stable in an open region if it exists and there is no delays, and for some nonzero delays the endemic equilibrium undergoes Hopf bifurcation and a periodic orbit emerges. Some numerical results are provided to support the theoretic results in this paper. These results suggest that prepatent periods in infection affect the prevalence of schistosomiasis, and it is an effective strategy on schistosomiasis control to lengthen in prepatent period on infected definitive hosts by drug treatment (or lengthen in prepatent period on infected intermediate snails by lower water temperature).

Keywords

A mathematical model Schistosoma japonicum transmission Dynamics Globally stable Periodic orbits 

2000 MR Subject Classification

34C25 92D25 58F14 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Anderson, R. and May, R., Helminth infections of humans: mathematical models, population dynamics, and control, Adv. Para., 24, 1985, 1–101.CrossRefGoogle Scholar
  2. [2]
    Anderson, R. and May, R., Infectious Diseases of Humans: Dynamics and Control, Oxford University Press, Oxford, New York, 1991.Google Scholar
  3. [3]
    Barbour, A., Modeling the transmission of schistosomiasis: an introductory view, Amer. J. Trop. Med. Hyg., 55(Suppl.), 1996, 135–143.Google Scholar
  4. [4]
    Beretta, E. and Kuang, Y., Geometric stability switch criteria in delay differential systems with delay dependent parameters, SIAM. J. Math. Anal., 33, 2002, 1144–1165.MATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    Castillo-Chavez, C., Feng, Z. and Xu, D., A schistosomiasis model with mating structure and time delay, Math. Biosci., 211, 2008, 333–341.MATHMathSciNetGoogle Scholar
  6. [6]
    Cooke, L., Stability analysis for a vector disease model, Rocky Mount, J. Math., 7, 1979, 253–263.MathSciNetGoogle Scholar
  7. [7]
    Driessche, P. and Watmough, J., Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180, 2002, 29–48.MATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    Hairston, G., An analysis of age-prevalence data by catalytic model, Bull. World Health Organ., 33, 1965, 163–175.Google Scholar
  9. [9]
    Hale, J. and Verduyn Lunel, S. M., Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993.MATHGoogle Scholar
  10. [10]
    Liang, S., Maszle, D. and Spear, R., A quantitative framework for a multi-group model of Schistosomiasis japonicum transmission dynamics and control in Sichuan, China, Acta Tropica, 82, 2002, 263–277.Google Scholar
  11. [11]
    Macdonald, G., The dynamics of helminth infections, with special reference to dchistosomes, Trans. R. Soc. Trop. Med. Hyg., 59, 1965, 489–506.CrossRefGoogle Scholar
  12. [12]
    Nasell, I. and Hirsch, W., The transmission dynamics of Schistosomiasis, Comm. Pure Appl. Math., 26, 1973, 395–453.MATHCrossRefMathSciNetGoogle Scholar
  13. [13]
    Pflüger, W., Roushdy, Z. and El Emam, M., The prepatent period and cercarial production of Schistosoma haematobium in Bulinus truncatus (Egyptian field strains) at different constant temperatures, Z. Parasitenkd, 70, 1984, 95–103.CrossRefGoogle Scholar
  14. [14]
    Wu, J. and Feng, Z., Mathematical models for schistosomiasis with delays and multiple definitive hosts, mathematical approaches for emerging and reemerging infectious diseases: models, methods, and theory (Minneapolis, MN, 1999), IMA Vol. Math. Appl., 126, Springer-Verlag, New York, 2002, 215–229.Google Scholar
  15. [15]
    Zhang, P., Feng, Z. and Milner, F., A schistosomiasis model with an age-structure in human hosts and its application to treatment strategies, Math. Biosci., 205(1), 2007, 83–107.MATHCrossRefMathSciNetGoogle Scholar
  16. [16]
    Zhou, X., Wang, L., Chen, M., et al, The public health significance and control of schistosomiasis in China then and now, Acta Tropica, 96, 2005, 97–105.CrossRefGoogle Scholar

Copyright information

© Editorial Office of CAM (Fudan University) and Springer Berlin Heidelberg 2010

Authors and Affiliations

  1. 1.Department of MathematicsShanghai Jiao Tong UniversityShanghaiChina

Personalised recommendations