Stability Analysis for the Disease Free Equilibrium of a Discrete Malaria Model with Two Delays

  • Chunqing Wu
  • Yanxin Zhang
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7390)


A discrete-time model is established to show the transmission of malaria between humans and mosquitoes with the incubation periods of parasites within both human and mosquito concerned. It is proved that the disease free equilibrium of the model is globally asymptotically stable when the basic reproduction number is less than 1 by constructing appropriate Lyapunov functions.


Malaria transmission Discrete time model Time delay Global stability Lyapunov function 


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  1. 1.
    Ruan, S.G., Xiao, D.M., Beier, J.C.: On the Delayed Ross-Macdonald Model for Malaria Transmission. Bull. Math. Biol. 70, 1098–1114 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Wu, C.Q., Jiang, Z.Y.: A Delayed Discrete Model and Control for Malaria Transmission. In: Proceedings of 4th International Conference on Bioinformatics and Biomedical Engineering. IEEE press, NewYork (2010)Google Scholar
  3. 3.
    World Health Organization, World Malaria Report (2008),
  4. 4.
    Hethcote, H.W.: The Mathematics of Infectious Disease. SIAM Review 42, 599–653 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Ross, R.: The Prevention of Malaria, 2nd edn. Murray, Lodon (1911)Google Scholar
  6. 6.
    Macdonald, G.: The Epidemiology and Control of Malaria. Oxford University Press, London (1957)Google Scholar
  7. 7.
    Auger, P., et al.: The Ross-Macdonald Model in Patchy Environment. Math. Biosci. 216(2), 123–131 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    An, D., Driessche, P., Watmough, J.: Reproduction Numbers and Sub-threshold Endemic Equilibria for Compartmental Models of Disease Transmission. Math. Biosci. 180, 29–48 (2002)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Orobeinikov, A., Wake, G.C.: Lyapunov Functions and Global Stability for SIR, SIRS, and SIS epidemiological models. Appl. Math. Lett. 15, 955–960 (2002)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Wu, Z.Y., Zhou, Y.C.: Advances in Mathematical Biology. Science Press of China, Beijing (2006) (in Chinese)Google Scholar
  11. 11.
    Wang, L., Wang, M.Q.: Ordinary Difference Equations. Xinjiang University Press, Urumqi (1991) (in Chinese)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Chunqing Wu
    • 1
  • Yanxin Zhang
    • 1
  1. 1.School of Mathematics and PhysicsChangzhou UniversityChangzhouChina

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