Journal of Mathematical Biology

, Volume 51, Issue 3, pp 247–267 | Cite as

Stability analysis of pathogen-immune interaction dynamics

  • Akiko Murase
  • Toru Sasaki
  • Tsuyoshi Kajiwara


The paper considers models of dynamics of infectious disease in vivo from the standpoint of the mathematical analysis of stability. The models describe the interaction of the target cells, the pathogens, and the humoral immune response. The paper mainly focuses on the interior equilibrium, whose components are all positive. If the model ignores the absorption of the pathogens due to infection, the interior equilibrium is always asymptotically stable. On the other hand, if the model does consider it, the interior equilibrium can be unstable and a simple Hopf bifurcation can occur. A sufficient condition that the interior equilibrium is asymptotically stable is obtained. The condition explains that the interior equilibrium is asymptotically stable when experimental parameter values are used for the model. Moreover, the paper considers the models in which uninfected cells are involved in the immune response to pathogens, and are removed by the immune complexes. The effect of the involvement strongly affects the stability of the interior equilibria. The results are shown with the aid of symbolic calculation software.

Key words or phrases

Infectious diseases Immune response, Stability Hopf bifurcation Symbolic calculation software 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Anderson, R.M., May, R.M., Gupta, S.: Non-linear phenomena in host-parasite interactions. Parasitology 99, S59–S79 (1989)Google Scholar
  2. 2.
    Deans, J.A., Cohen, S.: Immunology of malaria. Ann. Rev. Microbiol. 37, 25–49 (1983)CrossRefGoogle Scholar
  3. 3.
    Edelstein-Keshet, L.: Mathematical models in biology. McGraw-Hill, 1988Google Scholar
  4. 4.
    Fuchs, B.A., Levin, B.I.: Functions of a complex variable and some of their applications. Volume 2. Pergamon Press, 1961Google Scholar
  5. 5.
    Gravenor, M.B., McLean, A.R., Kwiatkowski, D.: The regulation of malaria parasitaemia. Parasitology 110, 115–122 (1995)PubMedGoogle Scholar
  6. 6.
    Hetzel, C., Anderson, R.M.: The within-host cellular dynamics of bloodstage malaria. Parasitology 113, 25–38 (1996)PubMedGoogle Scholar
  7. 7.
    Ho, D.D. et al.: Rapid turnover of plasma virions and CD4 lymphocytes in HIV-1 infection. Nature 373, 123–126 (1995)CrossRefPubMedGoogle Scholar
  8. 8.
    Liu, W.M.: Criterion of Hopf bifurcations without using eigenvalues. J. Math. Anal. Appl. 182, 250–256 (1994)CrossRefGoogle Scholar
  9. 9.
    Liu, W.:. Nonlinear oscillation in models of immune responses to persistent viruses. Theor. Popul. Biol. 52, 224–230 (1997)Google Scholar
  10. 10.
    Neumann, A.U. et al.: Hepatitis C viral dynamics in vivo and the antiviral efficacy of interferon-α therapy. Science 282, 103–107 (1998)CrossRefPubMedGoogle Scholar
  11. 11.
    Nowak, M.A., Bangham, C.R.M.: Population dynamics of immune responses to persistent viruses. Science 272, 74–79 (1996)PubMedGoogle Scholar
  12. 12.
    Nowak, M.A. et al.: Viral dynamics in hepatitis B virus infection. Proc. Natl. Acad. Sci. 93, 4398–4402 (1996)CrossRefPubMedGoogle Scholar
  13. 13.
    Perelson, A.S. et al.: HIV-1 dynamics in vivo. Science 271, 1582–1586 (1996)PubMedGoogle Scholar
  14. 14.
    Perelson, A.S., Nelson, P.W.: Mathematical analysis of HIV-1 dynamics in vivo. SIAM Review 41, 3–44 (1999)MathSciNetGoogle Scholar
  15. 15.
    Roitt, I., Brostoff, J., Male, D.: Immunology fifth edition. Mosby-Wolfe, 1998Google Scholar
  16. 16.
    Saul, A.: Models for the in-host dynamics of malaria revisited : errors in some basic models lead to large over-estimates of growth rates. Parasitology 117, 405–407 (1998)CrossRefPubMedGoogle Scholar
  17. 17.
    Wei, X. et al.: Viral dynamics in human immunodeficiency virus type 1 infection. Nature 373, 117–122 (1995)CrossRefPubMedGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Akiko Murase
    • 1
  • Toru Sasaki
    • 2
  • Tsuyoshi Kajiwara
    • 2
  1. 1.Higashi Hagi Junior High SchoolYamaguchiJapan
  2. 2.Department of Mathematical and Environmental SciencesOkayama UniversityOkayamaJapan

Personalised recommendations