Skip to main content

Advertisement

SpringerLink
Log in

A Dynamical Study of a Cellular Automata Model of the Spread of HIV in a Lymph Node

  • Original Article
  • Published:
Bulletin of Mathematical Biology Aims and scope Submit manuscript

Abstract

We conduct a mathematical study of a cellular automata model of the spread of the HIV virus in a lymph node. The model was proposed by Zorzenon dos Santos and Coutinho and captures the unique time scale of the viral spread. We give some rigorous mathematical results about the time scales and other dynamical aspects of the model as well as discuss parameter and model changes and their consequences.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Allouche, J., Courbage, M., Skordev, G., 2001. Notes on cellular automata. Cubo Mat. Educ. 3(2), 213–244.

    MATH  MathSciNet  Google Scholar 

  • Bernaschi, M., Castiglione, F., 2002. Selection of escape mutants from the immune recognition during HIV infection. Immunol. Cell Biol. 80, 307–313.

    Article  Google Scholar 

  • Blanchard, F., Tisseur, P., 2000. Some properties of cellular automata with equicontinuity points. Ann. Inst. Henri Poincaré Probab. Stat. 36(5), 569–582.

    Article  MATH  MathSciNet  Google Scholar 

  • Fauci, A., Pantaleo, G., Stanley, S., Weissman, D., 1996. Immunopathogenic mechanisms of HIV infection. Ann. Int. Med. 124(7), 654–663.

    Google Scholar 

  • Feinberg, M.B., 2002. The interface between the pathogenesis and treatment of HIV infection. In: The Human Immunodeficiency Virus: Biology, Immunology, and Therapy, pp. 384–440. Princeton University Press, Princeton.

    Google Scholar 

  • Gamber, E., 2006. Equicontinuity properties of D-dimensional cellular automata. Topol. Proc. 30(1), 197–222. Spring Topology and Dynamical Systems Conference.

    MATH  MathSciNet  Google Scholar 

  • Gamber Burkhead, E., 2008. A topological classification of D-dimensional cellular automata. Dyn. Syst. Int. J. to appear

  • Haase, A.T., 1999. Population biology of HIV-1 infection: Viral and CD4+  T cell demographics and dynamics in lymphatic tissues. Annu. Rev. Immunol. 17, 625–656.

    Article  Google Scholar 

  • Hawkins, J., Molinek, D., 2007. One-dimensional stochastic cellular automata. Topol. Proc. 31(2), 515–532.

    Google Scholar 

  • Hedlund, G.A., 1969. Endomorphisms and automorphisms of the shift dynamical system. Math. Syst. Theory 3, 320–375.

    Article  MATH  MathSciNet  Google Scholar 

  • Hubbs, J., 2006. A probabilistic cellular automaton model of the spread of the HIV virus. Master’s project. University of North Carolina at Chapel Hill.

  • Ilachinski, A., 2001. Cellular automata. A discrete universe. World Scientific, River Edge.

    MATH  Google Scholar 

  • Kari, J., 2005. Theory of cellular automata: A survey. Theor. Comput. Sci. 334, 3–33.

    Article  MATH  MathSciNet  Google Scholar 

  • Kitchens, B.P., 1998. Symbolic dynamics: one-sided, two-sided and countable state Markov shifts. Universitext, Springer, Berlin.

    MATH  Google Scholar 

  • Kůrka, P., 2001. Topological dynamics of cellular automata. In: Codes, Systems, and Graphical Models (Minneapolis, MN, 1999). IMA Vol. Math. Appl., vol. 123, pp. 447–485. Springer, Berlin

    Google Scholar 

  • Lind, D., Marcus, B., 1995. An introduction to symbolic dynamics and coding. Cambridge University Press, Cambridge.

    MATH  Google Scholar 

  • Nowak, M.A., May, R.M., 2000. Virus dynamics. Mathematical principles of immunology and virology. Oxford University Press, Oxford.

    MATH  Google Scholar 

  • Pantaleo, G., Fauci, A.S., 1996. Immunopathogenesis of HIV infection. Ann. Rev. Microbiol. 50, 825–854.

    Article  Google Scholar 

  • Pantaleo, G., Graziosi, C., Fauci, A., 1993. The immunopathogenesis of human immunodeficiency virus infection. New Engl. J. Med. 328(5), 327–335.

    Article  Google Scholar 

  • Schnittman, S.M., et al., 1989. The reservoir for HIV-1 in human peripheral blood is a T cell that maintains expression of CD4. Science 245(4915), 305–308.

    Article  Google Scholar 

  • Sinaĭ, Y.G., 1994. Topics in ergodic theory, Volume 44 of Princeton Mathematical Series. Princeton University Press, Princeton.

    MATH  Google Scholar 

  • Strain, M.C., Levine, H., 2002. Comment on “Dynamics of HIV infection: A cellular automata approach”. Phys. Rev. Lett. 89(21), 219805–1.

    Article  Google Scholar 

  • Walters, P., 1982. An introduction to ergodic theory. Springer, Berlin.

    MATH  Google Scholar 

  • Wolfram, S., 1984. Universality and complexity in cellular automata. Physica D 10(1–2), 1–35. Cellular automata (Los Alamos, NM, 1983).

    Article  MathSciNet  Google Scholar 

  • Wolfram, S., 2002. A new kind of science. Wolfram Media, Champaign.

    MATH  Google Scholar 

  • Zhang, Z.-Q., et al., 1998. Kinetics of CD4+T cell repopulation by lymphoid tissues after treatment of HIV-1 infection. Proc. Natl. Acad. Sci. 95, 1154–1159.

    Article  Google Scholar 

  • Zorzenon dos Santos, R.M., Coutinho, S., 2001. Dynamics of HIV infection: A cellular automata approach. Phys. Rev. Lett. 87(16), 168102-1–168102-4.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Computer Science, Meredith College, 3800 Hillsborough St., Raleigh, NC, 27607, USA

    E. G. Burkhead

  2. Department of Mathematics, University of North Carolina at Chapel Hill, CB #3250, Chapel Hill, NC, 27599-3250, USA

    J. M. Hawkins

  3. Department of Mathematics, Davidson College, P.O. Box 6999, Davidson, NC, 28035, USA

    D. K. Molinek

Authors
  1. E. G. Burkhead
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. M. Hawkins
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. D. K. Molinek
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to J. M. Hawkins.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Burkhead, E.G., Hawkins, J.M. & Molinek, D.K. A Dynamical Study of a Cellular Automata Model of the Spread of HIV in a Lymph Node. Bull. Math. Biol. 71, 25–74 (2009). https://doi.org/10.1007/s11538-008-9351-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11538-008-9351-0

Keywords

Search

Navigation