Journal of Mathematical Biology

, Volume 65, Issue 2, pp 263–291 | Cite as

Mathematical model of the primary CD8 T cell immune response: stability analysis of a nonlinear age-structured system

  • Emmanuelle Terry
  • Jacqueline Marvel
  • Christophe Arpin
  • Olivier Gandrillon
  • Fabien Crauste


The primary CD8 T cell immune response, due to a first encounter with a pathogen, happens in two phases: an expansion phase, with a fast increase of T cell count, followed by a contraction phase. This contraction phase is followed by the generation of memory cells. These latter are specific of the antigen and will allow a faster and stronger response when encountering the antigen for the second time. We propose a nonlinear mathematical model describing the T CD8 immune response to a primary infection, based on three nonlinear ordinary differential equations and one nonlinear age-structured partial differential equation, describing the evolution of CD8 T cell count and pathogen amount. We discuss in particular the roles and relevance of feedback controls that regulate the response. First we reduce our system to a system with a nonlinear differential equation with a distributed delay. We study the existence of two steady states, and we analyze the asymptotic stability of these steady states. Second we study the system with a discrete delay, and analyze global asymptotic stability of steady states. Finally, we show some simulations that we can obtain from the model and confront them to experimental data.


Immune response CD8 T cell Ordinary differential equations Delay equations 

Mathematics Subject Classification (2000)

34D20 34K60 35L60 35Q92 92C37 


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Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  • Emmanuelle Terry
    • 1
    • 2
  • Jacqueline Marvel
    • 3
  • Christophe Arpin
    • 3
  • Olivier Gandrillon
    • 2
    • 4
  • Fabien Crauste
    • 1
    • 2
  1. 1.Université de Lyon, Université Lyon 1, CNRS UMR 5208, Institut Camille JordanVilleurbanne-CedexFrance
  2. 2.INRIA Team Dracula, INRIA Center Grenoble Rhône-AlpesLyonFrance
  3. 3.INSERM U851 Université de Lyon, Université Lyon 1LyonFrance
  4. 4.Université de Lyon, Université Lyon 1, CNRS UMR 5534,Centre de Génétique et de Physiologie Moléculaire et CellulaireVilleurbanne-CedexFrance

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