Journal of Mathematical Biology

, Volume 65, Issue 2, pp 263–291

# 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
Article

## Abstract

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.

### Keywords

Immune response CD8 T cell Ordinary differential equations Delay equations

### Mathematics Subject Classification (2000)

34D20 34K60 35L60 35Q92 92C37

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## 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