Numerical solution of differential equations, modeling the evolution of some T-cells

We present a new mathematical model based on coupled reaction-diffusion partial differential equations describing the evolution of some T-cells in the immune system, with particular emphasis to satisfying the hypothesis on homeostasis of CD4+ cells described in [1]. The genesis of the model is described and framed into the Immunology setting, which was also conveyed into the choice of the proper boundary and initial conditions to impose in order to reach the expected global behavior of the solutions. The numerical treatment of the model is also presented, through proper numerical schemes ensuring that the numerical solution provides the prescribed qualitative behavior. Numerical experiments are also presented. References [1] Afonso R.M.Almeida, Ines F.Amado, Joseph Reynolds, Julien Berges, Grant Lythe, Carmen Molina-Paris and Antonio A.Freitas, Quorum-sensing in CD4+T cell homeostasis: a hypothesis and a model, 2012, Vol.3, Art.125 [2] R. Antia, T.G. Bergstrom, S. Pilyugin, S.M. Kaech, R. Ahmed, Models of CD8+ responses: What is the Antigen-Dependent Proliferation Program, J. theor. Biol. 221, 585-598 (2003). [3] P.S. Kim, D. Levy, P. P. Lee, Modeling and Simulation of the Immune System as a SelfRegulating Network, Methods Enzymol. 467, 79-109 (2009)