American Institute of Mathematical Sciences

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2010, 7(2): 277-300. doi: 10.3934/mbe.2010.7.277

A pressure model of immune response to mycobacterium tuberculosis infection in several space dimensions

Fabrizio Clarelli 1, and  Roberto Natalini 1,

1. 

Istituto per le Applicazioni del Calcolo “M. Picone”, CNR, c/o Dip. di Matematica, Università di Roma “Tor Vergata”, Via della Ricerca Scientifica, 1; I-00133 Roma, Italy, Italy

Received  January 2009 Revised  October 2009 Published  April 2010

Mycobacterium tuberculosis (Mtb) is a widely diffused infection. However, in general, the human immune system is able to contain it. In this work, we propose a mathematical model which describes the early immune response to the Mtb infection in the lungs, also including the possible evolution of the infection in the formation of a granuloma. The model is based on coupled reaction-diffusion-transport equations with chemotaxis, which take into account the interactions among bacteria, macrophages and chemoattractant. The novelty of this approach is in the modeling of the velocity field, proportional to the gradient of the pressure developed between the cells, which makes possible to deal with a full multidimensional description and efficient numerical simulations. We perform a linear stability analysis of the model and propose a robust implicit-explicit scheme to deal with long time simulations. Both in one and two-dimensions, we find that there are threshold values in the parameters space, between a contained infection and the uncontrolled bacteria growth, and the generation of granuloma-like patterns can be observed numerically.
Keywords: Internal velocity., Reaction-diffusion-advection, Multi-dimensional model, Mycobacterium tuberculosis, Chemotaxis.
Mathematics Subject Classification: Primary: 35K57, 92C17 ; Secondary: 65C20, 93A3.
Citation: Fabrizio Clarelli, Roberto Natalini. A pressure model of immune response to mycobacterium tuberculosis infection in several space dimensions. Mathematical Biosciences & Engineering, 2010, 7 (2) : 277-300. doi: 10.3934/mbe.2010.7.277
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