# American Institute of Mathematical Sciences

2011, 8(4): 973-986. doi: 10.3934/mbe.2011.8.973

## A mathematical model for cellular immunology of tuberculosis

 1 Departamento de Matemáticas y Estadística,, Universidad de Nariño, Pasto, Clle 18 - Cra 50, Colombia, Posgrado en Ciencias Matemáticas, UNAM, 04510 DF, Mexico 2 Departamento de Matemáticas, Facultad de Ciencias,, Universidad Nacional Autónoma de M ́exico, 04510, DF, Mexico 3 Departamento de Bioquímica, Instituto Nacional de Enfermedades Respiratorias ”Ismael Cosio Villegas”, Calzada Tlalpan 4502, Colonia sección XVI, 14080 DF, Mexico

Received  February 2010 Revised  February 2011 Published  August 2011

Tuberculosis (TB) is a global emergency. The World Health Organization reports about 9.2 million new infections each year, with an average of 1.7 million people killed by the disease. The causative agent is Mycobacterium tuberculosis (Mtb), whose main target are the macrophages, important immune system cells. Macrophages and T cell populations are the main responsible for fighting the pathogen. A better understanding of the interaction between Mtb, macrophages and T cells will contribute to the design of strategies to control TB. The purpose of this study is to evaluate the impact of the response of T cells and macrophages in the control of Mtb. To this end, we propose a system of ordinary differential equations to model the interaction among non-infected macrophages, infected macrophages, T cells and Mtb bacilli. Model analysis reveals the existence of two equilibrium states, infection-free equilibrium and the endemically infected equilibrium which can represent a state of latent or active infection, depending on the amount of bacteria.
Citation: Eduardo Ibarguen-Mondragon, Lourdes Esteva, Leslie Chávez-Galán. A mathematical model for cellular immunology of tuberculosis. Mathematical Biosciences & Engineering, 2011, 8 (4) : 973-986. doi: 10.3934/mbe.2011.8.973
##### References:
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##### References:
 [1] J. Alavez, J. Avendano, L. Esteva, J. Florez, J. Fuentes, G. García, G. Gómez and J. López, Within-host population dynamics of antibiotic-resistant M. tuberculosis, Mathematical Medicine and Biology, 24 (2007), 35-56. [2] M. Anis, S. Fulton, S. Reba, Y. Liu, C. Harding and W. Bomm, Modulation of pulmonary dendritic cell function during mycobacterial infection, Infect. Immun, 76 (2008), 671-677. [3] Centers for Disease Control and Precention, "TB Elimination. The Difference Between Latent TB Infection and Active TB Disease," March 2010. Available from: http://www.cdc.gov/tb/publications/factsheets/general/LTBIandActiveTB.pdf. [4] J. Chan and J. Flynn, The immunological aspects of latency in tuberculosis, Clin. Immunol, 110 (2004), 2-12. [5] J. Davis and L. Ramakrishnan, The role of the granuloma in expansion and dissemination of early tuberculous infection, Cell, 136 (2009), 37-49. [6] J. Egen, A. Rothfuchs, C. Feng, N. Winter, A. Sher and R. Germain, Macrophage and T cell dynamics during the development and disintegration of mycobacterial granulomas, Immunity, 28 (2008), 271-284. [7] A. Gallegos, E. Pamer and M. Glickman, Delayed protection by ESAT-6-specific effector CD4$^+$ T cells after airborne M. tuberculosis infection, J. Exp. Med, 205 (2008), 2359-2358. [8] J. Hale, "Ordinary Differential Equations," first edition, [9] D. Kirschner, D. Sud, C. Bigbee and J. L. Flynn, Contribution of CD8$^+$T cells to control of mycobacterium tuberculosis infection, The Journal of Immunology, 176 (2006), 4296-4314. [10] G. Magombedze, W. Garira and E. Mwenje, Modelling the human immune response mechanisms to Mycobacterium tuberculosis infection in the lungs, J. MBE, 3 (2006), 661-682. [11] S. Marino and D. Kirschner, The human immune response to the Mycobacterium tuberculosis in lung and lymph node, Journal of Theoretical Biology, 227 (2004), 463-486. [12] J. Davis and L. Ramakrishnan, The role of the granuloma in expansion and dissemination of early tuberculous infection, Cell, 136 (2007), 37-49. [13] J. Palomino, S. Cardoso and V. Ritacco, "Tuberculosis 2007, From Basic Science to Patient Care," first edition, [14] M. Sköld and S. M. Behar, Tuberculosis triggers a tissue-dependent program of differentiation and acquisition of effector functions by circulating monocytes, J. Immunology, 81 (2008), 6349-6360. [15] I. Sugawara, S. Mizuno, T. Tatsumi and T. Taniyama, Imaging of pulmonary granulomas using a photon imager, Jpn. J. Infect Dis., 59 (2006), 332-333. [16] M. Tsai, S. Chakravarty, G. Zhu, J. Xu, K. Tanaka, C. Koch, J. Tufariello, J. Flynn and J. Chan, Characterization of the tuberculous granuloma in murine and human lungs: Cellular composition and relative tissue oxygen tension, Cell Microbiology, 8 (2006), 218-232. [17] T. Ulrischs and S. Kaufmann, Cell mediated immune response in Tuberculosis, first edition, Lippincott Williams and Wilkins, New York, 2004. [18] WHO Report, "Global Tuberculosis Control - Surveillance, Planning, Financing," 2008. Available from: http://www.who.int/tb/publications/global_report/2008/pdf/fullreport.pdf. [19] J. Wigginton and D. Kischner, A model to predict cell mediated imune regulatory mechanisms during human infection with Mycobacterium tuberculosis, J. Immunology, 166 (2001), 1951-1967. [20] A. J. Wolf, B. Linas, G. J. Trevejo-Nuñez, E. Kincaid, T. Tamura, K. Takatsu and J. D. Ernst, Mycobacterium tuberculosis infects dendritic cells with high frequency and impairs their function in vivo, J. Immunology, 179 (2007), 2509-2519.
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