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Mathematical model for the growth of Mycobacterium tuberculosis in the granuloma

1. Departamento de Matemáticas y Estadística, Facultad de Ciencias Exactas y Naturales, Universidad de Nariño, Calle 18 Cra 50, Pasto, Colombia
2. Departamento de Matemáticas, Facultad de Ciencias, Universidad Nacional Autónoma de México, 04510 México DF, México
3. Departamento de Biología, Facultad de Ciencias Exactas y Naturales, Universidad de Nariño, Calle 18 Cra 50, Pasto, Colombia

In this work we formulate a model for the population dynamics of Mycobacterium tuberculosis (Mtb), the causative agent of tuberculosis (TB). Our main interest is to assess the impact of the competition among bacteria on the infection prevalence. For this end, we assume that Mtb population has two types of growth. The first one is due to bacteria produced in the interior of each infected macrophage, and it is assumed that is proportional to the number of infected macrophages. The second one is of logistic type due to the competition among free bacteria released by the same infected macrophages. The qualitative analysis and numerical results suggests the existence of forward, backward and S-shaped bifurcations when the associated reproduction number $R_0$ of the Mtb is less unity. In addition, qualitative analysis of the model shows that there may be up to three bacteria-present equilibria, two locally asymptotically stable, and one unstable.

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Keywords Ordinary differential equations; S-shaped bifurcation; tuberculosis; granuloma; macrophages and T cells

Citation: Eduardo Ibargüen-Mondragón, Lourdes Esteva, Edith Mariela Burbano-Rosero. Mathematical model for the growth of Mycobacterium tuberculosis in the granuloma. Mathematical Biosciences and Engineering, 2018, 15(2): 407-428. doi: 10.3934/mbe.2018018

References

  • [1] J. Alavez,R. Avendao,L. Esteva,J. A. Flores,J. L. Fuentes-Allen,G. Garca-Ramos,G. Gmez,J. Lpez Estrada, Population dynamics of antibiotic resistant M. tuberculosis, Math Med Biol, 24 (2007): 35-56.
  • [2] R. Antia,J. C. Koella,V. Perrot, Model of the Within-host dynamics of persistent mycobacterial infections, Proc R Soc Lond B, 263 (1996): 257-263.
  • [3] M. A. Behr,W. R. Waters, Is tuberculosis a lymphatic disease with a pulmonary portal?, Lancet, 14 (2004): 250-255.
  • [4] S. M. Blower,T. Chou, Modeling the emergence of the hot zones: Tuberculosis and the amplification dynamics of drug resistance, Nat Med, 10 (2004): 1111-1116.
  • [5] C. Castillo-Chávez,B. Song, Dynamical models of tuberculosis and their applications, Math Biosci Eng, 1 (2004): 361-404.
  • [6] T. Cohen,M. Murray, Modelling epidemics of multidrug-resistant M. tuberculosis of heterogeneous fitness, Nat Med, 10 (2004): 1117-1121.
  • [7] A. M. Cooper, Cell-mediated immune responses in tuberculosis, Annu Rev Immunol, 27 (2009): 393-422.
  • [8] C. Dye,M. A. Espinal, Will tuberculosis become resistant to all antibiotics?, Proc R Soc Lond B, 268 (2001): 45-52.
  • [9] F. R. Gantmacher, The Theory of Matrices, AMS Chelsea Publishing, Providence, RI, 1998.
  • [10] E. Guirado,L. S. Schlesinger, Modeling the Mycobacterium tuberculosis granuloma-the critical battlefield in host immunity and disease, Frontiers in Immunology, 4 (2013): 1-7.
  • [11] T. Gumbo,A. Louie,M. R. Deziel,L. M. Parsons,M. Salfinger,G. L. Drusano, Drusano, Selection of a moxifloxacin dose that suppresses drug resistance in Mycobacterium tuberculosis, by use of an in vitro pharmacodynamic infection model and mathematical modeling, J Infect Dis, 190 (2004): 1642-1651.
  • [12] E. G. Hoal-Van Helden,D. Hon,L. A. Lewis,N. Beyers,P. D. Van Helden, Mycobacterial growth in human macrophages: Variation according to donor, inoculum and bacterial strain, Cell Biol Int, 25 (2001): 71-81.
  • [13] E. Ibargüen-Mondragón,L. Esteva,L. Chávez-Galán, A mathematical model for cellular immunology of tuberculosis, Math Biosci Eng, 8 (2011): 973-986.
  • [14] E. Ibargüen-Mondragón,L. Esteva, Un modelo matemático sobre la dinámica del Mycobacterium tuberculosis en el granuloma, Revista Colombiana de Matemáticas, 46 (2012): 39-65.
  • [15] E. Ibargüen-Mondragón,J. P. Romero-Leiton,L. Esteva,E. M. Burbano-Rosero, Mathematical modeling of bacterial resistance to antibiotics by mutations and plasmids, J Biol Syst, 24 (2016): 129-146.
  • [16] E. Ibargüen-Mondragón,S. Mosqueraa,M. Cerón,E. M. Burbano-Rosero,S. P. Hidalgo-Bonilla,L. Esteva,J. P. Romero-Leiton, Mathematical modeling on bacterial resistance to multiple antibiotics caused by spontaneous mutations, BioSystems, 117 (2014): 60-67.
  • [17] S. Kaufmann, How can immunology contribute to the control of tuberculosis?, Nat Rev Immunol, 1 (2001): 20-30.
  • [18] D. Kirschner, Dynamics of Co-infection with M. tuberculosis and HIV-1, Theor Popul Biol, 55 (1999): 94-109.
  • [19] H. Koppensteiner,R. Brack-Werner,M. Schindler, Macrophages and their relevance in Human Immunodeficiency Virus Type Ⅰ infection, Retrovirology, 9 (2012): p82.
  • [20] Q. Li,C. C. Whalen,J. M. Albert,R. Larkin,L. Zukowsy,M. D. Cave,R. F. Silver, Differences in rate and variability of intracellular growth of a panel of Mycobacterium tuberculosis clinical isolates within monocyte model, Infect Immun, 70 (2002): 6489-6493.
  • [21] G. Magombedze,W. Garira,E. Mwenje, Modellingthe human immune response mechanisms to mycobacterium tuberculosis infection in the lungs, Math Biosci Eng, 3 (2006): 661-682.
  • [22] S. Marino,D. Kirschner, The human immune response to the Mycobacterium tuberculosis in lung and lymph node, J Theor Biol, 227 (2004): 463-486.
  • [23] J. Murphy,R. Summer,A. A. Wilson,D. N. Kotton,A. Fine, The prolonged life-span of alveolar macrophages, Am J Respir Cell Mol Biol, 38 (2008): 380-385.
  • [24] G. Pedruzzi,K. V. Rao,S. Chatterjee, Mathematical model of mycobacterium-host interaction describes physiology of persistence, J Theor Biol, 376 (2015): 105-117.
  • [25] L. Ramakrishnan, Revisiting the role of the granuloma in tuberculosis, Nat Rev Immunol, 12 (2012): 352-366.
  • [26] D. Russell, Who puts the tubercle in tuberculosis?, Nat Rev Microbiol, 5 (2007): 39-47.
  • [27] A. Saltelli,M. Ratto,S. Tarantola,F. Campolongo, Sensitivity analysis for chemical models, Chem Rev, 105 (2005): 2811-2828.
  • [28] M. Sandor,J. V. Weinstock,T. A. Wynn, Granulomas in schistosome and mycobacterial infections: A model of local immune responses, Trends Immunol, 24 (2003): 44-52.
  • [29] R. Shi,Y. Li,S. Tang, A mathematical model with optimal constrols for cellular immunology of tuberculosis, Taiwan J Math, 18 (2014): 575-597.
  • [30] D. Sud,C. Bigbee,J. L. Flynn,D. E. Kirschner, Contribution of CD8+ T cells to control of Mycobacterium tuberculosis infection, J Immunol, 176 (2006): 4296-4314.
  • [31] D. F. Tough,J. Sprent, Life span of naive and memory T cells, Stem Cells, 13 (1995): 242-249.
  • [32] M. C. Tsai,S. Chakravarty,G. Zhu,J. Xu,K. Tanaka,C. Koch,J. Tufariello,J. Flynn,J. Chan, Characterization of the tuberculous granuloma in murine and human lungs: cellular composition and relative tissue oxygen tension, Cell Microbiol, 8 (2006): 218-232.
  • [33] S. Umekia,Y. Kusunokia, Lifespan of human memory T-cells in the absence of T-cell receptor expression, Immunol Lettt, 62 (1998): 99-104.
  • [34] L. Westera,J. Drylewicz, Closing the gap between T-cell life span estimates from stable isotope-labeling studies in mice and humans, BLOOD, 122 (2013): 2205-2212.
  • [35] J. E. Wigginton,D. E. Kischner, A model to predict cell mediated immune regulatory mechanisms during human infection with Mycobacterium tuberculosis, J Immunol, 166 (2001): 1951-1967.
  • [36] Word Health Organization (WHO), Global tuberculosis report 2015,2003. Available from: http://apps.who.int/iris/bitstream/10665/191102/1/9789241565059_eng.pdf.
  • [37] Word Health Organization (WHO), Global tuberculosis report 2016,2003. Available from: http://apps.who.int/iris/bitstream/10665/250441/1/9789241565394-eng.pdf?ua=1.
  • [38] M. Zhang,J. Gong,Z. Yang,B. Samten,M. D. Cave,P. F. Barnes, Enhanced capacity of a widespread strain of Mycobacterium tuberculosis to grow in human monocytes, J Infect Dis, 179 (1998): 1213-1217.
  • [39] M. Zhang,S. Dhandayuthapani,V. Deretic, Molecular basis for the exquisite sensitivity of Mycobacterium tuberculosis to isoniazid, Proc Natl Acad Sci U S A, 93 (1996): 13212-13216.

 

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