Article Contents
Article Contents

# Mathematical model for the growth of Mycobacterium tuberculosis in the granuloma

• * Corresponding author: Eduardo Ibargüen-Mondragón
• 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.

Mathematics Subject Classification: Primary: 34D23, 93D20; Secondary: 65L05.

 Citation:

• Figure 1.  The flow diagram of macrophages, T cells and bacteria

Figure 2.  The graph of functions $g_1$ and $g_2$ defined in (20).

Figure 3.  Standard regression coefficients (SCR) for $R_0 = \frac{\nu}{\gamma_U + \mu_{B}}$ assuming the values given in Table 1 for $\nu, \gamma_{U} = \bar \gamma\displaystyle{\Lambda_U \over \mu_U}$ and $\mu_{B}$

Figure 4.  Standard regression coefficients (SCR) for $R_1 = \frac{\bar r \bar\beta {\Lambda_U \over \mu}}{\gamma_U + \mu_{B}}$ assuming the values given in Table 1 for $\bar r, \bar \beta, \displaystyle{\Lambda_U \over \mu_U}, \gamma_{U} = \bar \gamma\displaystyle{\Lambda_U \over \mu_U}$ and $\mu_{B}$.

Figure 5.  The numerical simulations of temporal course for bacteria with ten initial conditions show the stability of the bacteria-present equilibrium $P_2$ and the infection free equilibrium $P_0$ given in (47) when $\sigma = 0.24$, $\sigma_c = 0.319$, $R_0 = 0.4$, $R_0 = 0.34$, $R_1 = 1.5$, $g_1(B^{\max}) = 1.37\times 10^{312}$ and $g_2(B^{\max}) = 1.32\times 10^{942}$.

Figure 6.  The numerical simulations of temporal course for bacteria with ten initial conditions show the stability of the bacteria-present equilibria $P_1$ a$P_3$ given in (48) when $\sigma = 2.4\times 10^{-6}$, $\sigma_c = 0.003$, $R_0 = 0.0045$, $R^*_0 = 0.0043$, $R_1 = 0.43$.

Figure 7.  The stable infection free equilibrium $P_0$ bifurcates to the stable bacteria-present equilibrium $P_1$ in the value $R_0 = 1-R_1$.

Figure 8.  The results suggest forward and backward bifurcations, and a type of S-shaped bifurcation

Table 1.  Interpretation and values of the parameters. Data are deduced from the literature (references).

 Parameter Description Value Reference $\Lambda_U$ growth rate of unfected Mtb 600 -1000 day$^{-1}$ [19,23,30] $\bar\beta$ infection rate of Mtb $2.5*10^{-11}-2.5*10^{-7}$day$^{-1}$ [13,30] $\bar\alpha_T$ elim. rate of infected Mtb by T cell $2*10^{-5}-3*10^{-5}$ day$^{-1}$ [13,30] $\mu_U$ nat. death rate of $M_U$ 0028-0.0033 day$^{-1}$ [22,30] $\mu_I$ nat. death rate of $M_I$ 0.011 day$^{-1}$ [22,35,30] $\nu$ growth rate of Mtb 0.36 -0.52 day$^{-1}$ [12,20,38] $\mu_{B}$ natural death rate of Mtb 0.31 -0.52 day$^{-1}$ [39,30] $\bar \gamma_U$ elim. rate of Mtb by $M_U$ $1.2* 10^{-9} - 1.2*10^{-7}$ day$^{-1}$ [30] $K$ carrying cap. of Mtb in the gran. $10^8-10^9$ bacteria [7] $\bar k_I$ growth rate of T cells $8*10^{-3}$ day$^{-1}$ [11] $T_{max}$ maximum recruitment of T cells 5.000 day$^{-1}$ [11] $\mu_T$ natural death rate of T cells 0.33 day$^{-1}$ [35,30] $\bar r$ Average Mtb released by one $M_U$ 0.05-0.2 day$^{-1}$ [30,35]
•  [1] J. Alavez, R. Avendao, L. Esteva, J. A. Flores, J. L. Fuentes-Allen, G. Garca-Ramos, G. Gmez and J. Lpez Estrada, Population dynamics of antibiotic resistant M. tuberculosis, Math Med Biol, 24 (2007), 35-56. [2] R. Antia, J. C. Koella and V. Perrot, Model of the Within-host dynamics of persistent mycobacterial infections, Proc R Soc Lond B, 263 (1996), 257-263.  doi: 10.1098/rspb.1996.0040. [3] M. A. Behr and W. R. Waters, Is tuberculosis a lymphatic disease with a pulmonary portal?, Lancet, 14 (2004), 250-255.  doi: 10.1016/S1473-3099(13)70253-6. [4] S. M. Blower and T. Chou, Modeling the emergence of the hot zones: Tuberculosis and the amplification dynamics of drug resistance, Nat Med, 10 (2004), 1111-1116.  doi: 10.1038/nm1102. [5] C. Castillo-Chávez and B. Song, Dynamical models of tuberculosis and their applications, Math Biosci Eng, 1 (2004), 361-404.  doi: 10.3934/mbe.2004.1.361. [6] T. Cohen and 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.  doi: 10.1146/annurev.immunol.021908.132703. [8] C. Dye and M. A. Espinal, Will tuberculosis become resistant to all antibiotics?, Proc R Soc Lond B, 268 (2001), 45-52.  doi: 10.1098/rspb.2000.1328. [9] F. R. Gantmacher, The Theory of Matrices, AMS Chelsea Publishing, Providence, RI, 1998. [10] E. Guirado and L. S. Schlesinger, Modeling the Mycobacterium tuberculosis granuloma-the critical battlefield in host immunity and disease, Frontiers in Immunology, 4 (2013), 1-7.  doi: 10.3389/fimmu.2013.00098. [11] T. Gumbo, A. Louie, M. R. Deziel, L. M. Parsons, M. Salfinger and 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 and P. D. Van Helden, Mycobacterial growth in human macrophages: Variation according to donor, inoculum and bacterial strain, Cell Biol Int, 25 (2001), 71-81.  doi: 10.1006/cbir.2000.0679. [13] E. Ibargüen-Mondragón, L. Esteva and L. Chávez-Galán, A mathematical model for cellular immunology of tuberculosis, Math Biosci Eng, 8 (2011), 973-986.  doi: 10.3934/mbe.2011.8.973. [14] E. Ibargüen-Mondragón and 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 and E. M. Burbano-Rosero, Mathematical modeling of bacterial resistance to antibiotics by mutations and plasmids, J Biol Syst, 24 (2016), 129-146.  doi: 10.1142/S0218339016500078. [16] E. Ibargüen-Mondragón, S. Mosqueraa, M. Cerón, E. M. Burbano-Rosero, S. P. Hidalgo-Bonilla, L. Esteva and 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.  doi: 10.1038/35095558. [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 and M. Schindler, Macrophages and their relevance in Human Immunodeficiency Virus Type Ⅰ infection, Retrovirology, 9 (2012), p82.  doi: 10.1186/1742-4690-9-82. [20] Q. Li, C. C. Whalen, J. M. Albert, R. Larkin, L. Zukowsy, M. D. Cave and 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.  doi: 10.1128/IAI.70.11.6489-6493.2002. [21] G. Magombedze, W. Garira and E. Mwenje, Modellingthe human immune response mechanisms to mycobacterium tuberculosis infection in the lungs, Math Biosci Eng, 3 (2006), 661-682.  doi: 10.3934/mbe.2006.3.661. [22] S. Marino and D. Kirschner, The human immune response to the Mycobacterium tuberculosis in lung and lymph node, J Theor Biol, 227 (2004), 463-486.  doi: 10.1016/j.jtbi.2003.11.023. [23] J. Murphy, R. Summer, A. A. Wilson, D. N. Kotton and A. Fine, The prolonged life-span of alveolar macrophages, Am J Respir Cell Mol Biol, 38 (2008), 380-385.  doi: 10.1165/rcmb.2007-0224RC. [24] G. Pedruzzi, K. V. Rao and S. Chatterjee, Mathematical model of mycobacterium-host interaction describes physiology of persistence, J Theor Biol, 376 (2015), 105-117.  doi: 10.1016/j.jtbi.2015.03.031. [25] L. Ramakrishnan, Revisiting the role of the granuloma in tuberculosis, Nat Rev Immunol, 12 (2012), 352-366.  doi: 10.1038/nri3211. [26] D. Russell, Who puts the tubercle in tuberculosis?, Nat Rev Microbiol, 5 (2007), 39-47.  doi: 10.1038/nrmicro1538. [27] A. Saltelli, M. Ratto, S. Tarantola and F. Campolongo, Sensitivity analysis for chemical models, Chem Rev, 105 (2005), 2811-2828. [28] M. Sandor, J. V. Weinstock and 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 and S. Tang, A mathematical model with optimal constrols for cellular immunology of tuberculosis, Taiwan J Math, 18 (2014), 575-597.  doi: 10.11650/tjm.18.2014.3739. [30] D. Sud, C. Bigbee, J. L. Flynn and D. E. Kirschner, Contribution of CD8+ T cells to control of Mycobacterium tuberculosis infection, J Immunol, 176 (2006), 4296-4314. [31] D. F. Tough and J. Sprent, Life span of naive and memory T cells, Stem Cells, 13 (1995), 242-249.  doi: 10.1002/stem.5530130305. [32] M. C. 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 Microbiol, 8 (2006), 218-232.  doi: 10.1111/j.1462-5822.2005.00612.x. [33] S. Umekia and Y. Kusunokia, et al., Lifespan of human memory T-cells in the absence of T-cell receptor expression, Immunol Lettt, 62 (1998), 99-104.  doi: 10.1016/S0165-2478(98)00037-6. [34] L. Westera and J. Drylewicz, et al., Closing the gap between T-cell life span estimates from stable isotope-labeling studies in mice and humans, BLOOD, 122 (2013), 2205-2212.  doi: 10.1182/blood-2013-03-488411. [35] J. E. Wigginton and D. E. Kischner, A model to predict cell mediated immune regulatory mechanisms during human infection with Mycobacterium tuberculosis, J Immunol, 166 (2001), 1951-1967.  doi: 10.4049/jimmunol.166.3.1951. [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 and 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 and V. Deretic, Molecular basis for the exquisite sensitivity of Mycobacterium tuberculosis to isoniazid, Proc Natl Acad Sci U S A, 93 (1996), 13212-13216.  doi: 10.1073/pnas.93.23.13212.

Figures(8)

Tables(1)