Mathematical Model for the Growth of Mycobacterium Tuberculosis Infection in the Lungs
DOI:
https://doi.org/10.5614/cbms.2025.8.1.8Keywords:
Mathematical model, tuberculosis, Mtb bacterial, immune responseAbstract
In this work, we develop a population dynamics model of Mycobacterium tuberculosis (Mtb), the bacteria responsible for tuberculosis (TB), to evaluate the impact of bacterial competition on infection prevalence. We consider two types of Mtb population growth: The first is caused by bacteria that grow inside each infected macrophage and is believed to be correlated with the number of infected macrophages; The second is that extracellular bacteria grow through self-replication. In this study, we modeled the immune response to Mtb bacterial infection in the lungs using a five-dimensional differential equation system. This model represents changes in the number of healthy macrophages, infected macrophages, activated macrophages cells, extracellular bacterial particles, and naive T cells. Qualitative analysis and numerical results reveal the existence of two equilibrium points: disease-free equilibrium and endemic equilibrium, which represent latent or active tuberculosis based on the number of bacteria. In addition, a sensitive analysis of the model parameters shows that macrophages are not sufficient to control the initial invasion of Mtb. The immune system must therefore employ more complex defense mechanisms to contain Mtb infection, such as recruiting various elements of immune system and forming granulomas.
References
Alavez-Ramez, J., Castellanos, J.R.A., Esteva, L., Flores, J.A., Fuentes-Allen, J.L., Garc-Ramos, G., Gez, G. and Lez-Estrada, J., Within-host population dynamics of antibiotic-resistant M. tuberculosis, Mathematical Medicine and Biology, 24(1), pp. 35-56, 2007.
Antia, R., Koella, J.C. and Perrot, V., Models of the within-host dynamics of persistent mycobacterial infections, Proceedings of the Royal Society of London. Series B: Biological Sciences, 263(1368), pp. 257-263, 1996.
Anton, H. and Rorres, C., Elementary Linear Algebra: Applications Version, John Wiley & Sons, 2013.
Behr, M.A., Edelstein, P.H. and Ramakrishnan, L., Is Mycobacterium tuberculosis infection life long?, BMJ, 367, p. l5770, 2019.
Blower, S.M., McLean, A.R., Porco, T.C., Small, P.M., Hopewell, P.C., Sanchez, M.A. and Moss, A.R., The intrinsic transmission dynamics of tuberculosis epidemics, Nature Medicine, 1(8), pp. 815-821, 1995.
Bonecini-Almeida, M.G., Chitale, S., Boutsikakis, I., Geng, J., Doo, H., He, S. and Ho, J.L., Induction of in vitro human macrophage anti-Mycobacterium tuberculosis activity: requirement for IFN-? and primed lymphocytes, The Journal of Immunology, 160(9), pp. 4490-4499, 1998.
Carroll, K., Hobden, J., Miller, S., Morse, S., Mietzner, T., Detrick, B., Mitchell, T., McKerrow, J., Sakanari, J. and Jawetz, M., Adelberg?s Medical Microbiology, 27e, McGraw-Hill Education, United States, 2015.
Chakaya, J., Khan, M., Ntoumi, F., Aklillu, E., Fatima, R., Mwaba, P., Kapata, N., Mfinanga, S., Hasnain, S.E., Katoto, P.D., et al., Global tuberculosis report 2020?Reflections on the global TB burden, treatment and prevention efforts, International Journal of Infectious Diseases, 113, pp. S7-S12, 2021.
Chitnis, N., Hyman, J.M. and Cushing, J.M., Determining important parameters in the spread of malaria through the sensitivity analysis of a mathematical model, Bulletin of Mathematical Biology, 70, pp. 1272-1296, 2008.
Condos, R., Rom, W.N., Liu, Y.M. and Schluger, N.W., Local immune responses correlate with presentation and outcome in tuberculosis, American Journal of Respiratory and Critical Care Medicine, 157(3), pp. 729-735, 1998.
Danyun, H., Qian, W. and Wing-Cheong, L., Mathematical analysis of macrophage-bacteria interaction in tuberculosis infection, Discrete and Continuous Dynamical Systems-Series B, 23(8), pp. 3387-3413, 2018.
De Martino, M., Lodi, L., Galli, L. and Chiappini, E., Immune response to Mycobacterium tuberculosis: a narrative review, Frontiers in Pediatrics, 7, p. 350, 2019.
Du, Y., Wu, J. and Heffernan, J.M., A simple in-host model for Mycobacterium tuberculosis that captures all infection outcomes, Mathematical Population Studies, 24(1), pp. 37-63, 2017.
Egen, J.G., Rothfuchs, A.G., Feng, C.G., Winter, N., Sher, A. and Germain, R.N., Macrophage and T cell dynamics during the development and disintegration of mycobacterial granulomas, Immunity, 28(2), pp. 271-284, 2008.
Emery, J.C., Richards, A.S., Dale, K.D., McQuaid, C.F., White, R.G., Denholm, J.T. and Houben, R.M., Self-clearance of Mycobacterium tuberculosis infection: implications for lifetime risk and population at-risk of tuberculosis disease, Proceedings of the Royal Society B, 288(1943), p. 20201635, 2021.
Flesch, I. and Kaufmann, S., Activation of tuberculostatic macrophage functions by gamma interferon, interleukin-4, and tumor necrosis factor, Infection and Immunity, 58(8), pp. 2675-2677, 1990.
Gallegos, A.M., van Heijst, J.W., Samstein, M., Su, X., Pamer, E.G. and Glickman, M.S., A gamma interferon independent mechanism of CD4 T cell mediated control of M. tuberculosis infection in vivo, PLoS Pathogens, 7(5), p. e1002052, 2011.
Getahun, H., Matteelli, A., Chaisson, R.E. and Raviglione, M., Latent Mycobacterium tuberculosis infection, New England Journal of Medicine, 372(22), pp. 2127-2135, 2015.
Guirado, E. and Schlesinger, L.S., Modeling the Mycobacterium tuberculosis granuloma?The critical battlefield in host immunity and disease, Frontiers in Immunology, 4, p. 98, 2013.
Heffernan, J.M., Smith, R.J. and Wahl, L.M., Perspectives on the basic reproductive ratio, Journal of the Royal Society Interface, 2(4), pp. 281-293, 2005.
Houben, R.M. and Dodd, P.J., The global burden of latent tuberculosis infection: a re-estimation using mathematical modelling, PLoS Medicine, 13(10), p. e1002152, 2016.
Ibargn-Mondrag, E. and Esteva, L., Un modelo matemico sobre la dinica del Mycobacterium tuberculosis en el granuloma, Revista Colombiana de Matemicas, 46(1), pp. 39-65, 2012.
Ibargn-Mondrag, E. and Esteva, L., On the interactions of sensitive and resistant Mycobacterium tuberculosis to antibiotics, Mathematical Biosciences, 246(1), pp. 84-93, 2013.
Ibargn-Mondrag, E. and Esteva, L., On CTL response against Mycobacterium tuberculosis, Applied Mathematical Sciences, 8(48), pp. 2383-2389, 2014.
Ibargn-Mondrag, E. and Esteva, L., Simple mathematical models on macrophages and CTL responses against Mycobacterium tuberculosis, Journal of Biological Systems, 24(1), pp. 1-18, 2017.
Ibargn-Mondrag, E., Esteva, L. and Burbano-Rosero, E.M., Mathematical model for the growth of Mycobacterium tuberculosis in the granuloma, Mathematical Biosciences and Engineering, 15(2), pp. 407-428, 2018.
Ibarguen-Mondragon, E., Esteva, L. and Chez-Gal, L., A mathematical model for cellular immunology of tuberculosis, Mathematical Biosciences & Engineering, 8(4), pp. 973-986, 2011.
Ibargn-Mondrag, E., Mosquera, S., Cer, M., Burbano-Rosero, E.M., Hidalgo-Bonilla, S.P., Esteva, L. and Romero-Leit, J.P., Mathematical modeling on bacterial resistance to multiple antibiotics caused by spontaneous mutations, Biosystems, 117, pp. 60-67, 2014.
Ibargn-Mondrag, E., Romero-Leiton, J.P., Esteva, L. and Burbano-Rosero, E.M., Mathematical modeling of bacterial resistance to antibiotics by mutations and plasmids, Journal of Biological Systems, 24(1), pp. 129-146, 2016.
Janeway, C., Travers, P., Walport, M. and Shlomchik, M.J., Immunobiology: The Immune System in Health and Disease, Volume 2, Garland Pub., New York, 2001.
Kaufmann, S.H., How can immunology contribute to the control of tuberculosis?, Nature Reviews Immunology, 1(1), pp. 20-30, 2001.
Kirschner, D., Dynamics of co-infection with M. tuberculosis and HIV-1, Theoretical Population Biology, 55(1), pp. 94-109, 1999.
Lazarevic, V. and Flynn, J., CD8+ T cells in tuberculosis, American Journal of Respiratory and Critical Care Medicine, 166(8), pp. 1116-1121, 2002.
Magombedze, G., Garira, W. and Mwenje, E., Modelling the human immune response mechanisms to Mycobacterium tuberculosis infection in the lungs, Mathematical Biosciences & Engineering, 3(4), pp. 661-682, 2006.
Mahamed, D., Boulle, M., Ganga, Y., McArthur, C., Skroch, S., Oom, L., Catinas, O., Pillay, K., Naicker, M., Rampersad, S., et al., Intracellular growth of Mycobacterium tuberculosis after macrophage cell death leads to serial killing of host cells, ELife, 6, p. e22028, 2017.
Marino, S. and Kirschner, D.E., The human immune response to Mycobacterium tuberculosis in lung and lymph node, Journal of Theoretical Biology, 227(4), pp. 463-486, 2004.
Marino, S., Sud, D., Plessner, H., Lin, P.L., Chan, J., Flynn, J.L. and Kirschner, D.E., Differences in reactivation of tuberculosis induced from anti-TNF treatments are based on bioavailability in granulomatous tissue, PLoS Computational Biology, 3(10), p. e194, 2007.
Nachbar, J., Concave and convex functions, Lecture Notes for Economics, 4111, 2018.
Opoku, N.K.-D.O. and Mazandu, G.K., Modelling the human immune response dynamics during progression from Mycobacterium latent infection to disease, Applied Mathematical Modelling, 80, pp. 217-237, 2020.
World Health Organization et al., Global Tuberculosis Report 2022, Geneva, Switzerland: World Health Organization, 2023.
Pedruzzi, G., Rao, K.V. and Chatterjee, S., Mathematical model of Mycobacterium?host interaction describes physiology of persistence, Journal of Theoretical Biology, 376, pp. 105-117, 2015.
Queval, C.J., Brosch, R. and Simeone, R., The macrophage: A disputed fortress in the battle against Mycobacterium tuberculosis, Frontiers in Microbiology, 8, p. 2284, 2017.
Rodriguez-Herrera, A.E., Jordan-Salivia, G., et al., Macrophage-activating and tissue-damaging immune responses to M. tuberculosis, 1999.
Russell, D.G., Who puts the tubercle in tuberculosis?, Nature Reviews Microbiology, 5(1), pp. 39-47, 2007.
Shi, R., Li, Y. and Tang, S., A mathematical model with optimal controls for cellular immunology of tuberculosis, 2014.
Sud, D., Bigbee, C., Flynn, J.L. and Kirschner, D.E., Contribution of CD8+ T cells to control of Mycobacterium tuberculosis infection, The Journal of Immunology, 176(7), pp. 4296-4314, 2006.
Sugawara, I., Mizuno, S., Tatsumi, T. and Taniyama, T., Imaging of pulmonary granulomas using a photon imager, Japanese Journal of Infectious Diseases, 59(5), p. 332, 2006.
Tan, J.S., Canaday, D.H., Boom, W.H., Balaji, K.N., Schwander, S.K. and Rich, E.A., Human alveolar T lymphocyte responses to Mycobacterium tuberculosis antigens: role for CD4+ and CD8+ cytotoxic T cells and relative resistance of alveolar macrophages to lysis, Journal of Immunology (Baltimore, Md.: 1950), 159(1), pp. 290-297, 1997.
Van den Driessche, P. and Watmough, J., Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Mathematical Biosciences, 180(1-2), pp. 29-48, 2002.
Wigginton, J.E. and Kirschner, D., A model to predict cell-mediated immune regulatory mechanisms during human infection with Mycobacterium tuberculosis, The Journal of Immunology, 166(3), pp. 1951-1967, 2001.
Yang, X., Chen, L. and Chen, J., Permanence and positive periodic solution for the single-species nonautonomous delay diffusive models, Computers & Mathematics with Applications, 32(4), pp. 109-116, 1996.
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2025 Communication in Biomathematical Sciences
This work is licensed under a Creative Commons Attribution-NoDerivatives 4.0 International License.
