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On geometric conditions for reduction of the Moreau sweeping process to the Prandtl-Ishlinskii operator
Mathematical analysis of macrophage-bacteria interaction in tuberculosis infection
Department of Mathematics, City University of Hong Kong, Hong Kong SAR, China |
Tuberculosis (TB) is a leading cause of death from infectious disease. TB is caused mainly by a bacterium called Mycobacterium tuberculosis which often initiates in the respiratory tract. The interaction of macrophages and T cells plays an important role in the immune response during TB infection. Recent experimental results support that active TB infection may be induced by the dysfunction of Treg cell regulation that provides a balance between anti-TB T cell responses and pathology. To better understand the dynamics of TB infection and Treg cell regulation, we build a mathematical model using a system of differential equations that qualitatively and quantitatively characterizes the dynamics of macrophages, Th1 and Treg cells during TB infection. For sufficiently analyzing the interaction between immune response and bacterial infection, we separate our model into several simple subsystems for further steady state and stability studies. Using this system, we explore the conditions of parameters for three situations, recovery, latent disease and active disease, during TB infection. Our numerical simulations support that Th1 cells and Treg cells play critical roles in TB infection: Th1 cells inhibit the number of infected macrophages to reduce the chance of active disease; Treg cell regulation reduces the immune response to stabilize the dynamics of the system.
References:
[1] |
A. Chaudhry, R. M. Samstein, P. Treuting, Y. Liang, M. C. Pils, J.-M. Heinrich, R. S. Jack, F. T. Wunderlich, J. C. Bruning, W. Muller and A. Y. Rudensky,
Interleukin-10 signaling in regulatory T cells is required for suppression of Th17 cell-mediated inflammation, Immunity, 34 (2011), 566-578.
doi: 10.1016/j.immuni.2011.03.018. |
[2] |
R. Condos, W. N. Rom, Y. M. Liu and N. W. Schluger,
Local immune responses correlate with presentation and outcome in tuberculosis, American Journal of Respiratory and Critical Care Medicine, 157 (1998), 729-735.
doi: 10.1164/ajrccm.157.3.9705044. |
[3] |
I. E. A. Flesch and S. H. E. Kaufmann,
Activation of tuberculostatic macrophage functions by gamma interferon, interleukin-4, and tumor necrosis factor, Infection and Immunity, 58 (1990), 2675-2677.
|
[4] |
J. L. Flynn and J. Chan,
Immunology of tuberculosis, Annual Review of Immunology, 19 (2001), 93-129.
|
[5] |
F. R. Gantmacher,
Applications of the Theory of Matrices, Interscience Publishers Ltd., London, 1959. |
[6] |
A. M. Green, J. T. Mattila, C. L. Bigbee, K. S. Bongers, P. L. Lin and J. L. Flynn,
CD4(+) regulatory T cells in a cynomolgus macaque model of Mycobacterium tuberculosis infection, The Journal of Infectious Diseases, 202 (2010), 533-541.
|
[7] |
C. A. Janeway, P. Travers, M. Walport and M. Shlomchik,
Immunobiology: The Immune System in Health and Disease, New York: Garland Science, 2001. |
[8] |
M. Kursar, M. Koch, H.-W. Mittrücker, G. Nouailles, K. Bonhagen, T. Kamradt and S. H. E. Kaufmann,
Cutting Edge: Regulatory T cells prevent efficient clearance of Mycobacterium tuberculosis, Journal of Immunology, 178 (2007), 2661-2665.
doi: 10.4049/jimmunol.178.5.2661. |
[9] |
S. H. Lee,
Diagnosis and treatment of latent tuberculosis infection, Tuberculosis and Respiratory Diseases, 78 (2015), 56-63.
doi: 10.4046/trd.2015.78.2.56. |
[10] |
W. -C. Lo, V. Arsenescu, R. I. Arsenescu and A. Friedman, Inflammatory bowel disease: How effective is TNF-alpha suppression?
PLoS ONE, 12 (2017), e01708g5.
doi: 10.1371/journal.pone.0165782. |
[11] |
W.-C. Lo, R. I. Arsenescu and A. Friedman,
Mathematical model of the roles of T cells in inflammatory bowel disease, Bulletin of Mathematical Biology, 75 (2013), 1417-1433.
doi: 10.1007/s11538-013-9853-2. |
[12] |
K. J. Maloy and F. Powrie,
Intestinal homeostasis and its breakdown in inflammatory bowel disease, Nature, 474 (2011), 298-306.
doi: 10.1038/nature10208. |
[13] |
S. Marino, S. Pawar, C. L. Fuller, T. A. Reinhart, J. L. Flynn and D. E. Kirschner,
Dendritic cell trafficking and antigen presentation in the human immune response to Mycobacterium tuberculosis, Journal of Immunology, 173 (2004), 494-506.
doi: 10.4049/jimmunol.173.1.494. |
[14] |
S. Marino and D. E. Kirschner,
The human immune response to Mycobacterium tuberculosis in lung and lymph node, Journal of Theoretical Biology, 227 (2004), 463-486.
doi: 10.1016/j.jtbi.2003.11.023. |
[15] |
F. O. Martinez and S. Gordon, The M1 and M2 paradigm of macrophage activation: time for reassessment, F1000prime Reports, 6 (2014), p13.
doi: 10.12703/P6-13. |
[16] |
K. A. McDonough, Y. Kress and B. R. Bloom,
Pathogenesis of tuberculosis: Interaction of Mycobacterium tuberculosis with macrophages, Infection and Immunity, 61 (1993), 2763-2773.
|
[17] |
T. Mogues, M. E. Goodrich, L. Ryan, R. LaCourse and R. J. North,
The relative importance of T cell subsets in immunity and immunopathology of airborne Mycobacterium tuberculosis infection in mice, The Journal of Experimental Medicine, 193 (2001), 271-280.
|
[18] |
D. M. Nancy, C. P. Sara, M. V. Viviana, A. R. Carlos, R. Mauricio and F. G. Luis,
Regulatory T cell frequency and modulation of IFN-gamma and IL-17 in active and latent tuberculosis, Tuberculosis, 90 (2010), 252-261.
|
[19] |
G. Pedruzzi, K. V. S. Rao and S. Chatterjee,
Mathematical model of mycobacterium - host interaction describes physiology of persistence, Journal of Theoretical Biology, 376 (2015), 105-117.
doi: 10.1016/j.jtbi.2015.03.031. |
[20] |
E. Pienaar and M. Lerm,
A mathematical model of the initial interaction between Mycobacterium tuberculosis and macrophages, Journal of Theoretical Biology, 342 (2014), 23-32.
doi: 10.1016/j.jtbi.2013.09.029. |
[21] |
K. M. Quinn, R. S. McHugh, F. J. Rich, L. M. Goldsack, G. W. De Lisle, B. M. Buddle, B. Delahunt and J. R. Kirman,
Inactivation of CD4+CD25+ regulatory T cells during early mycobacterial infection increases cytokine production but does not affect pathogen load, Immunology and Cell Biology, 84 (2006), 467-474.
doi: 10.1111/j.1440-1711.2006.01460.x. |
[22] |
G. A. Rook, J. Steele, M. Ainsworth and B. R. Champion,
Activation of macrophages to inhibit proliferation of Mycobacterium tuberculosis: comparison of the effects of recombinant gamma-interferon on human monocytes and murine peritoneal macrophages, Immunology, 59 (1986), 333-338.
|
[23] |
P. Salgame,
Host innate and Th1 responses and the bacterial factors that control Mycobacterium tuberculosis infection, Current Opinion in Immunology, 17 (2005), 374-380.
doi: 10.1016/j.coi.2005.06.006. |
[24] |
S. K. Schwander, M. Torres, E. Sada, C. Carranza, E. Ramos, M. Tary-Lehmann, R. S. Wallis, J. Sierra and E. a. Rich,
Enhanced responses to Mycobacterium tuberculosis antigens by human alveolar lymphocytes during active pulmonary tuberculosis, The Journal of Infectious Diseases, 178 (1998), 1434-1445.
|
[25] |
D. K. Sojka, Y. H. Huang and D. J. Fowell,
Mechanisms of regulatory t-cell suppression - a diverse arsenal for a moving target, Immunology, 124 (2008), 13-22.
doi: 10.1111/j.1365-2567.2008.02813.x. |
[26] |
D. Sud, C. Bigbee, J. L. Flynn and D. E. Kirschner,
Contribution of CD8+ T cells to control of Mycobacterium tuberculosis infection, Journal of Immunology, 176 (2006), 4296-4314.
|
[27] |
J. Tan, D. Canaday, W. Boom, K. Balaji, S. Schwander and E. Rich,
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, 159 (1997), 290-297.
|
[28] |
K. Tsukaguchi, B. de Lange and W. H. Boom,
Differential regulation of IFN-gamma, TNF-alpha, and IL-10 production by CD4(+) alphabetaTCR+ T cells and vdelta2(+) gammadelta T cells in response to monocytes infected with Mycobacterium tuberculosis-H37Ra, Cellular Immunology, 194 (1999), 12-20.
|
[29] |
R. Van Furth, M. C. Diesselhoff-den Dulk and H. Mattie,
Quantitative study on the production and kinetics of mononuclear phagocytes during an acute inflammatory reaction, Journal of Experimental Medicine, 138 (1973), 1314-1330.
|
[30] |
I. Wergeland, J. Assmus and A. M. Dyrhol-Riise,
T regulatory cells and immune activation in Mycobacterium tuberculosis infection and the effect of preventive therapy, Scandinavian Journal of Immunology, 73 (2011), 234-242.
|
[31] |
J. E. Wigginton and D. Kirschner,
A model to predict cell-mediated immune regulatory mechanisms during human infection with Mycobacterium tuberculosis, Journal of Immunology, 166 (2001), 1951-1967.
doi: 10.4049/jimmunol.166.3.1951. |
[32] |
M. Zhang, J. Gong, Y. Lin and P. F. Barnes,
Growth of virulent and avirulent Mycobacterium tuberculosis strains in human macrophages, Infection and immunity, 66 (1998), 794-799.
|
show all references
References:
[1] |
A. Chaudhry, R. M. Samstein, P. Treuting, Y. Liang, M. C. Pils, J.-M. Heinrich, R. S. Jack, F. T. Wunderlich, J. C. Bruning, W. Muller and A. Y. Rudensky,
Interleukin-10 signaling in regulatory T cells is required for suppression of Th17 cell-mediated inflammation, Immunity, 34 (2011), 566-578.
doi: 10.1016/j.immuni.2011.03.018. |
[2] |
R. Condos, W. N. Rom, Y. M. Liu and N. W. Schluger,
Local immune responses correlate with presentation and outcome in tuberculosis, American Journal of Respiratory and Critical Care Medicine, 157 (1998), 729-735.
doi: 10.1164/ajrccm.157.3.9705044. |
[3] |
I. E. A. Flesch and S. H. E. Kaufmann,
Activation of tuberculostatic macrophage functions by gamma interferon, interleukin-4, and tumor necrosis factor, Infection and Immunity, 58 (1990), 2675-2677.
|
[4] |
J. L. Flynn and J. Chan,
Immunology of tuberculosis, Annual Review of Immunology, 19 (2001), 93-129.
|
[5] |
F. R. Gantmacher,
Applications of the Theory of Matrices, Interscience Publishers Ltd., London, 1959. |
[6] |
A. M. Green, J. T. Mattila, C. L. Bigbee, K. S. Bongers, P. L. Lin and J. L. Flynn,
CD4(+) regulatory T cells in a cynomolgus macaque model of Mycobacterium tuberculosis infection, The Journal of Infectious Diseases, 202 (2010), 533-541.
|
[7] |
C. A. Janeway, P. Travers, M. Walport and M. Shlomchik,
Immunobiology: The Immune System in Health and Disease, New York: Garland Science, 2001. |
[8] |
M. Kursar, M. Koch, H.-W. Mittrücker, G. Nouailles, K. Bonhagen, T. Kamradt and S. H. E. Kaufmann,
Cutting Edge: Regulatory T cells prevent efficient clearance of Mycobacterium tuberculosis, Journal of Immunology, 178 (2007), 2661-2665.
doi: 10.4049/jimmunol.178.5.2661. |
[9] |
S. H. Lee,
Diagnosis and treatment of latent tuberculosis infection, Tuberculosis and Respiratory Diseases, 78 (2015), 56-63.
doi: 10.4046/trd.2015.78.2.56. |
[10] |
W. -C. Lo, V. Arsenescu, R. I. Arsenescu and A. Friedman, Inflammatory bowel disease: How effective is TNF-alpha suppression?
PLoS ONE, 12 (2017), e01708g5.
doi: 10.1371/journal.pone.0165782. |
[11] |
W.-C. Lo, R. I. Arsenescu and A. Friedman,
Mathematical model of the roles of T cells in inflammatory bowel disease, Bulletin of Mathematical Biology, 75 (2013), 1417-1433.
doi: 10.1007/s11538-013-9853-2. |
[12] |
K. J. Maloy and F. Powrie,
Intestinal homeostasis and its breakdown in inflammatory bowel disease, Nature, 474 (2011), 298-306.
doi: 10.1038/nature10208. |
[13] |
S. Marino, S. Pawar, C. L. Fuller, T. A. Reinhart, J. L. Flynn and D. E. Kirschner,
Dendritic cell trafficking and antigen presentation in the human immune response to Mycobacterium tuberculosis, Journal of Immunology, 173 (2004), 494-506.
doi: 10.4049/jimmunol.173.1.494. |
[14] |
S. Marino and D. E. Kirschner,
The human immune response to Mycobacterium tuberculosis in lung and lymph node, Journal of Theoretical Biology, 227 (2004), 463-486.
doi: 10.1016/j.jtbi.2003.11.023. |
[15] |
F. O. Martinez and S. Gordon, The M1 and M2 paradigm of macrophage activation: time for reassessment, F1000prime Reports, 6 (2014), p13.
doi: 10.12703/P6-13. |
[16] |
K. A. McDonough, Y. Kress and B. R. Bloom,
Pathogenesis of tuberculosis: Interaction of Mycobacterium tuberculosis with macrophages, Infection and Immunity, 61 (1993), 2763-2773.
|
[17] |
T. Mogues, M. E. Goodrich, L. Ryan, R. LaCourse and R. J. North,
The relative importance of T cell subsets in immunity and immunopathology of airborne Mycobacterium tuberculosis infection in mice, The Journal of Experimental Medicine, 193 (2001), 271-280.
|
[18] |
D. M. Nancy, C. P. Sara, M. V. Viviana, A. R. Carlos, R. Mauricio and F. G. Luis,
Regulatory T cell frequency and modulation of IFN-gamma and IL-17 in active and latent tuberculosis, Tuberculosis, 90 (2010), 252-261.
|
[19] |
G. Pedruzzi, K. V. S. Rao and S. Chatterjee,
Mathematical model of mycobacterium - host interaction describes physiology of persistence, Journal of Theoretical Biology, 376 (2015), 105-117.
doi: 10.1016/j.jtbi.2015.03.031. |
[20] |
E. Pienaar and M. Lerm,
A mathematical model of the initial interaction between Mycobacterium tuberculosis and macrophages, Journal of Theoretical Biology, 342 (2014), 23-32.
doi: 10.1016/j.jtbi.2013.09.029. |
[21] |
K. M. Quinn, R. S. McHugh, F. J. Rich, L. M. Goldsack, G. W. De Lisle, B. M. Buddle, B. Delahunt and J. R. Kirman,
Inactivation of CD4+CD25+ regulatory T cells during early mycobacterial infection increases cytokine production but does not affect pathogen load, Immunology and Cell Biology, 84 (2006), 467-474.
doi: 10.1111/j.1440-1711.2006.01460.x. |
[22] |
G. A. Rook, J. Steele, M. Ainsworth and B. R. Champion,
Activation of macrophages to inhibit proliferation of Mycobacterium tuberculosis: comparison of the effects of recombinant gamma-interferon on human monocytes and murine peritoneal macrophages, Immunology, 59 (1986), 333-338.
|
[23] |
P. Salgame,
Host innate and Th1 responses and the bacterial factors that control Mycobacterium tuberculosis infection, Current Opinion in Immunology, 17 (2005), 374-380.
doi: 10.1016/j.coi.2005.06.006. |
[24] |
S. K. Schwander, M. Torres, E. Sada, C. Carranza, E. Ramos, M. Tary-Lehmann, R. S. Wallis, J. Sierra and E. a. Rich,
Enhanced responses to Mycobacterium tuberculosis antigens by human alveolar lymphocytes during active pulmonary tuberculosis, The Journal of Infectious Diseases, 178 (1998), 1434-1445.
|
[25] |
D. K. Sojka, Y. H. Huang and D. J. Fowell,
Mechanisms of regulatory t-cell suppression - a diverse arsenal for a moving target, Immunology, 124 (2008), 13-22.
doi: 10.1111/j.1365-2567.2008.02813.x. |
[26] |
D. Sud, C. Bigbee, J. L. Flynn and D. E. Kirschner,
Contribution of CD8+ T cells to control of Mycobacterium tuberculosis infection, Journal of Immunology, 176 (2006), 4296-4314.
|
[27] |
J. Tan, D. Canaday, W. Boom, K. Balaji, S. Schwander and E. Rich,
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, 159 (1997), 290-297.
|
[28] |
K. Tsukaguchi, B. de Lange and W. H. Boom,
Differential regulation of IFN-gamma, TNF-alpha, and IL-10 production by CD4(+) alphabetaTCR+ T cells and vdelta2(+) gammadelta T cells in response to monocytes infected with Mycobacterium tuberculosis-H37Ra, Cellular Immunology, 194 (1999), 12-20.
|
[29] |
R. Van Furth, M. C. Diesselhoff-den Dulk and H. Mattie,
Quantitative study on the production and kinetics of mononuclear phagocytes during an acute inflammatory reaction, Journal of Experimental Medicine, 138 (1973), 1314-1330.
|
[30] |
I. Wergeland, J. Assmus and A. M. Dyrhol-Riise,
T regulatory cells and immune activation in Mycobacterium tuberculosis infection and the effect of preventive therapy, Scandinavian Journal of Immunology, 73 (2011), 234-242.
|
[31] |
J. E. Wigginton and D. Kirschner,
A model to predict cell-mediated immune regulatory mechanisms during human infection with Mycobacterium tuberculosis, Journal of Immunology, 166 (2001), 1951-1967.
doi: 10.4049/jimmunol.166.3.1951. |
[32] |
M. Zhang, J. Gong, Y. Lin and P. F. Barnes,
Growth of virulent and avirulent Mycobacterium tuberculosis strains in human macrophages, Infection and immunity, 66 (1998), 794-799.
|










Parameter | Definition |
Bacteria growth rate | |
Infection rate | |
Death rate of bacteria by |
|
Deactivation rate of macrophage | |
Activation rate of macrophage | |
Bursting rate of |
|
Death rate of |
|
Death rate of |
|
Death rate of |
|
Death rate of |
|
Death rate of |
|
Ratio in |
|
Ratio in bacteria killing | |
Ratio in |
|
Half-saturation constant for |
|
Half-saturation constant for |
|
Half-saturation constant for |
|
Half-saturation constant for |
|
Half-saturation constant for |
|
Half-saturation constant for |
|
Estimated number of bacteria per macrophage | |
Number of bacteria releasing from macrophage by Th1 cell immunity |
Parameter | Definition |
Bacteria growth rate | |
Infection rate | |
Death rate of bacteria by |
|
Deactivation rate of macrophage | |
Activation rate of macrophage | |
Bursting rate of |
|
Death rate of |
|
Death rate of |
|
Death rate of |
|
Death rate of |
|
Death rate of |
|
Ratio in |
|
Ratio in bacteria killing | |
Ratio in |
|
Half-saturation constant for |
|
Half-saturation constant for |
|
Half-saturation constant for |
|
Half-saturation constant for |
|
Half-saturation constant for |
|
Half-saturation constant for |
|
Estimated number of bacteria per macrophage | |
Number of bacteria releasing from macrophage by Th1 cell immunity |
Steady states | Status | |
If Unstable steady state Stable or unstable steady state |
Latent disease (Stable Active disease (Unstable |
|
If Unstable steady state |
Active disease | |
Two steady states Unstable steady state Stable or unstable steady state |
Latent disease (Stable Active disease (Unstable |
|
If Stable steady state Unstable steady state |
Active disease or recovery (depends on initial values) |
|
If or One steady state Unstable steady state |
Active disease | |
If Two steady states Unstable steady state Stable steady state |
Latent disease | |
If Stable steady state |
Recovery | |
If Unstable steady state Stable steady state |
Latent disease | |
If Stable steady state |
Recovery |
Steady states | Status | |
If Unstable steady state Stable or unstable steady state |
Latent disease (Stable Active disease (Unstable |
|
If Unstable steady state |
Active disease | |
Two steady states Unstable steady state Stable or unstable steady state |
Latent disease (Stable Active disease (Unstable |
|
If Stable steady state Unstable steady state |
Active disease or recovery (depends on initial values) |
|
If or One steady state Unstable steady state |
Active disease | |
If Two steady states Unstable steady state Stable steady state |
Latent disease | |
If Stable steady state |
Recovery | |
If Unstable steady state Stable steady state |
Latent disease | |
If Stable steady state |
Recovery |
Parameter | Range for stability tests in Section 4 |
Values for the simulations in Figs. 4-6 and Figs. 8-10 |
Range for in Fig. 7 and Fig. 10 |
Unit | Reference |
[2,31] | |||||
[14,31] | |||||
- | [14] | ||||
- | Estimated | ||||
|
[14,31] | ||||
- | Estimated | ||||
[14] | |||||
- | Estimated | ||||
[3,14] | |||||
- | Estimated | ||||
- | |
Estimated | |||
Estimated | |||||
|
[14,29,31] | ||||
[14,29,31] | |||||
- | [14,29,31] | ||||
- | [14,31] | ||||
- | |
[14,31] | |||
- | |
[14] | |||
- | [14] | ||||
- | |
Estimated | |||
[31] | |||||
- | [31] | ||||
- | 1 | Estimated | |||
- | |
Estimated | |||
- | Estimated | ||||
- | Estimated | ||||
[14,32] | |||||
- | [14] |
Parameter | Range for stability tests in Section 4 |
Values for the simulations in Figs. 4-6 and Figs. 8-10 |
Range for in Fig. 7 and Fig. 10 |
Unit | Reference |
[2,31] | |||||
[14,31] | |||||
- | [14] | ||||
- | Estimated | ||||
|
[14,31] | ||||
- | Estimated | ||||
[14] | |||||
- | Estimated | ||||
[3,14] | |||||
- | Estimated | ||||
- | |
Estimated | |||
Estimated | |||||
|
[14,29,31] | ||||
[14,29,31] | |||||
- | [14,29,31] | ||||
- | [14,31] | ||||
- | |
[14,31] | |||
- | |
[14] | |||
- | [14] | ||||
- | |
Estimated | |||
[31] | |||||
- | [31] | ||||
- | 1 | Estimated | |||
- | |
Estimated | |||
- | Estimated | ||||
- | Estimated | ||||
[14,32] | |||||
- | [14] |
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