American Institute of Mathematical Sciences

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October  2018, 23(8): 3387-3413. doi: 10.3934/dcdsb.2018239

Mathematical analysis of macrophage-bacteria interaction in tuberculosis infection

Danyun He , Qian Wang and  Wing-Cheong Lo ,

Department of Mathematics, City University of Hong Kong, Hong Kong SAR, China

* Corresponding author: W.-C. LO

Received  November 2017 Revised  April 2018 Published  August 2018

Fund Project: W.-C. LO is supported by a CityU StUp Grant (No. 7200437).

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.

Keywords: Mathematical modeling, TB infection, T cells, macrophage, stability analysis.
Mathematics Subject Classification: Primary: 92B05, 92C42; Secondary: 92C37.
Citation: Danyun He, Qian Wang, Wing-Cheong Lo. Mathematical analysis of macrophage-bacteria interaction in tuberculosis infection. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3387-3413. doi: 10.3934/dcdsb.2018239
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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[4]

J. L. Flynn and J. Chan, Immunology of tuberculosis, Annual Review of Immunology, 19 (2001), 93-129.   Google Scholar

[5]

F. R. Gantmacher, Applications of the Theory of Matrices, Interscience Publishers Ltd., London, 1959.  Google Scholar

[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.   Google Scholar

[7]

C. A. Janeway, P. Travers, M. Walport and M. Shlomchik, Immunobiology: The Immune System in Health and Disease, New York: Garland Science, 2001. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[4]

J. L. Flynn and J. Chan, Immunology of tuberculosis, Annual Review of Immunology, 19 (2001), 93-129.   Google Scholar

[5]

F. R. Gantmacher, Applications of the Theory of Matrices, Interscience Publishers Ltd., London, 1959.  Google Scholar

[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.   Google Scholar

[7]

C. A. Janeway, P. Travers, M. Walport and M. Shlomchik, Immunobiology: The Immune System in Health and Disease, New York: Garland Science, 2001. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

Figure 1.  Model diagram of bacteria, T cells and macrophages interaction.The Sharp arrow means activation; blocked arrow means inhibition. When bacteria encountering resting macrophages, some macrophages will be activated and become activated macrophages ($M_a$), while some will be infected and become infected macrophages ($M_i$). $M_a$ may be deactivated in response to regulatory T cells (Treg). Th1 cells will induce the activation of $M_a$. $M_a$ and $M_r$ will inhibit the growth of bacteria. Bursting of $M_i$ will increase the concentration of bacteria. $M_i$ may be eliminated by Th1 cells. Th1 cells activation will increase in response to $M_i$ and $M_a$, and decrease by Treg cells' inhibition. Treg cells will be activated by Th1 cells and $M_a$
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Figure 2.  Model diagrams of subsystems for analysis.A) Two-equation model. The interaction between bacteria infection and resting macrophages protection. B) Three-equation model. Bacteria infection, macrophages protection with some resting macrophages are infected and become infected macrophages, which will burst and release bacteria
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Figure 3.  Phase plane portraits for the analysis of the two-equation system (7)-(8).A) Case 1 B) Case 2 C) Case 3. Star symbols indicate the steady states; red dotted lines represent $dM_r/dt =0$; blue solid lines represent $dB/dt=0$
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Figure 4.  The dynamics of $B$, $M_r$ and $M_i$ with different values of $\beta_{M_r}$ in the three-equation model (13)-(15).The initial conditions are $B(0)=100$, $M_r(0)=\alpha_{M_r}/\mu_{M_r}$ and $M_i(0)=0$. We set $\gamma_i=0.01 \ \text{day}^{-1}$ and the others parameters from Table 3. As the rate of $M_r$ recruitment by $M_i$ and $M_a$ increasing from 0.1 to 10, the number of $M_i$ and $B$ decreases, while the number of $M_r$ is relatively stable
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Figure 5.  Model 1: System (1)-(6) without involving Th1 cell and Treg cell.The initial conditions are $B(0)=100$, $M_r(0)=\alpha_{M_r}/\mu_{M_r}$ and $M_i(0)=M_a(0)=T_1(0)= T_r(0)=0$. We set $\alpha_{T_1}=\alpha_{T_r}=\beta_{T_r}=0 \ \text{day}^{-1}$ and the other parameters from Table 3. Resting macrophages are infected by bacteria and become infected macrophages, leading to a sharp increase in bacteria
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Figure 6.  Model 2: System (1)-(6) with Th1 cell but without Treg cell regulation. The initial conditions are $B(0)=100$, $M_r(0)=\alpha_{M_r}/\mu_{M_r}$ and $M_i(0)=M_a(0)=T_1(0)= T_r(0)=0$. We set $\alpha_{T_1}=5.2\times10^{-2} \ \text{day}^{-1}$, $\alpha_{T_r}=\beta_{T_r}=0 \ \text{day}^{-1}$ and the other parameters from Table 3. Without regulation from Treg cells, the population of bacteria, macrophages, and Th1 cells oscillates continuously
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Figure 7.  The effect of Treg cell regulation in Model 3.A) The change in the average of the OI with respect to the increases of $\alpha_{T_r}$, the $T_r$ activation rate by $M_a$ and $\beta_{T_r}$, the $T_r$ activation rate by $T_1$; B) The change in the average of the maximum concentration of bacteria with respect to increases of $\alpha_{T_r}$ and $\beta_{T_r}$. Each dot represents an average of the simulations with 500 sets of randomly selected parameters from the ranges listed in Table 3. For each simulation, the initial conditions are $B(0)=100$, $M_r(0)=\alpha_{M_r}/\mu_{M_r}$ and $M_i(0)=M_a(0)=T_1(0)= T_r(0)=0$. As $\alpha_{T_r}$ and $\beta_{T_r}$ increase, the oscillation of bacteria becomes smaller, while the maximum value of bacteria decreases first and then increases
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Figure 8.  Model 3: System (1)-(6) with Th1 cell control and low level of Treg cell regulation. The initial conditions are $B(0)=100$, $M_r(0)=\alpha_{M_r}/\mu_{M_r}$ and $M_i(0)=M_a(0)=T_1(0)= T_r(0)=0$. We set $\alpha_{T_1}=5.2\times10^{-2} \ \text{day}^{-1}$, $\alpha_{T_r}=\beta_{T_r}=0.1 \ \text{day}^{-1}$ and the other parameters from Table 3. With Treg cell regulation, the number of bacteria remains stable after a few cycles of oscillation
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Figure 9.  Model 4: System (1)-(6) with Th1 cell control and strong level of Treg cell regulation.The initial conditions are $B(0)=100$, $M_r(0)=\alpha_{M_r}/\mu_{M_r}$ and $M_i(0)=M_a(0)=T_1(0)= T_r(0)=0$. We set $\alpha_{T_1}=5.2\times10^{-2} \ \text{day}^{-1}$, $\alpha_{T_r}=\beta_{T_r}=1 \ \text{day}^{-1}$, $\beta_{M_r}=0.01 \ \text{day}^{-1}$ and the other parameters from Table 3. Strong Treg cell regulation causes sharp reductions of macrophages and Th1 cells, thus, the number of bacteria rapidly increases
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Figure 10.  The effect of enhaced macrophage recruitment in Model 4.A) The change of the average of the OI with respect to the increase of $\beta_{M_r}$; B) The change of the average of the maximum concentration of bacteria with respect to increase of $\beta_{M_r}$, the $M_r$ recruitment rate by $M_a$ and $M_i$; the insert shows the change of the average of the maximum concentration of bacteria when $\beta_{M_r}>0.2$. Each dot represents an average of the simulations with 500 sets of randomly selected parameters from the ranges listed in Table 3. For each simulation, We set $\alpha_{T_r}=\beta_{T_r}=1 \ \text{day}^{-1}$ and the initial conditions are $B(0)=100$, $M_r(0)=\alpha_{M_r}/\mu_{M_r}$ and $M_i(0)=M_a(0)=T_1(0)= T_r(0)=0$. C-D) The simulation results of $B$ with two different values of $\beta_{M_r}$. We set the initial conditions as in (A-B), $\alpha_{T_r}=\beta_{T_r}= 1 \ \text{day}^{-1}$ and the other parameters are listed in Table 3
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Table 1.  The definitions of the parameters used in the equations (1)-(6)
Parameter Definition
$\alpha_{M_r}$ $M_r$ source
$\alpha_B$ Bacteria growth rate
$\alpha_{T_1}$ $T_1$ activation rate by $M_i$ and $M_a$
$\alpha_{T_r}$ $T_r$ activation rate by $M_a$
$\beta_{M_r}$ $M_r$ recruitment by $M_i$ and $M_a$
$\beta_{T_r}$ $T_r$ activation rate by $T_1$
$k_{MB}$ Infection rate
$k_{MT}$ $T_1$ immunity rate
$k_{BM}$ Death rate of bacteria by $M_r$ and $M_a$
$\gamma_r$ Deactivation rate of macrophage
$\gamma_a$ Activation rate of macrophage
$\gamma_i$ Bursting rate of $M_i$
$\mu_{M_r}$ Death rate of $M_r$
$\mu_{M_i}$ Death rate of $M_i$
$\mu_{M_a}$ Death rate of $M_a$
$\mu_{T_1}$ Death rate of $T_1$
$\mu_{T_r}$ Death rate of $T_r$
$\omega_1$ Ratio in $M_r$ recruitment
$\omega_2$ Ratio in bacteria killing
$\omega_3$ Ratio in $T_1$ activation
$K_B$ Half-saturation constant for $B$ in infection
$Q_B$ Half-saturation constant for $B$ in macrophage activation
$K_1$ Half-saturation constant for $T_1$ in immunity
$K_r$ Half-saturation constant for $T_r$ in inhibition
$Q_r$ Half-saturation constant for $T_r$ in macrophage deactivation
$Q_1$ Half-saturation constant for $T_1$ in macrophage activation
$N$ Estimated number of bacteria per macrophage
$\bar{N}$ Number of bacteria releasing from macrophage by Th1 cell immunity
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Parameter Definition
$\alpha_{M_r}$ $M_r$ source
$\alpha_B$ Bacteria growth rate
$\alpha_{T_1}$ $T_1$ activation rate by $M_i$ and $M_a$
$\alpha_{T_r}$ $T_r$ activation rate by $M_a$
$\beta_{M_r}$ $M_r$ recruitment by $M_i$ and $M_a$
$\beta_{T_r}$ $T_r$ activation rate by $T_1$
$k_{MB}$ Infection rate
$k_{MT}$ $T_1$ immunity rate
$k_{BM}$ Death rate of bacteria by $M_r$ and $M_a$
$\gamma_r$ Deactivation rate of macrophage
$\gamma_a$ Activation rate of macrophage
$\gamma_i$ Bursting rate of $M_i$
$\mu_{M_r}$ Death rate of $M_r$
$\mu_{M_i}$ Death rate of $M_i$
$\mu_{M_a}$ Death rate of $M_a$
$\mu_{T_1}$ Death rate of $T_1$
$\mu_{T_r}$ Death rate of $T_r$
$\omega_1$ Ratio in $M_r$ recruitment
$\omega_2$ Ratio in bacteria killing
$\omega_3$ Ratio in $T_1$ activation
$K_B$ Half-saturation constant for $B$ in infection
$Q_B$ Half-saturation constant for $B$ in macrophage activation
$K_1$ Half-saturation constant for $T_1$ in immunity
$K_r$ Half-saturation constant for $T_r$ in inhibition
$Q_r$ Half-saturation constant for $T_r$ in macrophage deactivation
$Q_1$ Half-saturation constant for $T_1$ in macrophage activation
$N$ Estimated number of bacteria per macrophage
$\bar{N}$ Number of bacteria releasing from macrophage by Th1 cell immunity
Table 2.  Steady state analysis for the three-equation system (13)-(15). For the conditions, $L_M < L_B$ ($L_M>L_B$) corresponds to weak (strong) macrophage regulation when bacterial concentration is low; $R_M < R_B$ ($R_M>R_B$) corresponds to weak (strong) macrophage regulation when bacterial concentration is high
Steady states Status
$L_M < L_B$
$R_M < R_B$		 If $R_M < 0, $ Two steady states
Unstable steady state $(0, \alpha_{M_r}/\mu_{M_r}, 0)$
Stable or unstable steady state $(B^*, M_r^*, M_i^*)$		 Latent disease
(Stable $(B^*, M_r^*, M_i^*)$)
Active disease
(Unstable $(B^*, M_r^*, M_i^*)$)
If $R_M>0, $ One steady state
Unstable steady state $(0, \alpha_{M_r}/\mu_{M_r}, 0)$		 Active disease
$L_M < L_B$
$R_M>R_B$		 Two steady states
Unstable steady state $(0, \alpha_{M_r}/\mu_{M_r}, 0)$
Stable or unstable steady state $(B^*, M_r^*, M_i^*)$		 Latent disease
(Stable $(B^*, M_r^*, M_i^*)$)
Active disease
(Unstable $(B^*, M_r^*, M_i^*)$)
$L_M>L_B$
$R_M < R_B$		 If $R_M>0$ and $L_B>0$, Two steady states
Stable steady state $(0, \alpha_{M_r}/\mu_{M_r}, 0)$
Unstable steady state $(B^*, M_r^*, M_i^*)$		 Active disease or recovery
(depends on initial values)
If $R_M>0$ and $L_B < 0$,
or $R_M < 0$, $L_B < 0$ and $R_B/L_B < R_M/L_M$,
One steady state
Unstable steady state $(0, \alpha_{M_r}/\mu_{M_r}, 0)$		 Active disease
If $R_M < 0$, $L_B < 0$ and $R_B/L_B>R_M/L_M$,
Two steady states
Unstable steady state $(0, \alpha_{M_r}/\mu_{M_r}, 0)$
Stable steady state $(B^*, M_r^*, M_i^*)$		 Latent disease
If $R_M < 0$ and $L_B>0$, One steady state
Stable steady state $(0, \alpha_{M_r}/\mu_{M_r}, 0)$		 Recovery
$L_M>L_B$
$R_M>R_B$		 If $L_B < 0$, Two steady states
Unstable steady state $(0, \alpha_{M_r}/\mu_{M_r}, 0)$
Stable steady state $(B^*, M_r^*, M_i^*)$		 Latent disease
If $L_B>0$, One steady state
Stable steady state $(0, \alpha_{M_r}/\mu_{M_r}, 0)$		 Recovery
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Steady states Status
$L_M < L_B$
$R_M < R_B$		 If $R_M < 0, $ Two steady states
Unstable steady state $(0, \alpha_{M_r}/\mu_{M_r}, 0)$
Stable or unstable steady state $(B^*, M_r^*, M_i^*)$		 Latent disease
(Stable $(B^*, M_r^*, M_i^*)$)
Active disease
(Unstable $(B^*, M_r^*, M_i^*)$)
If $R_M>0, $ One steady state
Unstable steady state $(0, \alpha_{M_r}/\mu_{M_r}, 0)$		 Active disease
$L_M < L_B$
$R_M>R_B$		 Two steady states
Unstable steady state $(0, \alpha_{M_r}/\mu_{M_r}, 0)$
Stable or unstable steady state $(B^*, M_r^*, M_i^*)$		 Latent disease
(Stable $(B^*, M_r^*, M_i^*)$)
Active disease
(Unstable $(B^*, M_r^*, M_i^*)$)
$L_M>L_B$
$R_M < R_B$		 If $R_M>0$ and $L_B>0$, Two steady states
Stable steady state $(0, \alpha_{M_r}/\mu_{M_r}, 0)$
Unstable steady state $(B^*, M_r^*, M_i^*)$		 Active disease or recovery
(depends on initial values)
If $R_M>0$ and $L_B < 0$,
or $R_M < 0$, $L_B < 0$ and $R_B/L_B < R_M/L_M$,
One steady state
Unstable steady state $(0, \alpha_{M_r}/\mu_{M_r}, 0)$		 Active disease
If $R_M < 0$, $L_B < 0$ and $R_B/L_B>R_M/L_M$,
Two steady states
Unstable steady state $(0, \alpha_{M_r}/\mu_{M_r}, 0)$
Stable steady state $(B^*, M_r^*, M_i^*)$		 Latent disease
If $R_M < 0$ and $L_B>0$, One steady state
Stable steady state $(0, \alpha_{M_r}/\mu_{M_r}, 0)$		 Recovery
$L_M>L_B$
$R_M>R_B$		 If $L_B < 0$, Two steady states
Unstable steady state $(0, \alpha_{M_r}/\mu_{M_r}, 0)$
Stable steady state $(B^*, M_r^*, M_i^*)$		 Latent disease
If $L_B>0$, One steady state
Stable steady state $(0, \alpha_{M_r}/\mu_{M_r}, 0)$		 Recovery
Table 3.  Parameters used in the equations (1)-(6)
Parameter Range for
stability tests
in Section 4		 Values for the
simulations in
Figs. 4-6 and
Figs. 8-10		 Range for $OI$ tests
in Fig. 7 and Fig. 10		 Unit Reference
$\alpha_{M_r}$ $3.3\times10^{2}-1\times 10^3$ $5\times10^{2}$ $3.3\times10^{2}-1\times 10^3$ $\text{ml}^{-1} \text{day}^{-1}$ [2,31]
$\alpha_B$ $0.01-0.1$ $5\times10^{-2}$ $0.025-0.075$ $\text{day}^{-1}$ [14,31]
$\alpha_{T_1}$ - $5.2\times 10^{-2}$ $0.026-0.078$ $\text{day}^{-1}$ [14]
$\alpha_{T_r}$ - $1\times 10^{-1}$ $1$ $\text{day}^{-1}$ Estimated
$\beta_{M_r}$ $10^{-2}-10 $ $1\times 10^{-2}$ $10^{-2}-0.1$ $\text{day}^{-1}$ [14,31]
$\beta_{T_r}$ - $1\times 10^{-1}$ $1$ $\text{day}^{-1}$ Estimated
$k_{MB}$ $0.2-0.5$ $5\times 10^{-1}$ $0.2-0.5$ $\text{day}^{-1}$ [14]
$k_{MT}$ - $1\times 10^{-1} $ $0.08-0.12$ $\text{day}^{-1}$ Estimated
$k_{BM}$ $1.25\times 10^{-9}$
$-2\times 10^{-8}$		 $2\times 10^{-8}$ $1.25\times 10^{-9}-2\times 10^{-8}$ $\text{ml}\;\text{day}^{-1}$ [3,14]
$\gamma_r$ - $2\times 10^{-1}$ $0.1-0.5$ $\text{day}^{-1}$ Estimated
$\gamma_a$ - $4\times 10^{-1}$ $0.2-0.6$ $\text{day}^{-1}$ Estimated
$\gamma_i$ $0.01-0.1$ $4\times 10^{-2} $ $0.2-0.4$ $\text{day}^{-1}$ Estimated
$\mu_{M_r}$ $1\times 10^{-2} $ $1\times 10^{-2}$ $1\times 10^{-2}$ $\text{day}^{-1}$ [14,29,31]
$\mu_{M_i}$ $1\times 10^{-2}$ $1\times 10^{-2}$ $1\times 10^{-2}$ $\text{day}^{-1}$ [14,29,31]
$\mu_{M_a}$ - $1\times 10^{-2}$ $1\times 10^{-2}$ $\text{day}^{-1}$ [14,29,31]
$\mu_{T_1}$ - $3.33\times 10^{-1}$ $3.33\times 10^{-1}$ $\text{day}^{-1}$ [14,31]
$\mu_{T_r}$ - $3.33\times 10^{-1}$ $3.33\times 10^{-1}$ $\text{day}^{-1}$ [14,31]
$\omega_1$ - $7.14$ $7.14$ [14]
$\omega_2$ - $10$ $10$ [14]
$\omega_3$ - $10$ $10$ Estimated
$K_B$ $1\times 10^{7} $ $1\times 10^{7} $ $1\times 10^{7}$ $\text{ml}^{-1}$ [31]
$Q_B$ - $1\times 10^{6}$ $1\times 10^{6}$ $\text{ml}^{-1}$ [31]
$K_1$ - $1$ 1 Estimated
$K_r$ - $1\times 10^{5}$ $1\times 10^{5}$ $\text{ml}^{-1}$ Estimated
$Q_r$ - $1\times 10^{5}$ $1\times 10^{5}$ $\text{ml}^{-1}$ Estimated
$Q_1$ - $1\times 10^{2}$ $1\times 10^{2}$ $\text{ml}^{-1}$ Estimated
$N$ $50$ $50$ $50$ [14,32]
$\bar{N}$ - $10$ $10$ [14]
Table Options

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Parameter Range for
stability tests
in Section 4		 Values for the
simulations in
Figs. 4-6 and
Figs. 8-10		 Range for $OI$ tests
in Fig. 7 and Fig. 10		 Unit Reference
$\alpha_{M_r}$ $3.3\times10^{2}-1\times 10^3$ $5\times10^{2}$ $3.3\times10^{2}-1\times 10^3$ $\text{ml}^{-1} \text{day}^{-1}$ [2,31]
$\alpha_B$ $0.01-0.1$ $5\times10^{-2}$ $0.025-0.075$ $\text{day}^{-1}$ [14,31]
$\alpha_{T_1}$ - $5.2\times 10^{-2}$ $0.026-0.078$ $\text{day}^{-1}$ [14]
$\alpha_{T_r}$ - $1\times 10^{-1}$ $1$ $\text{day}^{-1}$ Estimated
$\beta_{M_r}$ $10^{-2}-10 $ $1\times 10^{-2}$ $10^{-2}-0.1$ $\text{day}^{-1}$ [14,31]
$\beta_{T_r}$ - $1\times 10^{-1}$ $1$ $\text{day}^{-1}$ Estimated
$k_{MB}$ $0.2-0.5$ $5\times 10^{-1}$ $0.2-0.5$ $\text{day}^{-1}$ [14]
$k_{MT}$ - $1\times 10^{-1} $ $0.08-0.12$ $\text{day}^{-1}$ Estimated
$k_{BM}$ $1.25\times 10^{-9}$
$-2\times 10^{-8}$		 $2\times 10^{-8}$ $1.25\times 10^{-9}-2\times 10^{-8}$ $\text{ml}\;\text{day}^{-1}$ [3,14]
$\gamma_r$ - $2\times 10^{-1}$ $0.1-0.5$ $\text{day}^{-1}$ Estimated
$\gamma_a$ - $4\times 10^{-1}$ $0.2-0.6$ $\text{day}^{-1}$ Estimated
$\gamma_i$ $0.01-0.1$ $4\times 10^{-2} $ $0.2-0.4$ $\text{day}^{-1}$ Estimated
$\mu_{M_r}$ $1\times 10^{-2} $ $1\times 10^{-2}$ $1\times 10^{-2}$ $\text{day}^{-1}$ [14,29,31]
$\mu_{M_i}$ $1\times 10^{-2}$ $1\times 10^{-2}$ $1\times 10^{-2}$ $\text{day}^{-1}$ [14,29,31]
$\mu_{M_a}$ - $1\times 10^{-2}$ $1\times 10^{-2}$ $\text{day}^{-1}$ [14,29,31]
$\mu_{T_1}$ - $3.33\times 10^{-1}$ $3.33\times 10^{-1}$ $\text{day}^{-1}$ [14,31]
$\mu_{T_r}$ - $3.33\times 10^{-1}$ $3.33\times 10^{-1}$ $\text{day}^{-1}$ [14,31]
$\omega_1$ - $7.14$ $7.14$ [14]
$\omega_2$ - $10$ $10$ [14]
$\omega_3$ - $10$ $10$ Estimated
$K_B$ $1\times 10^{7} $ $1\times 10^{7} $ $1\times 10^{7}$ $\text{ml}^{-1}$ [31]
$Q_B$ - $1\times 10^{6}$ $1\times 10^{6}$ $\text{ml}^{-1}$ [31]
$K_1$ - $1$ 1 Estimated
$K_r$ - $1\times 10^{5}$ $1\times 10^{5}$ $\text{ml}^{-1}$ Estimated
$Q_r$ - $1\times 10^{5}$ $1\times 10^{5}$ $\text{ml}^{-1}$ Estimated
$Q_1$ - $1\times 10^{2}$ $1\times 10^{2}$ $\text{ml}^{-1}$ Estimated
$N$ $50$ $50$ $50$ [14,32]
$\bar{N}$ - $10$ $10$ [14]
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