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Bifurcation analysis for an in-host Mycobacterium tuberculosis model
Key Laboratory of Eco-environments in Three Gorges Reservoir Region, School of Mathematics and Statistics, Southwest University, Chongqing 400715, China |
Tuberculosis infection is still a major threat to humans and it may progress slowly or rapidly to clearance, latent infection, or active disease. In this paper, considering T cells can perform acceleration effect on their own recruitment, an in-host model of Mycobacterium tuberculosis is studied. Focus type and elliptic type of nilpotent singularities of codimension 3 are analyzed in this four dimensional model. Complex dynamical behaviors such as homoclinic loop, saddle-node bifurcation of limit cycle and co-existence of two limit cycles are revealed by bifurcation analysis. Especially, the slow-fast periodic solution with large-amplitude or small-amplitude is observed in numerical simulations, which provides a perfect explanation for the reactivation of latent infection.
References:
[1] |
V. D. Alvarez-Jiménez, K. Leyva-Paredes, M. García-Martínez, L. Vázquez-Flores and I. Estrada-García, Extracellular vesicles released from Mycobacterium tuberculosis-infected neutrophils promote macrophage autophagy and decrease intracellular mycobacterial survival, Front. Immunol., 9 (2018), 272. Google Scholar |
[2] |
L. Cai, G. Chen and D. Xiao,
Multiparametric bifurcations of an epidemiological model with strong Allee effect, J. Math. Biol., 67 (2013), 185-215.
doi: 10.1007/s00285-012-0546-5. |
[3] |
C. Castillo-Chavez and B. Song,
Dynamical models of tuberculosis and their applications, Math. Biosci. Eng., 1 (2004), 361-404.
doi: 10.3934/mbe.2004.1.361. |
[4] |
J. Day, A. Friedman and L. S. Schlesinger, Modeling the immune rheostat of macrophages in the lung in response to infection, Proc. Natl. Acad. Sci. USA, 106 (2009), 11246-11251. Google Scholar |
[5] |
Y. Du, J. Wu and J. M. Heffernan,
A simple in-host model for Mycobacterium tuberculosis that captures all infection outcomes, Math. Popul. Stud., 24 (2017), 37-63.
doi: 10.1080/08898480.2015.1054220. |
[6] |
F. Dumortier, R. Roussarie, J. Sotomayor and H. Żaładek, Bifurcations of Planar Vector Fields, Nilpotent Singularities and Abelian Integrals, Lecture Notes in Mathematics, 1480. Springer-Verlag, Berlin, 1991.
doi: 10.1007/BFb0098353. |
[7] |
J. L. Flynn,
Immunology of tuberculosis and implications in vaccine development, Tuberculosis, 84 (2004), 93-101.
doi: 10.1016/j.tube.2003.08.010. |
[8] |
J. L. Flynn and J. Chan, Immunology of tuberculosis, Annu. Rev. Immunol., 19 (2001), 93-129. Google Scholar |
[9] |
D. Gammack, S. Ganguli, S. Marino, J. L. Segoviajuarez and D. E. Kirschner,
Understanding the immune response in tuberculosis using different mathematical models and biological scales, Multiscale Model. Simul., 3 (2005), 312-345.
doi: 10.1137/040603127. |
[10] |
C. Gong, J. J. Linderman and D. Kirschner,
A population model capture dynamics of tuberculosis granulomas predicts host infection outcomes, Math. Biosci. Eng., 12 (2015), 625-642.
doi: 10.3934/mbe.2015.12.625. |
[11] |
D. He, Q. Wang and W.-C. Lo,
Mathematical analysis of macrophage-bateria interaction in tuberculosis infection, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 3387-3413.
doi: 10.3934/dcdsb.2018239. |
[12] |
J. M. Heffernan, R. J. Smith and L. M. Wahl, Perspectives on the basic reproductive ratio, J. R. Soc. Interface, 2 (2005), 281-293. Google Scholar |
[13] |
E. Ibargüen-Mondragón, L. Esteva and E. M. Burbano-Rosero,
Mathematical model for the growth of Mycobacterium tuberculosis in the granuloma, Math. Biosci. Eng., 15 (2017), 407-428.
doi: 10.3934/mbe.2018018. |
[14] |
E. Ibarguenmondragon, L. Esteva and L. Chavezgalan,
A mathematical model for cellular immunology of tuberculosis, Math. Biosci. Eng., 8 (2011), 973-986.
doi: 10.3934/mbe.2011.8.973. |
[15] |
A. Kahnert, P. Seiler, M. Stein, S. Bandermann, K. Hahnke, H. J. Mollenkopf and S. H. E. Kaufmann, Alternative activation deprives macrophages of a coordinated defense program to Mycobacterium tuberculosis, Eur. J. Immunol., 36 (2006), 631-647. Google Scholar |
[16] |
P. L. Lin and J. L. Flynn,
Understanding latent tuberculosis: A moving target, J. Immunol., 185 (2010), 15-22.
doi: 10.4049/jimmunol.0903856. |
[17] |
S. Marino and D. E. Kirschner,
The human immune response to Mycobacterium tuberculosis in lung and lymph node, J. Theoret. Biol., 227 (2004), 463-486.
doi: 10.1016/j.jtbi.2003.11.023. |
[18] |
L. Ramakrishnan,
Revisiting the role of the granuloma in tuberculosis, Nat. Rev. Immunol., 12 (2012), 352-366.
doi: 10.1038/nri3211. |
[19] |
N. V. Serbina and J. L. Flynn,
Early emergence of CD8$^+$ T cells primed for production of type 1 cytokines in the lungs of Mycobacterium tuberculosis-infected mice, Infect. Immun., 67 (1999), 3980-3988.
doi: 10.1128/IAI.67.8.3980-3988.1999. |
[20] |
N. V. Serbina, C. C. Liu, C. A. Scanga and J. L. Flynn,
CD8$^+$ CTL from lungs of Mycobacterium tuberculosis-infected mice express perforin in vivo and Lyse infected macrophages, J. Immunol., 165 (2000), 353-363.
doi: 10.4049/jimmunol.165.1.353. |
[21] |
C. Shan, Y. Yi and H. Zhu,
Nilpotent singularities and dynamics in an SIR type of compartmental model with hospital resources, J. Differential Equations, 260 (2016), 4339-4365.
doi: 10.1016/j.jde.2015.11.009. |
[22] |
R. Shi, Y. Li and S. Tang,
A mathematical model with optimal control for cellular immunology of tuberculosis, Taiwanese J. Math., 18 (2014), 575-597.
doi: 10.11650/tjm.18.2014.3739. |
[23] |
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.
doi: 10.4049/jimmunol.177.8.5747-a. |
[24] |
M. Travar, M. Petkovic and A. Verhaz,
Type â…, â…¡, and â…¢ interferons: Regulating immunity to Mycobacterium tuberculosis infection, Arch. Immunol. Ther. Ex., 64 (2016), 19-31.
doi: 10.1007/s00005-015-0365-7. |
[25] |
J. E. Wigginton and D. Kirschner, A model to predict cell-mediated immune regulatory mechanisms during human infection with Mycobacterium tuberculosis, J. Immunol., 166 (2001), 1951-1967. Google Scholar |
[26] |
World Health Organization, Global tuberculosis report 2016, (2016), https://www.who.int/tb/publications/2016/en/. Google Scholar |
[27] |
World Health Organization, Global tuberculosis report 2019, (2019), https://www.who.int/tb/publications/global_report/en/. Google Scholar |
[28] |
P. Yu,
Closed-form conditions of bifurcation points for general differential equations, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 15 (2005), 1467-1483.
doi: 10.1142/S0218127405012582. |
[29] |
W. Zhang, Analysis of an in-host tuberculosis model for disease control, Appl. Math. Lett., 99 (2020), 105983, 7 pp.
doi: 10.1016/j.aml.2019.07.014. |
show all references
References:
[1] |
V. D. Alvarez-Jiménez, K. Leyva-Paredes, M. García-Martínez, L. Vázquez-Flores and I. Estrada-García, Extracellular vesicles released from Mycobacterium tuberculosis-infected neutrophils promote macrophage autophagy and decrease intracellular mycobacterial survival, Front. Immunol., 9 (2018), 272. Google Scholar |
[2] |
L. Cai, G. Chen and D. Xiao,
Multiparametric bifurcations of an epidemiological model with strong Allee effect, J. Math. Biol., 67 (2013), 185-215.
doi: 10.1007/s00285-012-0546-5. |
[3] |
C. Castillo-Chavez and B. Song,
Dynamical models of tuberculosis and their applications, Math. Biosci. Eng., 1 (2004), 361-404.
doi: 10.3934/mbe.2004.1.361. |
[4] |
J. Day, A. Friedman and L. S. Schlesinger, Modeling the immune rheostat of macrophages in the lung in response to infection, Proc. Natl. Acad. Sci. USA, 106 (2009), 11246-11251. Google Scholar |
[5] |
Y. Du, J. Wu and J. M. Heffernan,
A simple in-host model for Mycobacterium tuberculosis that captures all infection outcomes, Math. Popul. Stud., 24 (2017), 37-63.
doi: 10.1080/08898480.2015.1054220. |
[6] |
F. Dumortier, R. Roussarie, J. Sotomayor and H. Żaładek, Bifurcations of Planar Vector Fields, Nilpotent Singularities and Abelian Integrals, Lecture Notes in Mathematics, 1480. Springer-Verlag, Berlin, 1991.
doi: 10.1007/BFb0098353. |
[7] |
J. L. Flynn,
Immunology of tuberculosis and implications in vaccine development, Tuberculosis, 84 (2004), 93-101.
doi: 10.1016/j.tube.2003.08.010. |
[8] |
J. L. Flynn and J. Chan, Immunology of tuberculosis, Annu. Rev. Immunol., 19 (2001), 93-129. Google Scholar |
[9] |
D. Gammack, S. Ganguli, S. Marino, J. L. Segoviajuarez and D. E. Kirschner,
Understanding the immune response in tuberculosis using different mathematical models and biological scales, Multiscale Model. Simul., 3 (2005), 312-345.
doi: 10.1137/040603127. |
[10] |
C. Gong, J. J. Linderman and D. Kirschner,
A population model capture dynamics of tuberculosis granulomas predicts host infection outcomes, Math. Biosci. Eng., 12 (2015), 625-642.
doi: 10.3934/mbe.2015.12.625. |
[11] |
D. He, Q. Wang and W.-C. Lo,
Mathematical analysis of macrophage-bateria interaction in tuberculosis infection, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 3387-3413.
doi: 10.3934/dcdsb.2018239. |
[12] |
J. M. Heffernan, R. J. Smith and L. M. Wahl, Perspectives on the basic reproductive ratio, J. R. Soc. Interface, 2 (2005), 281-293. Google Scholar |
[13] |
E. Ibargüen-Mondragón, L. Esteva and E. M. Burbano-Rosero,
Mathematical model for the growth of Mycobacterium tuberculosis in the granuloma, Math. Biosci. Eng., 15 (2017), 407-428.
doi: 10.3934/mbe.2018018. |
[14] |
E. Ibarguenmondragon, L. Esteva and L. Chavezgalan,
A mathematical model for cellular immunology of tuberculosis, Math. Biosci. Eng., 8 (2011), 973-986.
doi: 10.3934/mbe.2011.8.973. |
[15] |
A. Kahnert, P. Seiler, M. Stein, S. Bandermann, K. Hahnke, H. J. Mollenkopf and S. H. E. Kaufmann, Alternative activation deprives macrophages of a coordinated defense program to Mycobacterium tuberculosis, Eur. J. Immunol., 36 (2006), 631-647. Google Scholar |
[16] |
P. L. Lin and J. L. Flynn,
Understanding latent tuberculosis: A moving target, J. Immunol., 185 (2010), 15-22.
doi: 10.4049/jimmunol.0903856. |
[17] |
S. Marino and D. E. Kirschner,
The human immune response to Mycobacterium tuberculosis in lung and lymph node, J. Theoret. Biol., 227 (2004), 463-486.
doi: 10.1016/j.jtbi.2003.11.023. |
[18] |
L. Ramakrishnan,
Revisiting the role of the granuloma in tuberculosis, Nat. Rev. Immunol., 12 (2012), 352-366.
doi: 10.1038/nri3211. |
[19] |
N. V. Serbina and J. L. Flynn,
Early emergence of CD8$^+$ T cells primed for production of type 1 cytokines in the lungs of Mycobacterium tuberculosis-infected mice, Infect. Immun., 67 (1999), 3980-3988.
doi: 10.1128/IAI.67.8.3980-3988.1999. |
[20] |
N. V. Serbina, C. C. Liu, C. A. Scanga and J. L. Flynn,
CD8$^+$ CTL from lungs of Mycobacterium tuberculosis-infected mice express perforin in vivo and Lyse infected macrophages, J. Immunol., 165 (2000), 353-363.
doi: 10.4049/jimmunol.165.1.353. |
[21] |
C. Shan, Y. Yi and H. Zhu,
Nilpotent singularities and dynamics in an SIR type of compartmental model with hospital resources, J. Differential Equations, 260 (2016), 4339-4365.
doi: 10.1016/j.jde.2015.11.009. |
[22] |
R. Shi, Y. Li and S. Tang,
A mathematical model with optimal control for cellular immunology of tuberculosis, Taiwanese J. Math., 18 (2014), 575-597.
doi: 10.11650/tjm.18.2014.3739. |
[23] |
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.
doi: 10.4049/jimmunol.177.8.5747-a. |
[24] |
M. Travar, M. Petkovic and A. Verhaz,
Type â…, â…¡, and â…¢ interferons: Regulating immunity to Mycobacterium tuberculosis infection, Arch. Immunol. Ther. Ex., 64 (2016), 19-31.
doi: 10.1007/s00005-015-0365-7. |
[25] |
J. E. Wigginton and D. Kirschner, A model to predict cell-mediated immune regulatory mechanisms during human infection with Mycobacterium tuberculosis, J. Immunol., 166 (2001), 1951-1967. Google Scholar |
[26] |
World Health Organization, Global tuberculosis report 2016, (2016), https://www.who.int/tb/publications/2016/en/. Google Scholar |
[27] |
World Health Organization, Global tuberculosis report 2019, (2019), https://www.who.int/tb/publications/global_report/en/. Google Scholar |
[28] |
P. Yu,
Closed-form conditions of bifurcation points for general differential equations, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 15 (2005), 1467-1483.
doi: 10.1142/S0218127405012582. |
[29] |
W. Zhang, Analysis of an in-host tuberculosis model for disease control, Appl. Math. Lett., 99 (2020), 105983, 7 pp.
doi: 10.1016/j.aml.2019.07.014. |





Range of |
Other conditions | Equilibria of system (2) |
No positive equilibrium | ||
No positive equilibrium | ||
One positive equilibrium | ||
One positive equilibrium | ||
One positive equilibrium | ||
One positive equilibrium3 | ||
Two positive equilibria1, 2 | ||
Three positive equilibria | ||
No positive equilibrium | ||
No positive equilibrium | ||
No positive equilibrium | ||
One positive equilibrium | ||
One positive equilibrium3 | ||
One positive equilibrium | ||
One positive equilibrium | ||
Two positive equilibria1, 2 | ||
Three positive equilibria | ||
No positive equilibrium | ||
No positive equilibrium | ||
One positive equilibrium | ||
One positive equilibrium | ||
One positive equilibrium2 | ||
Two positive equilibria | ||
No positive equilibrium | ||
No positive equilibrium | ||
One positive equilibrium2 | ||
Two positive equilibria | ||
1. Two positive equilibria1;2 means that there are a simple equilibrium and a equilibrium of multiplicity 2. One positive equilibrium2 means that there is a equilibrium of multiplicity 2. One positive equilibrium3 means that system has a equilibrium of multiplicity 3. |
Range of |
Other conditions | Equilibria of system (2) |
No positive equilibrium | ||
No positive equilibrium | ||
One positive equilibrium | ||
One positive equilibrium | ||
One positive equilibrium | ||
One positive equilibrium3 | ||
Two positive equilibria1, 2 | ||
Three positive equilibria | ||
No positive equilibrium | ||
No positive equilibrium | ||
No positive equilibrium | ||
One positive equilibrium | ||
One positive equilibrium3 | ||
One positive equilibrium | ||
One positive equilibrium | ||
Two positive equilibria1, 2 | ||
Three positive equilibria | ||
No positive equilibrium | ||
No positive equilibrium | ||
One positive equilibrium | ||
One positive equilibrium | ||
One positive equilibrium2 | ||
Two positive equilibria | ||
No positive equilibrium | ||
No positive equilibrium | ||
One positive equilibrium2 | ||
Two positive equilibria | ||
1. Two positive equilibria1;2 means that there are a simple equilibrium and a equilibrium of multiplicity 2. One positive equilibrium2 means that there is a equilibrium of multiplicity 2. One positive equilibrium3 means that system has a equilibrium of multiplicity 3. |
Para. | Range | Units | Source | Para. | Range | Units | Source |
1/ml day | [5, 13] | see text | 1/day | ||||
1/day | [5, 13] | 1/day | [5] | ||||
1/day | [5, 13] | 1/day | [13] | ||||
1/day | [5, 13] | 0-0.52 | 1/day | [4, 13] | |||
1/day | [5, 13] | 1/day | [5] | ||||
1/ml day | [5] | 1/day | [5] | ||||
1/day | [5] | 1/day | [13] |
Para. | Range | Units | Source | Para. | Range | Units | Source |
1/ml day | [5, 13] | see text | 1/day | ||||
1/day | [5, 13] | 1/day | [5] | ||||
1/day | [5, 13] | 1/day | [13] | ||||
1/day | [5, 13] | 0-0.52 | 1/day | [4, 13] | |||
1/day | [5, 13] | 1/day | [5] | ||||
1/ml day | [5] | 1/day | [5] | ||||
1/day | [5] | 1/day | [13] |
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