Bifurcation analysis for an in-host Mycobacterium tuberculosis model

Miaoran Yao; Yongxin Zhang; Wendi Wang

doi:10.3934/dcdsb.2020324

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April 2021, 26(4): 2299-2322. doi: 10.3934/dcdsb.2020324

Bifurcation analysis for an in-host Mycobacterium tuberculosis model

Miaoran Yao , Yongxin Zhang and Wendi Wang ^,

Key Laboratory of Eco-environments in Three Gorges Reservoir Region, School of Mathematics and Statistics, Southwest University, Chongqing 400715, China

^* Corresponding author: Wendi Wang

Dedicated to Professor Sze-Bi Hsu on the occasion of his 70th birthday

Received August 2020 Revised October 2020 Published April 2021 Early access November 2020

Fund Project: The study was supported by grant from the National Natural Science Foundation of China (12071381)

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.

Keywords: Immune response, backward bifurcation, slow-fast oscillation, Bogdanov-Takens bifurcation, codimension 3.

Mathematics Subject Classification: Primary:92D30;Secondary:34C23.

Citation: Miaoran Yao, Yongxin Zhang, Wendi Wang. Bifurcation analysis for an in-host Mycobacterium tuberculosis model. Discrete and Continuous Dynamical Systems - B, 2021, 26 (4) : 2299-2322. doi: 10.3934/dcdsb.2020324

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.
[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.
[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.
[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.
[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.
[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.
[26]	World Health Organization, Global tuberculosis report 2016, (2016), https://www.who.int/tb/publications/2016/en/.
[27]	World Health Organization, Global tuberculosis report 2019, (2019), https://www.who.int/tb/publications/global_report/en/.
[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.
[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.
[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.
[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.
[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.
[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.
[26]	World Health Organization, Global tuberculosis report 2016, (2016), https://www.who.int/tb/publications/2016/en/.
[27]	World Health Organization, Global tuberculosis report 2019, (2019), https://www.who.int/tb/publications/global_report/en/.
[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.

Figure 1. Backward and forward bifurcation. Blue (red) curves represent the stable (unstable) singularities and SN denotes saddle-node bifurcation. Parameters are taken as: $ \mu_U = 0.025,\; \beta = 1.1\times10^{-7},\; \alpha = 2.9\times10^{-5},\; K = 7\times10^{5},\; \delta = 2.7\times10^{-6},\; \mu_{T} = 0.008,\rho = 4.9\times10^{-7},\Lambda = 5100 $. (a) $ v = 0.41,\; \mu_{B} = 0.05 $. (b) $ v = 0.15,\; \mu_{B} = 0.12 $

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Figure 2. Bifurcation diagram of system (2). The black and green lines represent saddle-node bifurcation. The blue (red) line stands for subcritical (supercritical) Hopf bifurcation. CP, BT, and DH denote Cusp-bifurcation, Bogdanov-Takens bifurcation and Degenerate Hopf bifurcation respectively

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Figure 3. Bifurcation of positive equilibria. Blue (red) curve stands for stable (unstable) equilibrium. $ SN $, $ H^{+} $ and $ H_{i}^-\; (i = 1,2,3) $ represent saddle-node bifurcation, subcritical Hopf bifurcation and supercritical Hopf bifurcation respectively. The positive equilibrium coalesce with the boundary equilibrium at BP point

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Figure 4. Bifurcation of positive equilibria with periodic solution involved. Blue (red) curve stands for stable (unstable) equilibrium or periodic solution. $ SN_{lc} $, $ H^{+} $ and $ H_{i}^{-}k__ge (i=1,2,3) $ represent saddle-node bifurcation of limit cycle, subcritical Hopf bifurcation and supercritical Hopf bifurcation respectively. (d) We magnify the small neighborhood of $ H_{1}^- $ in (c)

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Figure 5. Time series diagram of large and small oscillation. Here $ \mu_U = 0.026 $, and other parameters are same as Figure 4

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Figure 6. A profile graph of functions $ F_1(B) $ and $ F_3(B) $ defined in (19) and (50). The blue (red) line denotes $ F_1(B) $ ($ F_3(B) $)

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Table 1. Distribution of the positive equilibria

Range of $ R_0,R_1 $	Other conditions	Equilibria of system (2)
$ R_0<R^*_0 $		No positive equilibrium
$ R_0\geq1,R_1\leq1 $		No positive equilibrium
$ R_0>1,R_1>1 $	$ c_2\geq0 $	One positive equilibrium
$ R_0\geq1,R_1>1 $	$ c_2<0,c_3\leq0 $	One positive equilibrium
	$ c_2<0,c_3>0,\Delta_1>0 $	One positive equilibrium
	$ c_2<0,c_3>0,P_1=0,\Delta_1=0 $	One positive equilibrium³
	$ c_2<0,c_3>0,P_1>0,\Delta_1=0 $	Two positive equilibria^{1, 2}
	$ c_2<0,c_3>0,\Delta_1<0 $	Three positive equilibria
${R_0} < 1$	${c_2} \ge 0$	No positive equilibrium
	${c_2} < 0,{R_1} \le 1$	No positive equilibrium
	${c_2} < 0,{R_0} = R_0^*,{R_1} = {R_2} > 1$	No positive equilibrium
$R_0^* < {R_0} < 1,{R_1} > 1 > {R_2}$	${c_2} < 0,{P_1} \le 0,{\Delta _1} > 0$	One positive equilibrium
	${c_2} < 0,{P_1} = 0,{\Delta _1} = 0$	One positive equilibrium³
	${c_2} < 0,{P_1} > 0,{\Delta _1} > 0$	One positive equilibrium
	${c_2} < 0,{P_1} > 0,{B^{2b}} \ge {B_{12}}$	One positive equilibrium
	${c_2} < 0,{P_1} > 0,{B^{2b}} < {B_{12}},{\Delta _1} = 0$	Two positive equilibria^{1, 2}
	${c_2} < 0,{P_1} > 0,{B^{2b}} < {B_{12}},{\Delta _1} < 0$	Three positive equilibria
$R_0^* < {R_0} < 1,{R_2} = 1$	${c_2} < 0,{P_1} \le 0$	No positive equilibrium
	${c_2} < 0,{P_1} > 0,{\Delta _1} > 0$	No positive equilibrium
	${c_2} < 0,{P_1} > 0,{\Delta _1} = 0,{B^{2b}} = {B_{12}}$	One positive equilibrium
	${c_2} < 0,{P_1} > 0,{\Delta _1} < 0,{B_{11}} > {B^{2b}} > {B_{12}}$	One positive equilibrium
	${c_2} < 0,{P_1} > 0,{\Delta _1} = 0,{B^{2b}} < {B_{12}}$	One positive equilibrium²
	${c_2} < 0,{P_1} > 0,{\Delta _1} < 0,{B^{2b}} < {B_{12}}$	Two positive equilibria
$R_0^* < {R_0} < 1,{R_2} > 1$	${c_2} < 0,{P_1} \le 0$	No positive equilibrium
	${c_2} < 0,{P_1} > 0,{\Delta _1} > 0$	No positive equilibrium
	${c_2} < 0,{P_1} > 0,{\Delta _1} = 0,{B^{2b}} < {B_{11}}$	One positive equilibrium²
	${c_2} < 0,{P_1} > 0,{\Delta _1} < 0,{B^{2b}} < {B_{11}}$	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.

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Range of $ R_0,R_1 $	Other conditions	Equilibria of system (2)
$ R_0<R^*_0 $		No positive equilibrium
$ R_0\geq1,R_1\leq1 $		No positive equilibrium
$ R_0>1,R_1>1 $	$ c_2\geq0 $	One positive equilibrium
$ R_0\geq1,R_1>1 $	$ c_2<0,c_3\leq0 $	One positive equilibrium
	$ c_2<0,c_3>0,\Delta_1>0 $	One positive equilibrium
	$ c_2<0,c_3>0,P_1=0,\Delta_1=0 $	One positive equilibrium³
	$ c_2<0,c_3>0,P_1>0,\Delta_1=0 $	Two positive equilibria^{1, 2}
	$ c_2<0,c_3>0,\Delta_1<0 $	Three positive equilibria
${R_0} < 1$	${c_2} \ge 0$	No positive equilibrium
	${c_2} < 0,{R_1} \le 1$	No positive equilibrium
	${c_2} < 0,{R_0} = R_0^*,{R_1} = {R_2} > 1$	No positive equilibrium
$R_0^* < {R_0} < 1,{R_1} > 1 > {R_2}$	${c_2} < 0,{P_1} \le 0,{\Delta _1} > 0$	One positive equilibrium
	${c_2} < 0,{P_1} = 0,{\Delta _1} = 0$	One positive equilibrium³
	${c_2} < 0,{P_1} > 0,{\Delta _1} > 0$	One positive equilibrium
	${c_2} < 0,{P_1} > 0,{B^{2b}} \ge {B_{12}}$	One positive equilibrium
	${c_2} < 0,{P_1} > 0,{B^{2b}} < {B_{12}},{\Delta _1} = 0$	Two positive equilibria^{1, 2}
	${c_2} < 0,{P_1} > 0,{B^{2b}} < {B_{12}},{\Delta _1} < 0$	Three positive equilibria
$R_0^* < {R_0} < 1,{R_2} = 1$	${c_2} < 0,{P_1} \le 0$	No positive equilibrium
	${c_2} < 0,{P_1} > 0,{\Delta _1} > 0$	No positive equilibrium
	${c_2} < 0,{P_1} > 0,{\Delta _1} = 0,{B^{2b}} = {B_{12}}$	One positive equilibrium
	${c_2} < 0,{P_1} > 0,{\Delta _1} < 0,{B_{11}} > {B^{2b}} > {B_{12}}$	One positive equilibrium
	${c_2} < 0,{P_1} > 0,{\Delta _1} = 0,{B^{2b}} < {B_{12}}$	One positive equilibrium²
	${c_2} < 0,{P_1} > 0,{\Delta _1} < 0,{B^{2b}} < {B_{12}}$	Two positive equilibria
$R_0^* < {R_0} < 1,{R_2} > 1$	${c_2} < 0,{P_1} \le 0$	No positive equilibrium
	${c_2} < 0,{P_1} > 0,{\Delta _1} > 0$	No positive equilibrium
	${c_2} < 0,{P_1} > 0,{\Delta _1} = 0,{B^{2b}} < {B_{11}}$	One positive equilibrium²
	${c_2} < 0,{P_1} > 0,{\Delta _1} < 0,{B^{2b}} < {B_{11}}$	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.

Table 2. Parameter range and source for simulation.

Para.	Range	Units	Source	Para.	Range	Units	Source
$ \Lambda $	$ 600-7000 $	1/ml day	[5, 13]	$ \mu_U $	see text	1/day
$ \sigma $	$ 0.011-0.5 $	1/day	[5, 13]	$ \mu_I $	$ 0-2 $	1/day	[5]
$ \beta $	$ 2.5\times10^{-11}-10^{-5} $	1/day	[5, 13]	$ v $	$ 0-0.52 $	1/day	[13]
$ N $	$ 0.05-100 $	1/day	[5, 13]	$ \mu_B $	0-0.52	1/day	[4, 13]
$ K $	$ 10^{5}-10^{10} $	1/day	[5, 13]	$ \theta $	$ 0.025-50 $	1/day	[5]
$ \delta $	$ 10^{-9}-10^{-6} $	1/ml day	[5]	$ \mu_T $	$ 0.01-0.33 $	1/day	[5]
$ \rho $	$ 10^{-8}-1 $	1/day	[5]	$ \alpha $	$ 2\times10^{-5}-3\times10^{-5} $	1/day	[13]

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Para.	Range	Units	Source	Para.	Range	Units	Source
$ \Lambda $	$ 600-7000 $	1/ml day	[5, 13]	$ \mu_U $	see text	1/day
$ \sigma $	$ 0.011-0.5 $	1/day	[5, 13]	$ \mu_I $	$ 0-2 $	1/day	[5]
$ \beta $	$ 2.5\times10^{-11}-10^{-5} $	1/day	[5, 13]	$ v $	$ 0-0.52 $	1/day	[13]
$ N $	$ 0.05-100 $	1/day	[5, 13]	$ \mu_B $	0-0.52	1/day	[4, 13]
$ K $	$ 10^{5}-10^{10} $	1/day	[5, 13]	$ \theta $	$ 0.025-50 $	1/day	[5]
$ \delta $	$ 10^{-9}-10^{-6} $	1/ml day	[5]	$ \mu_T $	$ 0.01-0.33 $	1/day	[5]
$ \rho $	$ 10^{-8}-1 $	1/day	[5]	$ \alpha $	$ 2\times10^{-5}-3\times10^{-5} $	1/day	[13]

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American Institute of Mathematical Sciences

Bifurcation analysis for an in-host Mycobacterium tuberculosis model

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Bifurcation analysis for an in-host Mycobacterium tuberculosis model

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