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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  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 & Continuous Dynamical Systems - B, 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. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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

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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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[18]

L. Ramakrishnan, Revisiting the role of the granuloma in tuberculosis, Nat. Rev. Immunol., 12 (2012), 352-366.  doi: 10.1038/nri3211.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

L. Ramakrishnan, Revisiting the role of the granuloma in tuberculosis, Nat. Rev. Immunol., 12 (2012), 352-366.  doi: 10.1038/nri3211.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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 equilibria1, 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 equilibria1, 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 equilibria1, 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 equilibria1, 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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