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

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

* 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.

Citation: Miaoran Yao, Yongxin Zhang, Wendi Wang. Bifurcation analysis for an in-host Mycobacterium tuberculosis model. Discrete & Continuous Dynamical Systems - B, 2021, 26 (4) : 2299-2322. doi: 10.3934/dcdsb.2020324
References:
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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. CaiG. 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. DayA. 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. DuJ. 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. GammackS. GanguliS. MarinoJ. 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. GongJ. 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. HeQ. 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. HeffernanR. 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-MondragónL. 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. IbarguenmondragonL. 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. KahnertP. SeilerM. SteinS. BandermannK. HahnkeH. 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. SerbinaC. C. LiuC. 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. ShanY. 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. ShiY. 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. SudC. BigbeeJ. 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. TravarM. 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 $
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
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
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)
Figure 4">Figure 5.  Time series diagram of large and small oscillation. Here $ \mu_U = 0.026 $, and other parameters are same as Figure 4
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) $)
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.
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]
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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