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Bifurcation analysis for an in-host Mycobacterium tuberculosis model

  • * Corresponding author: Wendi Wang

    * Corresponding author: Wendi Wang

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

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

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  • 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.

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

    Citation:

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  • 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 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.
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    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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