Bifurcation analysis for an in-host Mycobacterium tuberculosis model

Miaoran Yao; Yongxin Zhang; Wendi Wang

doi:10.3934/dcdsb.2020324

Article Contents

2021, Volume 26, Issue 4: 2299-2322. Doi: 10.3934/dcdsb.2020324

This issue Previous Article Positive solution branches of two-species competition model in open advective environments Next Article Asymptotic stability of two types of traveling waves for some predator-prey models

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
^* Corresponding author: Wendi Wang

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

Received: July 31, 2020

Revised: September 30, 2020

Early access: November 2020

Published: April 2021

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

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Abstract

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.

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

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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 $

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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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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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