Mathematical model for the growth of Mycobacterium tuberculosis in the granuloma

Eduardo Ibargüen-Mondragón; Lourdes Esteva; Edith Mariela Burbano-Rosero; Eduardo Ibargüen-Mondragón; Lourdes Esteva; Edith Mariela Burbano-Rosero

doi:10.3934/mbe.2018018

Mathematical Biosciences and Engineering

2018, Volume 15, Issue 2: 407-428. doi: 10.3934/mbe.2018018

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Mathematical model for the growth of Mycobacterium tuberculosis in the granuloma

1.
Departamento de Matemáticas y Estadística, Facultad de Ciencias Exactas y Naturales, Universidad de Nariño, Calle 18 Cra 50, Pasto, Colombia
2.
Departamento de Matemáticas, Facultad de Ciencias, Universidad Nacional Autónoma de México, 04510 México DF, México
3.
Departamento de Biología, Facultad de Ciencias Exactas y Naturales, Universidad de Nariño, Calle 18 Cra 50, Pasto, Colombia

Received: 27 July 2016 Accepted: 07 May 2017 Published: 01 April 2018
MSC : Primary: 34D23, 93D20; Secondary: 65L05

In this work we formulate a model for the population dynamics of Mycobacterium tuberculosis (Mtb), the causative agent of tuberculosis (TB). Our main interest is to assess the impact of the competition among bacteria on the infection prevalence. For this end, we assume that Mtb population has two types of growth. The first one is due to bacteria produced in the interior of each infected macrophage, and it is assumed that is proportional to the number of infected macrophages. The second one is of logistic type due to the competition among free bacteria released by the same infected macrophages. The qualitative analysis and numerical results suggests the existence of forward, backward and S-shaped bifurcations when the associated reproduction number $R_0$ of the Mtb is less unity. In addition, qualitative analysis of the model shows that there may be up to three bacteria-present equilibria, two locally asymptotically stable, and one unstable.
- Ordinary differential equations,
- S-shaped bifurcation,
- tuberculosis,
- granuloma,
- macrophages and T cells
Citation: Eduardo Ibargüen-Mondragón, Lourdes Esteva, Edith Mariela Burbano-Rosero. Mathematical model for the growth of Mycobacterium tuberculosis in the granuloma[J]. Mathematical Biosciences and Engineering, 2018, 15(2): 407-428. doi: 10.3934/mbe.2018018

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Abstract

In this work we formulate a model for the population dynamics of Mycobacterium tuberculosis (Mtb), the causative agent of tuberculosis (TB). Our main interest is to assess the impact of the competition among bacteria on the infection prevalence. For this end, we assume that Mtb population has two types of growth. The first one is due to bacteria produced in the interior of each infected macrophage, and it is assumed that is proportional to the number of infected macrophages. The second one is of logistic type due to the competition among free bacteria released by the same infected macrophages. The qualitative analysis and numerical results suggests the existence of forward, backward and S-shaped bifurcations when the associated reproduction number $R_0$ of the Mtb is less unity. In addition, qualitative analysis of the model shows that there may be up to three bacteria-present equilibria, two locally asymptotically stable, and one unstable.

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