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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

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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