# American Institute of Mathematical Sciences

April  2018, 15(2): 407-428. doi: 10.3934/mbe.2018018

## 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

* Corresponding author: Eduardo Ibargüen-Mondragón

Grant No 182-01/11/201, Vicerrectoría de Investigaciones, Posgrados y Relaciones Internacionales de la Universidad de Nariño.

Received  July 27, 2016 Accepted  May 07, 2017 Published  January 2018

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.

Citation: Eduardo Ibargüen-Mondragón, Lourdes Esteva, Edith Mariela Burbano-Rosero. Mathematical model for the growth of Mycobacterium tuberculosis in the granuloma. Mathematical Biosciences & Engineering, 2018, 15 (2) : 407-428. doi: 10.3934/mbe.2018018
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Grant No 182-01/11/201, Vicerrectoría de Investigaciones, Posgrados y Relaciones Internacionales de la Universidad de Nariño.

##### References:
The flow diagram of macrophages, T cells and bacteria
The graph of functions $g_1$ and $g_2$ defined in (20).
for $\nu, \gamma_{U} = \bar \gamma\displaystyle{\Lambda_U \over \mu_U}$ and $\mu_{B}$">Figure 3.  Standard regression coefficients (SCR) for $R_0 = \frac{\nu}{\gamma_U + \mu_{B}}$ assuming the values given in Table 1 for $\nu, \gamma_{U} = \bar \gamma\displaystyle{\Lambda_U \over \mu_U}$ and $\mu_{B}$
for $\bar r, \bar \beta, \displaystyle{\Lambda_U \over \mu_U}, \gamma_{U} = \bar \gamma\displaystyle{\Lambda_U \over \mu_U}$ and $\mu_{B}$.">Figure 4.  Standard regression coefficients (SCR) for $R_1 = \frac{\bar r \bar\beta {\Lambda_U \over \mu}}{\gamma_U + \mu_{B}}$ assuming the values given in Table 1 for $\bar r, \bar \beta, \displaystyle{\Lambda_U \over \mu_U}, \gamma_{U} = \bar \gamma\displaystyle{\Lambda_U \over \mu_U}$ and $\mu_{B}$.
The numerical simulations of temporal course for bacteria with ten initial conditions show the stability of the bacteria-present equilibrium $P_2$ and the infection free equilibrium $P_0$ given in (47) when $\sigma = 0.24$, $\sigma_c = 0.319$, $R_0 = 0.4$, $R_0 = 0.34$, $R_1 = 1.5$, $g_1(B^{\max}) = 1.37\times 10^{312}$ and $g_2(B^{\max}) = 1.32\times 10^{942}$.
The numerical simulations of temporal course for bacteria with ten initial conditions show the stability of the bacteria-present equilibria $P_1$ a$P_3$ given in (48) when $\sigma = 2.4\times 10^{-6}$, $\sigma_c = 0.003$, $R_0 = 0.0045$, $R^*_0 = 0.0043$, $R_1 = 0.43$.
The stable infection free equilibrium $P_0$ bifurcates to the stable bacteria-present equilibrium $P_1$ in the value $R_0 = 1-R_1$.
The results suggest forward and backward bifurcations, and a type of S-shaped bifurcation
Interpretation and values of the parameters. Data are deduced from the literature (references).
 Parameter Description Value Reference $\Lambda_U$ growth rate of unfected Mtb 600 -1000 day$^{-1}$ [19,23,30] $\bar\beta$ infection rate of Mtb $2.5*10^{-11}-2.5*10^{-7}$day$^{-1}$ [13,30] $\bar\alpha_T$ elim. rate of infected Mtb by T cell $2*10^{-5}-3*10^{-5}$ day$^{-1}$ [13,30] $\mu_U$ nat. death rate of $M_U$ 0028-0.0033 day$^{-1}$ [22,30] $\mu_I$ nat. death rate of $M_I$ 0.011 day$^{-1}$ [22,35,30] $\nu$ growth rate of Mtb 0.36 -0.52 day$^{-1}$ [12,20,38] $\mu_{B}$ natural death rate of Mtb 0.31 -0.52 day$^{-1}$ [39,30] $\bar \gamma_U$ elim. rate of Mtb by $M_U$ $1.2* 10^{-9} - 1.2*10^{-7}$ day$^{-1}$ [30] $K$ carrying cap. of Mtb in the gran. $10^8-10^9$ bacteria [7] $\bar k_I$ growth rate of T cells $8*10^{-3}$ day$^{-1}$ [11] $T_{max}$ maximum recruitment of T cells 5.000 day$^{-1}$ [11] $\mu_T$ natural death rate of T cells 0.33 day$^{-1}$ [35,30] $\bar r$ Average Mtb released by one $M_U$ 0.05-0.2 day$^{-1}$ [30,35]
 Parameter Description Value Reference $\Lambda_U$ growth rate of unfected Mtb 600 -1000 day$^{-1}$ [19,23,30] $\bar\beta$ infection rate of Mtb $2.5*10^{-11}-2.5*10^{-7}$day$^{-1}$ [13,30] $\bar\alpha_T$ elim. rate of infected Mtb by T cell $2*10^{-5}-3*10^{-5}$ day$^{-1}$ [13,30] $\mu_U$ nat. death rate of $M_U$ 0028-0.0033 day$^{-1}$ [22,30] $\mu_I$ nat. death rate of $M_I$ 0.011 day$^{-1}$ [22,35,30] $\nu$ growth rate of Mtb 0.36 -0.52 day$^{-1}$ [12,20,38] $\mu_{B}$ natural death rate of Mtb 0.31 -0.52 day$^{-1}$ [39,30] $\bar \gamma_U$ elim. rate of Mtb by $M_U$ $1.2* 10^{-9} - 1.2*10^{-7}$ day$^{-1}$ [30] $K$ carrying cap. of Mtb in the gran. $10^8-10^9$ bacteria [7] $\bar k_I$ growth rate of T cells $8*10^{-3}$ day$^{-1}$ [11] $T_{max}$ maximum recruitment of T cells 5.000 day$^{-1}$ [11] $\mu_T$ natural death rate of T cells 0.33 day$^{-1}$ [35,30] $\bar r$ Average Mtb released by one $M_U$ 0.05-0.2 day$^{-1}$ [30,35]
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