Mathematical analysis of macrophage-bacteria interaction in tuberculosis infection

Danyun He; Qian Wang; Wing-Cheong Lo

doi:10.3934/dcdsb.2018239

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2018, Volume 23, Issue 8: 3387-3413. Doi: 10.3934/dcdsb.2018239 open access

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Mathematical analysis of macrophage-bacteria interaction in tuberculosis infection

Department of Mathematics, City University of Hong Kong, Hong Kong SAR, China

* Corresponding author: W.-C. LO
* Corresponding author: W.-C. LO

Received: October 31, 2017

Revised: March 31, 2018

Early access: August 2018

Published: August 2018

W.-C. LO is supported by a CityU StUp Grant (No. 7200437)

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Abstract

Tuberculosis (TB) is a leading cause of death from infectious disease. TB is caused mainly by a bacterium called Mycobacterium tuberculosis which often initiates in the respiratory tract. The interaction of macrophages and T cells plays an important role in the immune response during TB infection. Recent experimental results support that active TB infection may be induced by the dysfunction of Treg cell regulation that provides a balance between anti-TB T cell responses and pathology. To better understand the dynamics of TB infection and Treg cell regulation, we build a mathematical model using a system of differential equations that qualitatively and quantitatively characterizes the dynamics of macrophages, Th1 and Treg cells during TB infection. For sufficiently analyzing the interaction between immune response and bacterial infection, we separate our model into several simple subsystems for further steady state and stability studies. Using this system, we explore the conditions of parameters for three situations, recovery, latent disease and active disease, during TB infection. Our numerical simulations support that Th1 cells and Treg cells play critical roles in TB infection: Th1 cells inhibit the number of infected macrophages to reduce the chance of active disease; Treg cell regulation reduces the immune response to stabilize the dynamics of the system.

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Mathematics Subject Classification: Primary: 92B05, 92C42; Secondary: 92C37.

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Figure 1. Model diagram of bacteria, T cells and macrophages interaction.The Sharp arrow means activation; blocked arrow means inhibition. When bacteria encountering resting macrophages, some macrophages will be activated and become activated macrophages ($M_a$), while some will be infected and become infected macrophages ($M_i$). $M_a$ may be deactivated in response to regulatory T cells (Treg). Th1 cells will induce the activation of $M_a$. $M_a$ and $M_r$ will inhibit the growth of bacteria. Bursting of $M_i$ will increase the concentration of bacteria. $M_i$ may be eliminated by Th1 cells. Th1 cells activation will increase in response to $M_i$ and $M_a$, and decrease by Treg cells' inhibition. Treg cells will be activated by Th1 cells and $M_a$

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Figure 2. Model diagrams of subsystems for analysis.A) Two-equation model. The interaction between bacteria infection and resting macrophages protection. B) Three-equation model. Bacteria infection, macrophages protection with some resting macrophages are infected and become infected macrophages, which will burst and release bacteria

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Figure 3. Phase plane portraits for the analysis of the two-equation system (7)-(8).A) Case 1 B) Case 2 C) Case 3. Star symbols indicate the steady states; red dotted lines represent $dM_r/dt =0$; blue solid lines represent $dB/dt=0$

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Figure 4. The dynamics of $B$, $M_r$ and $M_i$ with different values of $\beta_{M_r}$ in the three-equation model (13)-(15).The initial conditions are $B(0)=100$, $M_r(0)=\alpha_{M_r}/\mu_{M_r}$ and $M_i(0)=0$. We set $\gamma_i=0.01 \ \text{day}^{-1}$ and the others parameters from Table 3. As the rate of $M_r$ recruitment by $M_i$ and $M_a$ increasing from 0.1 to 10, the number of $M_i$ and $B$ decreases, while the number of $M_r$ is relatively stable

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Figure 5. Model 1: System (1)-(6) without involving Th1 cell and Treg cell.The initial conditions are $B(0)=100$, $M_r(0)=\alpha_{M_r}/\mu_{M_r}$ and $M_i(0)=M_a(0)=T_1(0)= T_r(0)=0$. We set $\alpha_{T_1}=\alpha_{T_r}=\beta_{T_r}=0 \ \text{day}^{-1}$ and the other parameters from Table 3. Resting macrophages are infected by bacteria and become infected macrophages, leading to a sharp increase in bacteria

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Figure 6. Model 2: System (1)-(6) with Th1 cell but without Treg cell regulation. The initial conditions are $B(0)=100$, $M_r(0)=\alpha_{M_r}/\mu_{M_r}$ and $M_i(0)=M_a(0)=T_1(0)= T_r(0)=0$. We set $\alpha_{T_1}=5.2\times10^{-2} \ \text{day}^{-1}$, $\alpha_{T_r}=\beta_{T_r}=0 \ \text{day}^{-1}$ and the other parameters from Table 3. Without regulation from Treg cells, the population of bacteria, macrophages, and Th1 cells oscillates continuously

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Figure 7. The effect of Treg cell regulation in Model 3.A) The change in the average of the OI with respect to the increases of $\alpha_{T_r}$, the $T_r$ activation rate by $M_a$ and $\beta_{T_r}$, the $T_r$ activation rate by $T_1$; B) The change in the average of the maximum concentration of bacteria with respect to increases of $\alpha_{T_r}$ and $\beta_{T_r}$. Each dot represents an average of the simulations with 500 sets of randomly selected parameters from the ranges listed in Table 3. For each simulation, the initial conditions are $B(0)=100$, $M_r(0)=\alpha_{M_r}/\mu_{M_r}$ and $M_i(0)=M_a(0)=T_1(0)= T_r(0)=0$. As $\alpha_{T_r}$ and $\beta_{T_r}$ increase, the oscillation of bacteria becomes smaller, while the maximum value of bacteria decreases first and then increases

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Figure 8. Model 3: System (1)-(6) with Th1 cell control and low level of Treg cell regulation. The initial conditions are $B(0)=100$, $M_r(0)=\alpha_{M_r}/\mu_{M_r}$ and $M_i(0)=M_a(0)=T_1(0)= T_r(0)=0$. We set $\alpha_{T_1}=5.2\times10^{-2} \ \text{day}^{-1}$, $\alpha_{T_r}=\beta_{T_r}=0.1 \ \text{day}^{-1}$ and the other parameters from Table 3. With Treg cell regulation, the number of bacteria remains stable after a few cycles of oscillation

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Figure 9. Model 4: System (1)-(6) with Th1 cell control and strong level of Treg cell regulation.The initial conditions are $B(0)=100$, $M_r(0)=\alpha_{M_r}/\mu_{M_r}$ and $M_i(0)=M_a(0)=T_1(0)= T_r(0)=0$. We set $\alpha_{T_1}=5.2\times10^{-2} \ \text{day}^{-1}$, $\alpha_{T_r}=\beta_{T_r}=1 \ \text{day}^{-1}$, $\beta_{M_r}=0.01 \ \text{day}^{-1}$ and the other parameters from Table 3. Strong Treg cell regulation causes sharp reductions of macrophages and Th1 cells, thus, the number of bacteria rapidly increases

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Figure 10. The effect of enhaced macrophage recruitment in Model 4.A) The change of the average of the OI with respect to the increase of $\beta_{M_r}$; B) The change of the average of the maximum concentration of bacteria with respect to increase of $\beta_{M_r}$, the $M_r$ recruitment rate by $M_a$ and $M_i$; the insert shows the change of the average of the maximum concentration of bacteria when $\beta_{M_r}>0.2$. Each dot represents an average of the simulations with 500 sets of randomly selected parameters from the ranges listed in Table 3. For each simulation, We set $\alpha_{T_r}=\beta_{T_r}=1 \ \text{day}^{-1}$ and the initial conditions are $B(0)=100$, $M_r(0)=\alpha_{M_r}/\mu_{M_r}$ and $M_i(0)=M_a(0)=T_1(0)= T_r(0)=0$. C-D) The simulation results of $B$ with two different values of $\beta_{M_r}$. We set the initial conditions as in (A-B), $\alpha_{T_r}=\beta_{T_r}= 1 \ \text{day}^{-1}$ and the other parameters are listed in Table 3

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Table 1. The definitions of the parameters used in the equations (1)-(6)

Parameter	Definition
$\alpha_{M_r}$	$M_r$ source
$\alpha_B$	Bacteria growth rate
$\alpha_{T_1}$	$T_1$ activation rate by $M_i$ and $M_a$
$\alpha_{T_r}$	$T_r$ activation rate by $M_a$
$\beta_{M_r}$	$M_r$ recruitment by $M_i$ and $M_a$
$\beta_{T_r}$	$T_r$ activation rate by $T_1$
$k_{MB}$	Infection rate
$k_{MT}$	$T_1$ immunity rate
$k_{BM}$	Death rate of bacteria by $M_r$ and $M_a$
$\gamma_r$	Deactivation rate of macrophage
$\gamma_a$	Activation rate of macrophage
$\gamma_i$	Bursting rate of $M_i$
$\mu_{M_r}$	Death rate of $M_r$
$\mu_{M_i}$	Death rate of $M_i$
$\mu_{M_a}$	Death rate of $M_a$
$\mu_{T_1}$	Death rate of $T_1$
$\mu_{T_r}$	Death rate of $T_r$
$\omega_1$	Ratio in $M_r$ recruitment
$\omega_2$	Ratio in bacteria killing
$\omega_3$	Ratio in $T_1$ activation
$K_B$	Half-saturation constant for $B$ in infection
$Q_B$	Half-saturation constant for $B$ in macrophage activation
$K_1$	Half-saturation constant for $T_1$ in immunity
$K_r$	Half-saturation constant for $T_r$ in inhibition
$Q_r$	Half-saturation constant for $T_r$ in macrophage deactivation
$Q_1$	Half-saturation constant for $T_1$ in macrophage activation
$N$	Estimated number of bacteria per macrophage
$\bar{N}$	Number of bacteria releasing from macrophage by Th1 cell immunity

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Table 2. Steady state analysis for the three-equation system (13)-(15). For the conditions, $L_M < L_B$ ($L_M>L_B$) corresponds to weak (strong) macrophage regulation when bacterial concentration is low; $R_M < R_B$ ($R_M>R_B$) corresponds to weak (strong) macrophage regulation when bacterial concentration is high

	Steady states	Status
$L_M < L_B$ $R_M < R_B$	If $R_M < 0, $ Two steady states Unstable steady state $(0, \alpha_{M_r}/\mu_{M_r}, 0)$ Stable or unstable steady state $(B^, M_r^, M_i^*)$	Latent disease (Stable $(B^, M_r^, M_i^)$) Active disease (Unstable $(B^, M_r^, M_i^)$)
$L_M < L_B$ $R_M < R_B$	If $R_M>0, $ One steady state Unstable steady state $(0, \alpha_{M_r}/\mu_{M_r}, 0)$	Active disease
$L_M < L_B$ $R_M>R_B$	Two steady states Unstable steady state $(0, \alpha_{M_r}/\mu_{M_r}, 0)$ Stable or unstable steady state $(B^, M_r^, M_i^*)$	Latent disease (Stable $(B^, M_r^, M_i^)$) Active disease (Unstable $(B^, M_r^, M_i^)$)
$L_M>L_B$ $R_M < R_B$	If $R_M>0$ and $L_B>0$, Two steady states Stable steady state $(0, \alpha_{M_r}/\mu_{M_r}, 0)$ Unstable steady state $(B^, M_r^, M_i^*)$	Active disease or recovery (depends on initial values)
	If $R_M>0$ and $L_B < 0$, or $R_M < 0$, $L_B < 0$ and $R_B/L_B < R_M/L_M$, One steady state Unstable steady state $(0, \alpha_{M_r}/\mu_{M_r}, 0)$	Active disease
	If $R_M < 0$, $L_B < 0$ and $R_B/L_B>R_M/L_M$, Two steady states Unstable steady state $(0, \alpha_{M_r}/\mu_{M_r}, 0)$ Stable steady state $(B^, M_r^, M_i^*)$	Latent disease
	If $R_M < 0$ and $L_B>0$, One steady state Stable steady state $(0, \alpha_{M_r}/\mu_{M_r}, 0)$	Recovery
$L_M>L_B$ $R_M>R_B$	If $L_B < 0$, Two steady states Unstable steady state $(0, \alpha_{M_r}/\mu_{M_r}, 0)$ Stable steady state $(B^, M_r^, M_i^*)$	Latent disease
$L_M>L_B$ $R_M>R_B$	If $L_B>0$, One steady state Stable steady state $(0, \alpha_{M_r}/\mu_{M_r}, 0)$	Recovery

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Table 3. Parameters used in the equations (1)-(6)

Parameter	Range for stability tests in Section 4	Values for the simulations in Figs. 4-6 and Figs. 8-10	Range for $OI$ tests in Fig. 7 and Fig. 10	Unit	Reference
$\alpha_{M_r}$	$3.3\times10^{2}-1\times 10^3$	$5\times10^{2}$	$3.3\times10^{2}-1\times 10^3$	$\text{ml}^{-1} \text{day}^{-1}$	[2,31]
$\alpha_B$	$0.01-0.1$	$5\times10^{-2}$	$0.025-0.075$	$\text{day}^{-1}$	[14,31]
$\alpha_{T_1}$	-	$5.2\times 10^{-2}$	$0.026-0.078$	$\text{day}^{-1}$	[14]
$\alpha_{T_r}$	-	$1\times 10^{-1}$	$1$	$\text{day}^{-1}$	Estimated
$\beta_{M_r}$	$10^{-2}-10 $	$1\times 10^{-2}$	$10^{-2}-0.1$	$\text{day}^{-1}$	[14,31]
$\beta_{T_r}$	-	$1\times 10^{-1}$	$1$	$\text{day}^{-1}$	Estimated
$k_{MB}$	$0.2-0.5$	$5\times 10^{-1}$	$0.2-0.5$	$\text{day}^{-1}$	[14]
$k_{MT}$	-	$1\times 10^{-1} $	$0.08-0.12$	$\text{day}^{-1}$	Estimated
$k_{BM}$	$1.25\times 10^{-9}$ $-2\times 10^{-8}$	$2\times 10^{-8}$	$1.25\times 10^{-9}-2\times 10^{-8}$	$\text{ml}\;\text{day}^{-1}$	[3,14]
$\gamma_r$	-	$2\times 10^{-1}$	$0.1-0.5$	$\text{day}^{-1}$	Estimated
$\gamma_a$	-	$4\times 10^{-1}$	$0.2-0.6$	$\text{day}^{-1}$	Estimated
$\gamma_i$	$0.01-0.1$	$4\times 10^{-2} $	$0.2-0.4$	$\text{day}^{-1}$	Estimated
$\mu_{M_r}$	$1\times 10^{-2} $	$1\times 10^{-2}$	$1\times 10^{-2}$	$\text{day}^{-1}$	[14,29,31]
$\mu_{M_i}$	$1\times 10^{-2}$	$1\times 10^{-2}$	$1\times 10^{-2}$	$\text{day}^{-1}$	[14,29,31]
$\mu_{M_a}$	-	$1\times 10^{-2}$	$1\times 10^{-2}$	$\text{day}^{-1}$	[14,29,31]
$\mu_{T_1}$	-	$3.33\times 10^{-1}$	$3.33\times 10^{-1}$	$\text{day}^{-1}$	[14,31]
$\mu_{T_r}$	-	$3.33\times 10^{-1}$	$3.33\times 10^{-1}$	$\text{day}^{-1}$	[14,31]
$\omega_1$	-	$7.14$	$7.14$		[14]
$\omega_2$	-	$10$	$10$		[14]
$\omega_3$	-	$10$	$10$		Estimated
$K_B$	$1\times 10^{7} $	$1\times 10^{7} $	$1\times 10^{7}$	$\text{ml}^{-1}$	[31]
$Q_B$	-	$1\times 10^{6}$	$1\times 10^{6}$	$\text{ml}^{-1}$	[31]
$K_1$	-	$1$	1		Estimated
$K_r$	-	$1\times 10^{5}$	$1\times 10^{5}$	$\text{ml}^{-1}$	Estimated
$Q_r$	-	$1\times 10^{5}$	$1\times 10^{5}$	$\text{ml}^{-1}$	Estimated
$Q_1$	-	$1\times 10^{2}$	$1\times 10^{2}$	$\text{ml}^{-1}$	Estimated
$N$	$50$	$50$	$50$		[14,32]
$\bar{N}$	-	$10$	$10$		[14]

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