Mathematical analysis of macrophage-bacteria interaction in tuberculosis infection

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.

Bifurcation analysis for an in-host Mycobacterium tuberculosis model

“…Furthermore, the Bogdanov-Takens bifurcation of codimension 3 in system (2) induces richer dynamical behaviors such as the degenerate Hopf bifurcation, two limit cycles, saddle-node of limit cycle, a homoclinic loop, large-amplitude oscillations and small-amplitude oscillations. These phenomena are not observed in [5,11,13,29]. It should be noted that as the parameters stay in the neighborhood of the SN lc , there exist a stable periodic solution, an unstable periodic solution and a stable equilibrium.…”

“…Firstly, we find that system (2) has a stable immune-free equilibrium when the reproduction number of immune response R 1 is less than unity, which illustrates that people with weak or deficient acquired immunity are more likely to develop active tuberculosis [7]. Furthermore, comparing to [5,11,13,22,29], there emerge bifurcation phenomena such as homocilnic bifurcation, Bogdanov-Takens bifurcation of codimension 2 and 3 and saddle-node bifurcation of limit cycle. These findings provide more insights how the latency can be a dynamic process and may rapidly progress to reactivation by a large oscillation, and how a slow-fast periodic solution mimics the progression of the disease which stays long in the low infection status and then reaches a high infection status, as advocated by [8,16] for tuberculosis.…”

“…Various mathematical models have been applied to understand the dynamics of Mtb [5,11,13,14,17,22,25,29]. Recently, considering the principal features of cellular immunology against Mtb and the new discovery that Mtb can grow vigorously in the micro-environment of granulomas, Ibarguen-Mondragon et al [13] extended their previous studies [14] and established a model as follows:…”