Boundedness and asymptotic stability in a model for tuberculosis granuloma formation

TL;DR

This paper analyzes a complex mathematical model for tuberculosis granuloma formation, proving global existence and exponential convergence of solutions under certain conditions related to initial data size and reproduction number.

Contribution

It establishes boundedness and asymptotic stability of solutions for the tuberculosis model with specific parameter constraints.

Findings

Solutions exist globally under small initial data.

Solutions converge exponentially to equilibrium when R0<1.

Model behavior depends on the reproduction number R0.

Abstract

This paper deals with a problem which describes tuberculosis granuloma formation \begin{align*} \begin{cases} u_t = \Delta u - \nabla \cdot (u \nabla v) - uv - u + \beta, &x \in \Omega,\ t>0, \\ v_t = \Delta v + v -uv + \mu w, &x \in \Omega,\ t>0, \\ w_t = \Delta w + uv - wz - w, &x \in \Omega,\ t>0, \\ z_t = \Delta z - \nabla \cdot (z \nabla w) + f(w)z -z, &x \in \Omega,\ t>0 \end{cases} \end{align*} under homogeneous Neumann boundary conditions and initial conditions, where $\Omega \subset \mathbb{R}^n$ ($n\ge 2$) is a smooth bounded domain, $\beta,\mu>0$ and $f$ is some function, and shows that if initial data are small in some sense then the solution $(u,v,w,z)$ of the problem exists globally and convergences to $(\beta,0,0,0)$ exponentially when $\beta>1$ and the reproduction number $R_0 := \frac{\mu \beta + 1}{\beta}$ satisfies $R_0<1$.