Boundedness and asymptotic stability in a model for tuberculosis granuloma formation

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 $R_0<1$.