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 = Δu - \nabla \cdot (u \nabla v) - uv - u + β, &x \in Ω,\ t>0, \ v_t = Δv + v -uv + μw, &x \in Ω,\ t>0, \ w_t = Δw + uv - wz - w, &x \in Ω,\ t>0, \ z_t = Δz - \nabla \cdot (z \nabla w) + f(w)z -z, &x \in Ω,\ t>0 \end{cases} \end{align*} under homogeneous Neumann boundary conditions and initial conditions, where $\mathrm{\Omega }\subset {\mathbb{R}}^n$ ($n\ge 2$) 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 $R_0<1$.