  We study the system \begin{align*} \label{prob:star} \tag{$\star$}
  \begin{cases}
  u_t = \Delta u - \nabla \cdot (u \nabla v) - u - f(u) w + \kappa, \\
  v_t = \Delta v - v + f(u) w, \\
  w_t = \Delta w - w + v,
  \end{cases} \end{align*} which models the virus dynamics in an early stage of
an HIV infection, in a smooth, bounded domain $\Omega \subset \mathbb R^n, n
\in \mathbb N,$ for a parameter $\kappa \ge 0$ and a given function $f \in
C^1([0, \infty))$ satisfying $f \ge 0$, $f(0) = 0$ and $f(s) \le K_f s^\alpha$
for all $s \ge 1$, some $K_f \gt 0$ and $\alpha \in \mathbb R$.
  We prove that whenever \begin{align*}
  \alpha \lt \frac2n, \end{align*} solutions to \eqref{prob:star} exist
globally and are bounded. The proof mainly relies on smoothing estimates for
the Neumann heat semigroup and (in the case $\alpha \gt 1$) on a functional
inequality.
  Furthermore, we provide some indication why the exponent $\frac2n$ could be
essentially optimal.
