Boundedness of solutions to a virus infection model with saturated chemotaxis

We show global existence and boundedness of classical solutions to a virus infection model with chemotaxis in bounded smooth domains of arbitrary dimension and for any sufficiently regular nonnegative initial data and homogeneous Neumann boundary conditions. More precisely, the system considered is $$\begin{cases}\begin{split} & u_t=\Delta u - \nabla\cdot(\frac{u}{(1+u)^{\alpha}}\nabla v) - uw + \kappa - u, \\ & v_t=\Delta v + uw - v, \\ & w_t=\Delta w - w + v, \end{split}\end{cases}$$ with $\kappa\ge 0$, and solvability and boundedness of the solution are shown under the condition that $$\begin{cases} \alpha > \frac 12 + \frac{n^2}{6n+4}, &\text{if } \quad 1 \leq n \leq 4 \\ \alpha > \frac {n}4, &\text{if } \quad n \geq 5. \end{cases}$$