Summary: The chemotaxis May-Nowak model \[ \begin{cases} u_t=D_u \Delta u-\nabla \cdot (uf\left( u\right) \nabla v)-g\left( u\right) w+r-u, & x\in \Omega,\quad t>0,\\ v_t=D_v \Delta v+g\left( u\right) w-v, & x\in \Omega,\quad t>0,\\ w_t=D_w \Delta w+v-w, & x\in \Omega ,\quad t>0, \end{cases} \] is considered in a bounded domain \(\Omega \subset \mathbb{R}^n\) (\(n\ge 1\)) with homogeneous Neumann boundary conditions and the parameters \(D_u,D_v,D_w,r>0\). The chemotactic sensitivity function and the conversion function are given by \(f\left( s\right) =K_f\left( 1+s\right)^{-\alpha}\) and \(g(s) = K_g s^{\beta}\) for all \(s > 0\), respectively, where \(K_f\in \mathbb{R}\), \(K_g, \alpha, \beta >0\). The global boundedness of solution is shown if the following case holds: \[ \alpha > \max \left\{ \frac{n\beta}{4}, \frac{\beta}{2}, \frac{n(n+2)}{6n+8}\beta +\frac{1}{2} \right\}. \] Moreover, for the large time behavior of the global smooth bounded solution, the basic reproduction number \(R_0:=K_gr^{\beta}\) has an important effect [X. Lai and X. Zou, Bull. Math. Biol. 76, No. 11, 2806–2833 (2014; Zbl 1329.92074); W. Wang et al., Nonlinear Anal., Real World Appl. 33, 253–283 (2017; Zbl 1352.92094)], and system has the infection-free steady state if \(0<R_0<1\) and the coexistence equilibrium steady state if \(R_0>1\) [A. Korobeinikov, Bull. Math. Biol. 66, No. 4, 879–883 (2004; Zbl 1334.92409)]. By constructing an appropriate energy function, under the conditions that \(K_f\) and \(K_g\) are appropriately mild, it is shown that:

●

If \(R_0\in (0,1)\), then any global bounded solution converges to \(\left( r, 0, 0\right)\) as \(t\rightarrow \infty \);

●

If \(R_0\in (1,\infty ), \beta =1\), then any global bounded solution converges to \(\bigg ( \frac{1}{K_g}, r-\frac{1}{K_g}, r-\frac{1}{K_g} \bigg )\) as \(t\rightarrow \infty \).