This paper deals with the Neumann initial-boundary value problem for the chemotaxis-SIS epidemic model with nonlinear incidence rate \(\beta (x)u^pv^q\ (p, q>0)\) \(\begin{aligned} \left\{ \begin{array}{ll} u_t=d_1\Delta u+\chi \nabla \cdot (u\nabla v)-\beta (x)u^pv^q+\gamma (x)v,~~\ \ \ & x \in \Omega ,\ t>0,\\[2ex] v_t=d_2\Delta v+\beta (x)u^pv^q-\gamma (x)v,~~\ \ \ & x \in \Omega ,\ t>0 \end{array} \right. \end{aligned}\) in a smooth bounded domain \(\Omega \subset \mathbb {R}^n\ (n \ge 1)\) , where \(d_1> 0,\ d_2 > 0\) , \(\chi \in \mathbb {R}\) , \(0 < \beta \in C^1(\overline{\Omega }),\) and \( 0<\gamma \in C^1(\overline{\Omega })\) . We prove that the problem admits a unique global classical solution if \( (u_0, v_0) \in L^{\infty }(\Omega ) \times L^1(\Omega )\) is suitably small and \(\Vert v_0\Vert _{L^{\infty }(\Omega )} + \Vert \nabla v_0\Vert _{L^{m}(\Omega )}\le K\) for each \(K>0\) and \(m>n\) . Moreover, we establish the large time behavior of small-data solutions.
