Stabilization in a chemotaxis-May–Nowak model with exposed state

Abstract

This paper deals with the Neumann initial-boundary value problem for the chemotaxis-May–Nowak model with exposed state for virus dynamics

$$\begin{aligned} \left\{ \begin{array}{lll} u_t=\Delta u-\chi \nabla \cdot \left( u\nabla v\right) +\kappa -d_1u-\beta uw,\\ z_t=\Delta z+\beta uw-(d_2+c)z,\\ v_t=\Delta v+cz-d_3v,\\ w_t=\Delta w+kv-d_4w\\ \end{array}\right. \end{aligned}$$

in a smooth bounded domain $\Omega \subset {\mathbb {R}}^n$ ($n\ge 1$). Here $\chi \in {\mathbb {R}}$ and $\kappa$, $\beta$, c, k, $d_i$ $(i=1, 2, 3, 4)$ are positive constants. We prove that for sufficiently smooth initial data the problem possesses a unique global classical solution, which is uniformly bounded with small size of $|\chi |$. Moreover, the large time behavior of the obtained solution is investigated with respect to the basic reproduction number as a threshold.