Global boundedness of a density-dependent SIRS epidemic model with nonlinear infection mechanism

In this paper, we study the following density-dependent SIRS epidemic model with power-like infection mechanism$ \begin{align*} \left\{ \begin{aligned} &u_t = \Delta(d_1(v)u)-\beta(x) u^qv^p+\gamma w, \quad &&x\in\Omega, t>0, \\ &v_t = d_2\Delta v+\beta(x) u^qv^p-(\delta+\alpha)v, \quad &&x\in\Omega, t>0, \\ &w_t = \Delta(d_3(v)w)+\delta v-\gamma w, \quad &&x\in\Omega, t>0, \end{aligned} \right. \end{align*} $in a bounded smooth domain $ \Omega\subset \mathbb{R}^n (n\geq2) $. Here $ p $, $ q $, $ \gamma $, $ \delta $, $ \alpha>0 $ are constants. We prove that the problem possesses a global classical solution in $ n $ dimension $ (n\geq2) $ when $ q<\frac{2}{n} $, $ p\leq1 $. Moreover, we show that the above solution will converge to the steady state $ (u^*, 0, 0) $, where $ u^* = \vert\Omega\vert^{-1}\left(\int_{\Omega}(u_0+v_0+w_0)-\alpha\int_{0}^{\infty}\int_{\Omega} v\right) $.