Asymptotic stability of disease-free equilibria in a diffusive logistic SIR epidemic model with saturated incidence and treatment rates

This paper deals with the Neumann initial-boundary value problem for the diffusive logistic epidemic model with saturated incidence and treatment rates,$ \begin{align*} \begin{cases} \dfrac{\partial S}{\partial t} = d_S\Delta S-\dfrac{\beta SI}{1+\alpha I}+rS\left(1-\dfrac{S}{K}\right),\quad &x \in \Omega, \ t>0, \\ \dfrac{\partial I}{\partial t} = d_I\Delta I+\dfrac{\beta SI}{1+\alpha I}-\dfrac{\lambda I}{1+\varepsilon I},\quad &x \in \Omega, \ t>0, \end{cases} \end{align*} $where $ \Omega \subset \mathbb{R}^N $ ($ N \in \mathbb{N} $) is a bounded domain with smooth boundary and $ d_S, d_I, r, K, \alpha, \beta, \varepsilon, \lambda >0 $ are constants. The main result of this paper asserts that the disease-free equilibrium $ (K,0) $ is (globally) asymptotically stable under some conditions for $ K, \alpha, \beta, \varepsilon $ and $ \lambda $, and initial data if $ \mathcal{R}_0: = \frac{K\beta}{\lambda}\le1 $.