In this paper, we study a Susceptible–Vaccinated–Infected–Recovered (SVIR) epidemic model in a spatially heterogeneous environment under the Dirichlet boundary condition. We define the basic reproduction number ${\Re }_0$ by the spectral radius of the next generation operator, and show that it is a threshold parameter. The disease extinction and persistence in the case of a bounded domain are considered. More precisely, we show that the disease-free equilibrium is globally asymptotically stable if ${\Re }_0<1$; the system is uniformly persistent and an endemic equilibrium exists if ${\Re }_0>1$. To verify our theoretical results, we perform some numerical simulations, using the Fredholm discretization method to identify ${\Re }_0$.

在本文中，我们研究了狄利克雷边界条件下空间异质环境中的易感-接种-感染-恢复（SVIR）流行病模型。我们定义基本再生数${\Re }_0$由下一代算子的谱半径，并表明它是一个阈值参数。考虑了有界域情况下的疾病灭绝和持续性。更准确地说，我们证明了无病平衡是全局渐近稳定的，如果${\Re }_0<1$; 该系统是一致持久的并且存在地方性平衡，如果${\Re }_0>1$. 为了验证我们的理论结果，我们进行了一些数值模拟，使用 Fredholm 离散化方法来识别${\Re }_0$.