Influence of spatial heterogeneous environment on long-term dynamics of a reaction-diffusion SVIR epidemic model with relaps

In this paper by adding the factors of disease relapse and vaccination in the space hetero-geneous environment, we establish and discuss a class of reaction-diffusion SVIR model with relapse and a varying external source in spatial heterogeneous environment. By applying a different method than the Lyapunov function, we study the long-term dynamic behavior of this model by means of global exponential attractor theory and gradient flow method. The global asymptotic stability and the persistence of epidemic are proved. To test the validity of our theoretical results, we choose some specific epidemic disease with some more practical and more definitive official data to simulate the global stability and exponential attraction of the model. The simulation results showed that the factors of disease relapse, vaccination and spatial heterogeneity had a great influence on the persists uniformly of the disease.