In this paper we introduce a method of global exponential attractor in the reaction-diffusion epidemic model in spatial heterogeneous environment to study the spread trend and long-term dynamic behavior of the COVID-19 epidemic. First, we prove the existence of the global exponential attractor of general dissipative evolution systems. Then, by using the existence theorem, the global asymptotic stability and the persistence of epidemic are discussed. Finally, combine with the official data of the COVID-19 and the national control strategy, some numerical simulations on the stability and global exponential attractiveness of the COVID-19 epidemic are given. Simulations show that the spread trend of the epidemic is in line with our theoretical results, and the preventive measures taken by the Chinese government are effective.