Dynamic analysis of a delayed COVID-19 epidemic with home quarantine in temporal-spatial heterogeneous via global exponential attractor method.

As the COVID-19 epidemic has entered the normalization stage, the task of prevention and control remains very arduous. This paper constructs a time delay reaction-diffusion model that is closer to the actual spread of the COVID-19 epidemic, including relapse, time delay, home quarantine and temporal-spatial heterogeneous environment that affect the spread of COVID-19. These factors increase the number of equations and the coupling between equations in the system, making it difficult to apply the methods commonly used to discuss global dynamics, such as the Lyapunov function method. Therefore, we use the global exponential attractor theory in the infinite-dimensional dynamic system to study the spreading trend of the COVID-9 epidemic with relapse, time delay, home quarantine in a temporal-spatial heterogeneous environment. Using our latest results of global exponential attractor theory, the global asymptotic stability and the persistence of the COVID-19 epidemic are discussed. We find that due to the influence of relapse in the in temporal-spatial heterogeneity environment, the principal eigenvalue can describe the spread of the epidemic more accurately than the usual basic reproduction number . That is, the non-constant disease-free equilibrium is globally asymptotically stable when and the COVID-19 epidemic is persisting uniformly when . Combine with the latest official data of the COVID-19 and the prevention and control strategies of different countries, some numerical simulations on the stability and global exponential attractiveness of the spread of the COVID-19 epidemic in China and the USA are given. The simulation results fully reflect the impact of the temporal-spatial heterogeneous environment, relapse, time delay and home quarantine strategies on the spread of the epidemic, revealing the significant differences in epidemic prevention strategies and control effects between the East and the West. The results of this study provide a theoretical basis for the current epidemic prevention and control.