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In this paper, a new spatio-temporal model is formulated to study the spread of coronavirus infection (COVID-19) in a spatially heterogeneous environment with the impact of vaccination. Initially, a detailed qualitative analysis of the spatio-temporal model is presented. The existence, uniqueness, positivity, and boundedness of the model solution are investigated. Local asymptotical stability of the diffusive COVID-19 model at steady state is carried out using well-known criteria. Moreover, a suitable nonlinear Lyapunov functional is constructed for the global asymptotical stability of the spatio-temporal model. Further, the model is solved numerically based on uniform and non-uniform initial conditions. Two different numerical schemes named: finite difference operator-splitting and mesh-free operator-splitting based on multi-quadratic radial basis functions are implemented in the numerical study. The impact of diffusion as well as some pharmaceutical and non-pharmaceutical control measures, i.e., reducing an effective contact causing infection transmission, vaccination rate and vaccine waning rate on the disease dynamics is presented in a spatially heterogeneous environment. Furthermore, the impact of the aforementioned interventions is investigated with and without diffusion on the incidence of disease. The simulation results conclude that the random motion of individuals has a significant impact on the disease dynamics and helps in setting a better control strategy for disease eradication. [ABSTRACT FROM AUTHOR]

This paper investigates the role of large outbreaks on the persistence of Covid-19 over time. Using data from 650 European regions in 14 countries, I first show that winter school holidays in late February/early March 2020 (weeks 8, 9 and 10) led to large regional outbreaks of Covid-19 in the spring with the spread being 60% and up-to over 90% higher compared to regions with earlier school holidays. While the impact of these initial large outbreaks fades away over the summer months, it systematically reappears from the fall as regions with school holidays in weeks 8, 9 and 10 had 30–70% higher spread. This suggests that following a large outbreak, there is a strong element of underlying (latent) regional persistence of Covid-19. The strong degree of persistence highlights the long-term benefits of effective (initial) containment policies, as once a large outbreak has occurred, Covid-19 persists. This result emphasizes the need for vaccinations against Covid-19 in regions that have recently experienced large outbreaks but are well below herd-immunity, to avoid a new surge of cases. [ABSTRACT FROM AUTHOR]