The paper explores how local spatial structure influences the success of epidemiological invasions. It emphasizes the role of spatial correlations in determining invasion dynamics, particularly in disease spread through structured networks. By modeling these correlations, researchers can predict invasion thresholds and statistical properties of epidemics without large-scale simulations. The study focuses on the SIR model, which describes the spread of infectious diseases through a network of individuals. Key concepts include the basic reproductive ratio $ R_0 $, which determines whether an epidemic can invade a population, and the final size of an epidemic, which reflects the total number of individuals infected. The paper introduces a method to model spatial correlations using pair-wise equations, allowing for the analysis of network structures. It shows that the structure of a network, characterized by the average number of neighbors $ n $ and the interconnectedness $ \phi $, significantly affects the spread of diseases. When $ \phi $ is high, indicating a more interconnected network, the basic reproductive ratio $ R_0 $ is lower, reducing the likelihood of an epidemic. Conversely, when $ \phi $ is low, the network is less interconnected, and $ R_0 $ is higher. The study also examines how vaccination affects these dynamics. Vaccination reduces the number of susceptible individuals, thereby lowering the basic reproductive ratio and increasing the threshold for invasion. The paper demonstrates that the final size of an epidemic is influenced by both the network structure and the initial distribution of susceptible individuals. It shows that in highly interconnected networks, the final size of an epidemic is smaller, and the invasion threshold is higher. The paper concludes that spatial structure plays a crucial role in determining the success or failure of epidemics. By incorporating spatial correlations into models, researchers can better predict the spread of diseases and the effectiveness of interventions like vaccination. The study highlights the importance of considering network structure in epidemiological models and provides a framework for understanding how spatial heterogeneity influences disease dynamics.The paper explores how local spatial structure influences the success of epidemiological invasions. It emphasizes the role of spatial correlations in determining invasion dynamics, particularly in disease spread through structured networks. By modeling these correlations, researchers can predict invasion thresholds and statistical properties of epidemics without large-scale simulations. The study focuses on the SIR model, which describes the spread of infectious diseases through a network of individuals. Key concepts include the basic reproductive ratio $ R_0 $, which determines whether an epidemic can invade a population, and the final size of an epidemic, which reflects the total number of individuals infected. The paper introduces a method to model spatial correlations using pair-wise equations, allowing for the analysis of network structures. It shows that the structure of a network, characterized by the average number of neighbors $ n $ and the interconnectedness $ \phi $, significantly affects the spread of diseases. When $ \phi $ is high, indicating a more interconnected network, the basic reproductive ratio $ R_0 $ is lower, reducing the likelihood of an epidemic. Conversely, when $ \phi $ is low, the network is less interconnected, and $ R_0 $ is higher. The study also examines how vaccination affects these dynamics. Vaccination reduces the number of susceptible individuals, thereby lowering the basic reproductive ratio and increasing the threshold for invasion. The paper demonstrates that the final size of an epidemic is influenced by both the network structure and the initial distribution of susceptible individuals. It shows that in highly interconnected networks, the final size of an epidemic is smaller, and the invasion threshold is higher. The paper concludes that spatial structure plays a crucial role in determining the success or failure of epidemics. By incorporating spatial correlations into models, researchers can better predict the spread of diseases and the effectiveness of interventions like vaccination. The study highlights the importance of considering network structure in epidemiological models and provides a framework for understanding how spatial heterogeneity influences disease dynamics.