The paper by M. J. Keeling explores the impact of local spatial structure on the success of epidemiological invasions. It emphasizes that the early spatial correlations developed during the invasion process significantly influence the invasion dynamics. The author models these correlations between individuals to understand the role of spatial heterogeneity without the need for large-scale simulations. The study focuses on the spread of infectious diseases through a structured network, determining invasion thresholds and statistical properties of a single epidemic. Key findings include: 1. **Spatial Correlations and Intraspecific Competition**: The limited spread of invading organisms leads to greater intraspecific competition compared to homogeneously mixed models. 2. **Basic Reproductive Ratio ($R_0$)**: The basic reproductive ratio is lower in network models than in mean-field models, indicating reduced invasion potential. 3. **Final Size of Epidemics**: The final size of an epidemic is also reduced in network models, with the effect being more pronounced as the network becomes more interconnected. 4. **Vaccination**: The vaccination threshold, which depends on the network structure and the aggregation of susceptibles, increases with the number of neighbors and decreases with increasing interconnectedness. The paper provides a theoretical framework for understanding how spatial structure affects the dynamics of infectious diseases and offers insights into the role of spatial correlations in ecological invasions.The paper by M. J. Keeling explores the impact of local spatial structure on the success of epidemiological invasions. It emphasizes that the early spatial correlations developed during the invasion process significantly influence the invasion dynamics. The author models these correlations between individuals to understand the role of spatial heterogeneity without the need for large-scale simulations. The study focuses on the spread of infectious diseases through a structured network, determining invasion thresholds and statistical properties of a single epidemic. Key findings include: 1. **Spatial Correlations and Intraspecific Competition**: The limited spread of invading organisms leads to greater intraspecific competition compared to homogeneously mixed models. 2. **Basic Reproductive Ratio ($R_0$)**: The basic reproductive ratio is lower in network models than in mean-field models, indicating reduced invasion potential. 3. **Final Size of Epidemics**: The final size of an epidemic is also reduced in network models, with the effect being more pronounced as the network becomes more interconnected. 4. **Vaccination**: The vaccination threshold, which depends on the network structure and the aggregation of susceptibles, increases with the number of neighbors and decreases with increasing interconnectedness. The paper provides a theoretical framework for understanding how spatial structure affects the dynamics of infectious diseases and offers insights into the role of spatial correlations in ecological invasions.