Final size of an SIR epidemic in heterogeneous networks through local and global propagations

One route of transmission pathway may underestimate the infectious capacity, and each individual in a population could have heterogeneous number of contacts. To address these issues, we consider an SIR epidemic model with both local and global contacts. We mainly investigate the relationship between basic reproduction number and degree correlation coefficient in bimodal degree distribution networks and the case of three degree values in correlated networks. The results reveal that variations in the basic reproduction number depend not only on the degree correlation coefficient in the network but also on the spatial distribution characteristics of the connection pattern. Moreover, our research indicates that as the network structure becomes more complex, the relationship between the basic reproduction number and the degree correlation coefficient exhibits nonlinear and more complex characteristics. For degree uncorrelated networks, we calculate the basic reproduction number and final size. In bimodal networks, when the average degree is held constant, the basic reproduction number and final epidemic size exhibit a negative correlation with increasing degree heterogeneity. Additionally, basing on the edge-based compartmental method, we derive a low-dimensional nonlinear ODE system for degree uncorrelated networks and verify the existence and uniqueness of the final size solution through numerical simulations. Finally, we observe that transmission mechanisms and infection rates differentially affect both the basic reproduction number and final epidemic size.