Final epidemic size of a two-community SIR model with asymmetric coupling

Communities are commonly not isolated but interact asymmetrically with each other, allowing the propagation of infectious diseases within the same community and between different communities. To reveal the impact of asymmetrical interactions and contact heterogeneity on disease transmission, we formulate a two-community SIR epidemic model, in which each community has its contact structure while communication between communities occurs through temporary commuters. We derive an explicit formula for the basic reproduction number ℛ_0 , give an implicit equation for the final epidemic size z, and analyze the relationship between them. Unlike the typical positive correlation between ℛ_0 and z in the classic SIR model, we find a negatively correlated relationship between counterparts of our model deviating from homogeneous populations. Moreover, we investigate the impact of asymmetric coupling mechanisms on ℛ_0 . The results suggest that, in scenarios with restricted movement of susceptible individuals within a community, ℛ_0 does not follow a simple monotonous relationship, indicating that an unbending decrease in the movement of susceptible individuals may increase ℛ_0 . We further demonstrate that network contacts within communities have a greater effect on ℛ_0 than casual contacts between communities. Finally, we develop an epidemic model without restriction on the movement of susceptible individuals, and the numerical simulations suggest that the increase in human flow between communities leads to a larger ℛ_0 .