Newman presents a study on the spread of epidemic disease on networks, focusing on the SIR (Susceptible-Infectious-Removed) model. The paper shows that a wide range of standard epidemiological models can be solved exactly on various network structures. It addresses both uniform and non-uniform transmission probabilities, as well as structured populations, such as sexually transmitted diseases in a bipartite network of men and women. The study confirms the accuracy of exact solutions through numerical simulations. The paper introduces the SIR model, where individuals are divided into three states: susceptible, infectious, and removed. The model is extended to account for varying transmission probabilities and times, and it is shown that the overall transmissibility T, which is the average probability of transmission, determines the spread of the disease. The model is equivalent to a bond percolation model, where the disease spreads through occupied edges in a network. The paper then presents exact solutions for the size distribution of outbreaks on networks with arbitrary degree distributions. It uses generating functions to derive these solutions, showing that the average outbreak size and the presence of an epidemic depend on the transmissibility T and the network structure. The epidemic transition occurs when T reaches a critical value T_c, which depends on the network's degree distribution. The study also considers structured populations, such as bipartite networks, where individuals are divided into two groups (e.g., men and women). The model is extended to account for different transmission probabilities between the two groups, and it is shown that the epidemic transition depends on the product of the transmissibilities in each direction. The paper concludes that the spread of disease on networks is influenced by network topology, and that the critical transmissibility T_c for an epidemic to occur depends on the network's structure. The results are validated through numerical simulations and show that the model accurately predicts the spread of disease on networks with varying degrees of connectivity.Newman presents a study on the spread of epidemic disease on networks, focusing on the SIR (Susceptible-Infectious-Removed) model. The paper shows that a wide range of standard epidemiological models can be solved exactly on various network structures. It addresses both uniform and non-uniform transmission probabilities, as well as structured populations, such as sexually transmitted diseases in a bipartite network of men and women. The study confirms the accuracy of exact solutions through numerical simulations. The paper introduces the SIR model, where individuals are divided into three states: susceptible, infectious, and removed. The model is extended to account for varying transmission probabilities and times, and it is shown that the overall transmissibility T, which is the average probability of transmission, determines the spread of the disease. The model is equivalent to a bond percolation model, where the disease spreads through occupied edges in a network. The paper then presents exact solutions for the size distribution of outbreaks on networks with arbitrary degree distributions. It uses generating functions to derive these solutions, showing that the average outbreak size and the presence of an epidemic depend on the transmissibility T and the network structure. The epidemic transition occurs when T reaches a critical value T_c, which depends on the network's degree distribution. The study also considers structured populations, such as bipartite networks, where individuals are divided into two groups (e.g., men and women). The model is extended to account for different transmission probabilities between the two groups, and it is shown that the epidemic transition depends on the product of the transmissibilities in each direction. The paper concludes that the spread of disease on networks is influenced by network topology, and that the critical transmissibility T_c for an epidemic to occur depends on the network's structure. The results are validated through numerical simulations and show that the model accurately predicts the spread of disease on networks with varying degrees of connectivity.