SIR dynamics in random networks with communities

This paper investigates the effects of the community structure of a network on the spread of an epidemic. To this end, we first establish a susceptible-infected-recovered (SIR) model in a two-community network with an arbitrary joint degree distribution. The network is formulated as a probability generating function. We also obtain the sufficient conditions for disease outbreak and extinction, which involve the first-order and second-order moments of the degree distribution. As an example, we then study the effect of community structure on epidemic spread in a complex network with a Poisson joint degree distribution. The numerical solutions of the SIR model well agree with stochastic simulations based on the Monte Carlo method, confirming that the model is reliable and accurate. Finally, by strengthening the community structure in the simulation, i.e. fixing the total degree distribution and reducing the number ratio of the external edges, we can increase or decrease the final cumulative epidemic incidence depending on the transmissibility of the virus between humans and the community structure at that point. Why community structure can affect disease dynamics in a complicated way is also discussed. In any case, for large-scale epidemics, strengthening the community structure to reduce the size of disease is undoubtedly an effective way.