The epidemic model based on the approximation for third-order motifs on networks.

The spread of an infectious disease may depend on the structure of the network. To study the influence of the structure parameters of the network on the spread of the epidemic, we need to put these parameters into the epidemic model. The method of moment closure introduces structure parameters into the epidemic model. In this paper, we present a new moment closure epidemic model based on the approximation of third-order motifs in networks. The order of a motif defined in this paper is determined by the number of the edges in the motif, rather than by the number of nodes in the motif as defined in the literature. We provide a general approach to deriving a set of ordinary differential equations that describes, to a high degree of accuracy, the spread of an infectious disease. Using this method, we establish a susceptible-infected-recovered (SIR) model. We then calculate the basic reproduction number of the SIR model, and find that it decreases as the clustering coefficient increases. Finally, we perform some simulations using the proposed model to study the influence of the clustering coefficient on the final epidemic size, the maximum number of infected, and the peak time of the disease. The numerical simulations based on the SIR model in this paper fit the stochastic simulations based on the Monte Carlo method well at different levels of clustering. Our results show that the clustering coefficient poses impediments to the spread of disease under an SIR model.