Analytical solution for the spread of epidemic diseases in community clustered network

Abstract

We present a bond percolation model for community clustered networks with an arbitrarily specified joint degree distribution. Our model is based on the Probability Generating Function (PGF) method for multitype networks, but incorporate the free-excess degree distribution, which makes it applicable for clustered networks. In the context of contact network epidemiology, our model serves as a special case of community clustered networks which are more appropriate for modelling the disease transmission in community networks with clustering effects. Beyond the percolation threshold, we are able to obtain the probability that a randomly chosen community-$i$ node leads to the giant component. The probability refers to the probability that an individual in a community will be affected from the infective disease. Besides that, we also establish method to calculate the size of the giant component and the average small-component size (excluding the giant component). When the clustering effect is taken into account through the free-excess degree distribution, the model shows that the clustering effect will decrease the size of the giant component. In short, our model enables one to carry out numerical calculations to simulate the disease transmission in community networks with different community structure effects and clustering effects.