Spreading of Infections on Network Models: Percolation Clusters and Random Trees

We discuss network models as a general and suitable framework for describing the spreading of an infectious disease within a population. We discuss two types of finite random structures as building blocks of the network, one based on percolation concepts and the second one on random tree structures. We study, as is done for the SIR model, the time evolution of the number of susceptible (S), infected (I) and recovered (R) individuals, in the presence of a spreading infectious disease, by incorporating a healing mechanism for infecteds. In addition, we discuss in detail the implementation of lockdowns and how to simulate them. For percolation clusters, we present numerical results based on site percolation on a square lattice, while for random trees we derive new analytical results, which are illustrated in detail with a few examples. It is argued that such hierarchical networks can complement the well-known SIR model in most circumstances. We illustrate these ideas by revisiting USA COVID-19 data.

Automated summary

The article discusses network models as a framework for describing the spread of infectious diseases, using two types of random structures as building blocks. It analyzes the time evolution of susceptible, infected, and recovered individuals during the spread of an infectious disease, and discusses the implementation of lockdowns and simulation methods. The article also presents numerical and analytical results for percolation clusters and random trees, concluding that hierarchical networks can complement the SIR model in most circumstances.