Abstract
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.
Highlights
The interest in understanding and modeling epidemics spreading in humans has surged conspicuously since the outbreak of the COVID-19 phenomenon
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
We have presented a general network model of a population built upon random finite clusters, representing highly correlated individuals belonging to a social group or community
Summary
The interest in understanding and modeling epidemics spreading in humans has surged conspicuously since the outbreak of the COVID-19 phenomenon (see e.g., [1,2,3,4,5,6,7,8,9]). The clusters are constructed or designed in such a way that their internal spreading dynamics can be solved ‘exactly’, either numerically or analytically, and their contributions to the total infections in the network can be added sequentially say, such that one can mimic the time evolution of the infectious disease in a real situation. We solve the spreading dynamics on these minor clusters, in order to introduce the SIR categories in the problem and discuss how to implement the healing times for infected individuals.
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