Probabilistic Procedures for SIR and SIS Epidemic Dynamics on Erdös-Rényi Contact Networks

Abstract

This paper introduces the mathematical formalization of two probabilistic procedures for susceptible-infected-recovered (SIR) and susceptible-infected-susceptible (SIS) infectious diseases epidemic models, over Erdös-Rényi contact networks. In our approach, we consider the epidemic threshold, for both models, defined by the inverse of the spectral radius of the associated adjacency matrices, which expresses the network topology. The epidemic threshold dynamics are analyzed, depending on the global dynamics of the network structure. The main contribution of this work is the relationship established between the epidemic threshold and the topological entropy of the Erdös-Rényi contact networks. In addition, a relationship between the basic reproduction number and the topological entropy is also stated. The trigger of the infectious state is studied, where the probability value of the stability of the infected state after the first instant, depending on the degree of the node in the seed set, is proven. Some numerical studies are included and illustrate the implementation of the probabilistic procedures introduced, complementing the discussion on the choice of the seed set.

Keywords

• infectious disease
• SIR and SIS models
• Erdös-Rényi networks
• epidemic threshold dynamics
• topological entropy