The Susceptible-Infected-Susceptible model and its approximations
An undirected graph G(N,L) with N nodes and L links. The links are represented in a symmetric adjacency matrix A, in the matrix, if there is a link between nodes i and j, then aij =aji =1, if not, aij = 0. The state of a node i is specified by a Bernoulli random variable Xi ∈ {0, 1}: Xi = 0 for a uninfected node and Xi =1 for an infected node. A node can be in one of the two states: infected, with probability v (t) = Pr[X (t) = 1] or uninfected, with probability 1 − v (t).
There are several approximations of SIS epidemics:
The N-intertwined approximation and the Pastor-Satorras and Vespignani HMF approximation
the Pastor-Satorras and Vespignani HMF approximation:
“Pastor-Satorras and Vespignani studied the susceptible-infected-susceptible epidemic on networks and proposed the heterogeneous mean-field approximation, in which the degree distribution plays an important role. Highly connected nodes are statistically significant and the strong fluctuations in the degree distribution cannot be neglected.”(Susceptible-infected-susceptible model: A comparison of N-intertwined and heterogeneous mean-field approximations, Cong Li, Ruud van de Bovenkamp, and Piet Van Mieghem)
The N-intertwined epidemic approximation:
“The N-intertwined epidemic approximation is derived by separately observing each node. Every node i at time t in the network is in one of two states: infected, with probability Prob[Xi (t ) = 1], and healthy, with probability Prob[Xi (t ) = 0]. Since a node can only be in one of two states, Prob[Xi(t) = 0] + Prob[Xi(t) = 1] = 1. Since the curing and infection processes are Poisson processes, the whole epidemic process is a Markov process.” (Susceptible-infected-susceptible model: A comparison of N-intertwined and heterogeneous mean-field approximations, Cong Li, Ruud van de Bovenkamp, and Piet Van Mieghem)
Citation:
“Susceptible-infected-susceptible model: A comparison of N-intertwined and heterogeneous mean-field approximations”, Cong Li, Ruud van de Bovenkamp, and Piet Van Mieghem
“Epidemic phase transition of the SIS type in networks”, P. Van Mieghem
“Effect of the interconnected network structure on the epidemic threshold”, Huijuan Wang, Qian Li, Gregorio D’Agostino, Shlomo Havlin, H. Eugene Stanley, and Piet Van Mieghem
“The N-intertwined SIS epidemic network model” Piet Van Mieghem