This paper discusses the dynamics of the epidemic spreading susceptible-infected-recovery (SIR) model on the stationary growth networks, relating them to the node-connectivity distribution that characterizes the network. We introduce the interaction Markov chains mean-field equations and the stochastic numerical approach to examine the threshold (steady state) and time-independent behaviour for the epidemic model on such network. Analytical methods and simulated experiments show there exhibits a critical threshold for the infinite size networks with the exponent less than or equal to 3 below which it cannot diffuse in such type of the system. For the BA networks, we present analytical and Monte Carlo calculations and compare the results with those obtained by the numerical method, which indicates stochastic numerical approach (SNA) can save memory and get the fast exploration.
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