Dynamical Properties Of SIR And SEIR Models On Scale-Free Networks

This dissertation analyzes the dynamical properties of two class of epidemic models on scale-free networks, with emphasis on the SIR and SEIR models with piecewise linear infectivity and various immunization strategies with time delays. We obtain the epidemic threshold for SIR model with piecewise linear infectivity, and get the conditions for positive threshold. Then we analyze various SIR models with immunization strategies and calculate the corresponding thresholds, meanwhile, we do some numerical simulations and comparisons, and find that target immunization is more effective than proportional, acquaintance and active immunizations under the same immunization probability. While any kind of immunization strategy has certain effect on disease control.Then we study a kind of model with exposed period. Based on known results about SEIR model, we put forward a new epidemic model with time delays during immunization by introducing the birth and death rates, natural mortality, and immunization rate. We obtain the besic reproduction number of this model. Furthermore, by constructing Lyapunov function and applying LaSalle invariance principle, we prove that when the basic regeneration number is not greater than 1 the disease-free equilibrium is globally stable, hence the disease will eventually disappear; when the basic regeneration number is greater than 1 there exists a unique endemic equilibrium; and under certain conditions concerning delay, immunization rate, and natural mortality, the endemic equilibrium is locally asymptotically stable.