Modeling the dynamics of epidemic spreading on homogenous and heterogeneous networks

This paper proposes two modified susceptible-infected-recovered-susceptible models on homogenous and heterogeneous networks, respectively. In the study of the homogenous network model, it is proved that if the basic reproduction number of the model is less than one, then the disease-free equilibrium is locally asymptotically stable and becomes globally asymptotically stable under the condition that the threshold value is less than one. Otherwise, if is more than one, the endemic equilibrium is locally asymptotically stable and becomes globally asymptotically stable under the assumption that the total population will tend to a specific plane. In the study of the heterogeneous network model, this paper discusses the existences of the disease-free equilibrium and endemic equilibrium of the model. It is proved that if the threshold value is less than one, then the disease-free equilibrium is globally asymptotically stable. Otherwise, if is more than one, the system is permanent. A series of numerical experiments are given to illustrate the theoretical results. We also numerically predict the effect of vaccination ratio on the size of HBV-infected mainland Chinese population.