Nonlinear model of epidemic spreading in a complex social network.

The epidemic spreading in a human society is a complex process, which can be described on the basis of a nonlinear mathematical model. In such an approach the complex and hierarchical structure of social network (which has implications for the spreading of pathogens and can be treated as a complex network), can be taken into account. In our model each individual has one of the four permitted states: susceptible, infected, infective, unsusceptible or dead. This refers to the SEIR model used in epidemiology. The state of an individual changes in time, depending on the previous state and the interactions with other individuals. The description of the interpersonal contacts is based on the experimental observations of the social relations in the community. It includes spatial localization of the individuals and hierarchical structure of interpersonal interactions. Numerical simulations were performed for different types of epidemics, giving the progress of a spreading process and typical relationships (e.g. range of epidemic in time, the epidemic curve). The spreading process has a complex and spatially chaotic character. The time dependence of the number of infective individuals shows the nonlinear character of the spreading process. We investigate the influence of the preventive vaccinations on the spreading process. In particular, for a critical value of preventively vaccinated individuals the percolation threshold is observed and the epidemic is suppressed.