In the talk we present a simple extension of the configuration model to weighted networks, and state some asymptotic properties of the network model. The weights may be used for some stochastic process taking place on the network; for example an epidemic where the probability of transmission between two individuals depends on the weight of the connected edge (the weight for example reflecting social distance). We also consider the case where individuals (nodes) are heterogeneous in he sense that the transmission probability depends on the infectivity of the infector and the susceptibility of the victim. We end with analysis of some empirical networks: movement of patients in hospitals in Stockholm (for mrsa), workplaces and households of the Swedish population (for influenza), and census of sex-contacts (for STDs).