In this paper we explore the potential of the pairwise-type modelling approach to be
extended to weighted networks where nodal degree and weights are not independent. As a
baseline or null model for weighted networks, we consider undirected, heterogenous
networks where edge weights are randomly distributed. We show that the pairwise model
successfully captures the extra complexity of the network, but does this at the cost of
limited analytical tractability due the high number of equations. To circumvent this
problem, we employ the edge-based modelling approach to derive models corresponding to two
different cases, namely for degree-dependent and randomly distributed weights. These
models are more amenable to compute important epidemic descriptors, such as early growth
rate and final epidemic size, and produce similarly excellent agreement with simulation.
Using a branching process approach we compute the basic reproductive ratio for both models
and discuss the implication of random and correlated weight distributions on this as well
as on the time evolution and final outcome of epidemics. Finally, we illustrate that the
two seemingly different modelling approaches, pairwise and edge-based, operate on similar
assumptions and it is possible to formally link the two.