In industrialized nations, potable water is often provided through sophisticated water distribution systems. If pathogenic bacteria are introduced into the water distribution network, the presence of a biofilm can lead to biofilm-assisted retention of the pathogens, affecting the potability of the water. To study the dynamics of planktonic and biofilm-bound pathogens within the large network of pipes in a water distribution system, we develop a network model governing the concentration of introduced pathogens within the bulk fluid and the biofilms lining the pipes. Under time-constant flow regimes within the network, we prove that the long-time behaviour of the entire network is dependent on the Lyapunov exponents for each connection in the network when viewed in isolation and the network connectivity. An efficient algorithm is developed for predicting the long-time behaviour of the pathogen population within large networks using the network's topological ordering. The algorithm's predictions are validated using numerical simulations of the full non-linear system on a range of water distribution network sizes.