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The ability of disease to invade a community network that is connected by environmental pathogen movement is examined. Each community is modeled by a susceptible–infectious–recovered (SIR) framework that includes an environmental pathogen reservoir, and the communities are connected by pathogen movement on a strongly connected, weighted, directed graph. Disease invasibility is determined by the basic reproduction number $$\mathcal{{R}}_0$$ for the domain. The domain $$\mathcal{{R}}_0$$ is computed through a Laurent series expansion, with perturbation parameter corresponding to the ratio of the pathogen decay rate to the rate of water movement. When movement is fast relative to decay, $$\mathcal{{R}}_0$$ is determined by the product of two weighted averages of the community characteristics. The weights in these averages correspond to the network structure through the rooted spanning trees of the weighted, directed graph. Clustering of disease “hot spots” influences disease invasibility. In particular, clustering hot spots together according to a generalization of the group inverse of the Laplacian matrix facilitates disease invasion.

A mathematical model for cholera is formulated that incorporates hyperinfectivity and temporary immunity using distributed delays. The basic reproduction number \(\mathcal{R}_{0}\) is defined and proved to give a sharp threshold that determines whether or not the disease dies out. The case of constant temporary immunity is further considered with two different infectivity kernels. Numerical simulations are carried out to show that when \(\mathcal{R}_{0}>1\), the unique endemic equilibrium can lose its stability and oscillations occur. Using cholera data from the literature, the quantitative effects of hyperinfectivity and temporary immunity on oscillations are investigated numerically.