# Disease invasion on community networks with environmental pathogen movement

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## Abstract

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.

## Keywords

Cholera Waterborne disease Basic reproduction number Spanning trees Group inverse## Mathematics Subject Classification

05C20 15A09 92D30## Notes

### Acknowledgments

JHT and MCE acknowledge support from the National Science Foundation (OCE-1115881) and the Mathematical Biosciences Institute (DMS-0931642). The research of PvdD is partially supported through a Discovery Grant from the Natural Science and Engineering Research Council of Canada (NSERC). ZS acknowledges support from the University of Central Florida through a start-up grant. The authors are grateful to the anonymous reviewers for their thoughtful, constructive comments. This paper was improved by discussions at a Research in Teams meeting (13rit168) held at the Banff International Research Station.

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