  The field of epidemiology has presented fascinating and relevant questions
for mathematicians, primarily concerning the spread of viruses in a community.
The importance of this research has greatly increased over time as its
applications have expanded to also include studies of electronic and social
networks and the spread of information and ideas. We study virus propagation on
a non-linear hub and spoke graph (which models well many airline networks). We
determine the long-term behavior as a function of the cure and infection rates,
as well as the number of spokes n. For each n we prove the existence of a
critical threshold relating the two rates. Below this threshold, the virus
always dies out; above this threshold, all non-trivial initial conditions
iterate to a unique non-trivial steady state. We end with some generalizations
to other networks.
