We discuss two recent results concerning disease modeling on networks. The
infection is assumed to spread via contagion (e.g., transmission over the edges of
an underlying network). In the first scenario, we observe the infection status of
individuals at a particular time instance and the goal is to identify a confidence
set of nodes that contain the source of the infection with high probability. We
show that when the underlying graph is a tree with certain regularity properties
and the structure of the graph is known, confidence sets may be constructed with
cardinality independent of the size of the infection set. In the scenario, the goal is
to infer the network structure of the underlying graph based on knowledge of the
infected individuals. We develop a hypothesis test based on permutation testing,
and describe a sufficient condition for the validity of the hypothesis test based on
automorphism groups of the graphs involved in the hypothesis test.
This is joint work with Justin Khim (UPenn).