  Predictive contagion models are ubiquitous in epidemiology, social sciences,
engineering, and management. This paper formulates a prescriptive contagion
analytics model where a decision-maker allocates shared resources across
multiple segments of a population, each governed by continuous-time dynamics.
We define four real-world problems under this umbrella: vaccine distribution,
vaccination centers deployment, content promotion, and congestion mitigation.
These problems feature a large-scale mixed-integer non-convex optimization
structure with constraints governed by ordinary differential equations,
combining the challenges of discrete optimization, non-linear optimization, and
continuous-time system dynamics. This paper develops a branch-and-price
methodology for prescriptive contagion analytics based on: (i) a set
partitioning reformulation; (ii) a column generation decomposition; (iii) a
state-clustering algorithm for discrete-decision continuous-state dynamic
programming; and (iv) a tri-partite branching scheme to circumvent
non-linearities. Extensive experiments show that the algorithm scales to very
large and otherwise-intractable instances, outperforming state-of-the-art
benchmarks. Our methodology provides practical benefits in contagion systems;
in particular, it can increase the effectiveness of a vaccination campaign by
an estimated 12-70%, resulting in 7,000 to 12,000 extra saved lives over a
three-month horizon mirroring the COVID-19 pandemic. We provide an open-source
implementation of the methodology in an online repository to enable
replication.
