Epidemic Model-Based Benchmark for Optimal Control on Networks

Network dynamical systems add an additional challenge of scale to optimal control schemes. There are many options of overcoming it, such as approximations and heuristics based on mean field games, neural networks, or reinforcement learning, or the actual structure of the networks, each with its own advantages and tradeoffs.

Metapopulation epidemic models, where each population is an entity on a map, such as a city or a district, are a convenient option for benchmarking varying optimal control schemes: these can be designed with varying number of nodes (dimension), have a natural per-node optimal control, e.g. the “lockdown level,” and a straightforward visualization option of choropleth maps.

In this talk, we will describe a procedure for generating plausible instances of such models with from 1 to circa 64,000 nodes based on publicly available census data for the contiguous U.S., each with the network of short-range travel (commute) and long-range travel (airplane), the latter derived from publicly available passenger flight statistics---along with a formal aggregation routine enabling a view of the same geography at different resolutions.

As a showcase, we designed a “baseline” optimal control scheme for three instances covering Oregon and Washington states: a 2-node instance on state level, a 75-node on county level, and a 2,072-node instance made of “atomic” population units, the census tracts, which are put through a metapopulation SIR model with per-node “lockdown level” optimal control on a 180-day time horizon, with the objective of minimizing the cumulative number of infections and the square of this lockdown control; the results are compared with the “no-lockdown” model.
The optimal control was derived through the Pontryagin Maximum Principle and numerically computed by the forward-backward sweep method, which converges within 5 seconds on the 2- and 75-node instances and within 40 seconds on the 2,072-node one.