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Designing an Interacting Particle System : Optimization of the Piecewise-Homogeneous Contact Process

We consider an optimization problem centered around a population-growth model similar to the onedimensional, finite contact process, but in which the reproduction rate varies spatially over the population. The objective is to choose the reproduction-rate-profile to maximize the time until the population is extinct. We first extend existing results about the contact process that describe the asymptotic growth of the extinction time with the population size to processes in which the reproduction rate is not constant but rather piecewise constant. We then formulate the optimization problem and characterize its solution in terms of statistics of the original contact process. We also examine the analogous problem for the simpler biased voter model, and obtain similar but more lucid results.