Designing a contact process: the piecewise-homogeneous process on a finite set with applications

We consider how to choose the reproduction rates in a one-dimensional contact process on a finite set to maximize the growth rate of the extinction time with the population size. The constraints are an upper bound on the average reproduction rate, and that the rate profile must be piecewise constant. We show that the optimum growth rate is achieved by a rate profile with at most two rates, and we characterize the solution in terms of a “spatial correlation length” of the supercritical process. We examine the analogous problem for the simpler biased voter model, for which we completely characterize the optimum profile. The contact process proofs make use of a planar-graph duality in the graphical representation, due to Durrett and Schonmann.