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

Abstract

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. r 2004 Elsevier B.V. All rights reserved. MSC: primary 60K35; secondary 60K30; 82C26; 92D25; 92D30

How to distribute antidote to control epidemics

“…A slightly different aspect on the spread of viruses was studied by Wagner and Anantharam [26]. They raised the question of whether an inhomogeneous spread rate β xy for the different edges {x, y} of the underlying graph might be more efficient for the virus.…”

On traversable length inside semi-cylinder in 2d supercritical bond percolation

“…For every interval I N,i , the birth rates for all vertices in I N,i ∩ Z are assumed to be equal to λ i . One of results in [10] affirms that if all λ i 's are larger than the critical point λ c of the (original) 1-dimensional contact process,…”

“…On the basis of the argument in [4], Wagner and Anantharam [10] studied the extinction time σ CP N for the 1-dimensional contact process with piecewise homogeneous birth rates and an identical death rate on a finite interval. The precise definition is as follows: Let the death rates for all vertices be identically equal to the normalized rate 1.…”

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We investigate a limit theorem on traversable length inside semi-cylinder in the 2-dimensional supercritical Bernoulli bond percolation, which gives an extension of Theorem 2 in [5]. This type of limit theorems was originally studied for the extinction time for the 1-dimensional contact process on a finite interval in [10]. Actually, our main result Theorem 2.1 is stated under a rather general 2-dimensional bond percolation setting.

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Epidemic Spreading With External Agents

“…Analyzing epidemic behavior is complemented by works seeking to control such processes for various purposes. For example, various authors have looked at designing algorithms for optimal seeding in networks to maximize the spread of an SIR epidemic [7], for ensuring long-lasting SIS epidemics by distributing virulence across edges [27], and for efficient routing over spatial networks with fixed long-range links [28].…”