Boundary Modified Contact Processes

Abstract

We introduce a one dimensional contact process for which births to the right of the rightmost particle and to the left of the leftmost particle occur at rate λ _e (where e is for external). Other births occur at rate λ _i (where i is for internal). Deaths occur at rate 1. The case λ _e=λ _i is the well known basic contact process for which there is a critical value λ _c>1 such that if the birth rate is larger than λ _c the process has a positive probability of surviving. Our main motivation here is to understand the relative importance of the external birth rates. We show that if λ _e≤1 then the process always dies out while if λ _e>1 and if λ _i is large enough then the process may survive. We also show that if λ _i<λ _c the process dies out for all λ _e. To extend this notion to d>1 we introduce a second process that has an epidemiological interpretation. For this process each site can be in one of three states: infected, a susceptible that has never been infected, or a susceptible that has been infected previously. Furthermore, the rates at which the two types of susceptible become infected are different. We obtain some information about the phase diagram about this case as well.