Boundary Modified Contact Processes

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 λe (where e is for external). Other births occur at rate λi (where i is for internal). Deaths occur at rate 1. The case λe = λi is the well known basic contact process for which there is a critical value λc > 1 such that if the birth rate is larger than λc 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 λe ≤ 1 then the process always dies out while if λe > 1 and if λi is large enough then the process may survive. We also show that if λi < λc the process dies out for all λe. 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.

Duke Scholars

Author Richard Timothy Durrett Mathematics