A contact process with a semi-infected state on the complete graph

Abstract

ABSTRACTIn this paper, we are concerned with a contact process with a semi-infected state on the complete graph Cn with n vertices. Our model is a special case of a general model introduced by Schinazi in 2003. In our model, each vertex is in one of three states, namely, “healthy,” “semi-infected,” or “fully-infected.” Only fully-infected vertices can infect others. A healthy vertex becomes semi-infected when being infected while a semi-infected vertex becomes fully-infected when being further infected. Each (semi- and fully-) infected vertex becomes healthy at constant rate. Our main result shows a phase transition for the waiting time until extinction of the fully-infected vertices. Conditioned on all the vertices are fully-infected when t = 0, we show that fully-infected vertices survive for exp {O(n)} units of time when the infection rate λ > 4 while they die out in O(log n) units of time when λ < 4.