A contact process with a semi-infected state on the complete graph. Issue 2 (4th March 2018) - Xue, Xiaofeng
ABSTRACT: In 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.