The contact process with semi-infected state on the complete graph

In this paper we are concerned with the contact process with semi-infected state on the complete graph $C_n$ with $n$ vertices. In our model, each vertex is in one of three states that `healthy', `semi-infected' or `wholly-infected'. Only wholly-infected vertices can infect others. A healthy vertex becomes semi-infected when being infected while a semi-infected vertex becomes wholly-infected when being further infected. Each (semi- and wholly-) infected vertex becomes healthy at constant rate. Our main result shows the phase transition for the time wholly-infected vertices wait for to die out. Conditioned on all the vertices are wholly-infected when $t=0$, we show that wholly-infected vertices survive for $\exp\{O(n)\}$ units of time when the infection rate $\lambda>4$ while die out in $O(\log n)$ units of time when $\lambda<4$.