Invariant measure for the contact process with modified border in the non-attractive region

We investigate a modified one-dimensional contact process with varying infection rates. Specifically, the infection spreads at rate $\lambda_e$ along the boundaries of the infected region and at rate $\lambda_i$ elsewhere. We establish the existence of an invariant measure when $\lambda_i = \lambda_c$ and $\lambda_e > \lambda_c$, where $\lambda_c$ is the critical parameter of the standard contact process. Moreover, we show that, when viewed from the rightmost infected site, the process converges weakly to this invariant measure. Finally, we prove that along the critical curve within the attractive region of the phase space, the infection almost surely dies out.