  We are concerned with the supercritical contact process modified so that
first infection occurs at a lower rate, it is known that the process survives
with positive probability. Regarding the rightmost infected of the process
started from one site infected and conditioned to survive, we specify a
sequence of space-time points at which its behaviour regenerates and thus
obtain the corresponding strong law and central limit theorem. We also extend
complete convergence to this modified case.
