Bed bug bites….how long after the bites does the itching start?

I think I might have been bitted by bed bugs. Does anyone know how long after the biting the itch starts? For example, I first noticed little bites yesterday afternoon, on my face, arms, shoulders and legs. Today, the itch is much worse. Does this sound like bed bugs? I have been on vacation in a tropical country and returned to my home (clean and no one in it while I was away) five days ago. Is it possible that the itch wouldnt start for a few days?