Well it can't be that early because any investigation shows innocent the 1st 2 days and the sheriff and Dr can't communicate till Day 3.
Think that you read the rules wrong, Ice. See below for role descriptions of Doc/Sheriff. It looks like they communicate immediately.
You are the town’s Doctor. You are trying to develop an antibiotic to protect the citizens of ACville. You have a weak serum and it has limited duration so it cannot be used on the same person more than once. At the onset of night, you may protect one person with it and it will save them from being turned or killed by the zombies, starting with night #3. You may communicate with the Sherriff.
You are the town’s Sherriff. Your job is to locate and eliminate the Zombies. You can do an investigation each night in your search to rid the town of this evil. You may communicate with the Doctor.