I don't think they put out a public alert on the evening of the 9th. The first public notification was put out at approximately 2:30 PM on the 10th, giving initial details and asking the public for any info they might have.
Between 9:24 PM on the 9th (the time she was reported missing by her advisor) and 2:30 PM on the 10th, they checked her lab office and apartment, checked with local and regional hospitals to see if she had been admitted, checked with taxi companies to see if someone matching her description had purchased a ride, and checked at Illinois Terminal (Amtrak and long distance bus depot) to see if she had purchased a train or bus ticket and departed from the terminal. I believe they also used K-9 units to search the grounds around Orchard Downs (university housing where she was living). While that was going on, they interviewed her friends/coworkers to figure out what she was doing that day, then subsequently contacted the manager at One North (complex where she was going to rent an apartment) and viewed his text messages with her.
So, by early afternoon on the 10th, they had determined she wasn't in an accident and in the hospital, she hadn't apparently left town on her own, and had figured out where she was going on the afternoon of the 9th and what was the last contact anyone had with her -and figured out she never made it to her intended destination. That's the point they realized there was a high likelihood they had a very serious situation on their hands, and began to notify the public and ask for any info people might have. I assume that while that notice was being put out, they were already gathering security camera footage and starting to review it to see if they could spot her and track her movements .