Hi Greg, 

You said that it was a "thing" and not a "person" that "ceases" in  the "midst of the week".  Actually, it is the "person " of Jesus Christ who is cut-off  by being crucified in the midst of the 70th week.  That event  is what causes  God to cease to accept  the blood sacrifice of animals for the atonement  of sins after Jesus shed His  blood for the remission of sins for all  mankind. 

Daniel 9:27

"And he (Jesus) shall confirm the covenant with  many ( confirm the Abrahamic Covenant with many) for one week ( the 70th week of Daniel ), an d in the midst of the ( 70th ) week he (Jesus) shall cause the sacrifice ( of animals ) and oblation to cease (  being accepted  by God because Jesus has n ow shed His blood)......

Blessings to you.