Usually when the deer 'disappear' as you said, the dreaded breeding phase is taking its effect. Thus the mature bucks are holed up with their does and the woods is literally a dead zone concerning deer movement.

What can be learned? By assuming that the breeding phase is causing the sudden screech in deer movement (and the majority of times it is) you can count 28 days before and after these days to determine when your the first and last wave of estrous does came in and will come into heat. Also- if you believe the moon strongly effects the rut as much as I do, you can determine how long after the rutting moon the breeding phase is, thus telling you how long the seek and chase phase is as well.

Again, assuming the breeding stage of the rut is caused the lack of deer movement you can learn a lot of the buck to doe ratio of your herd as well. With an unbalanced buck to doe ratio the rut is going to be lonnng... and... drawnn.. out, less frenzied and simply not as fun to hunt. However, with a balanced buck to doe ratio the rut becomes more predictable (with reference to the three stages) and you can again better gauge the rut according to the rutting moon in the years to come.