The individual has 60 days to decide whether to elect COBRA continuation coverage. (This 60 day period allows for the employee to change their election as many times as they would like). The person then has 45 days after electing coverage to pay the initial premium.
Using your example, they "elected" coverage on Day 1 (Aug 1), but have 60 days to change their mind. August 1 premium is due, but there is a 45-day period for them to pay, which is September 15. Therefore only August premium is due at this time. (There is a provision that allows for COBRA participants to pay less than the 100% due and still be covered, but let's not bring that into play for this example).
Now the curve ball. Since this person has 60 days to choose, they can change their mind, tell you they do not want COBRA, and then on the 60th day (Oct 1) elect coverage.
If I may make a suggestion, you may want to invest in either a cobra software package or have an outside firm do it for you. COBRA is still an employer risk, but either route should help you. Your health insurance broker should be able to direct you to the best alternative based on your needs.
Here is a good website to visit for information
http://www.dol.gov/dol/topic/health-plans/cobra.htmGood luck.