Maybe you can clone the issue befor you close the old task. It involves manual tasks unless you do coding.
If you are doing coding, you can create a new issue type and have a workflow post function for that issue types which creates a copy of the task when the original task is resolved.
I ran across this answer about a commercial plugin that might work for you as well.
I posted a description of an implementation of periodic tasks at:
This is not exactly what you are asking for but it is related and it has been helpful in our organization.
...word of mouth, so by 2016, we were working with several other entities on campus to implement Jira Service Desk . The Atlassian motto of “for every team” has really come true for us in this case. We...
Connect with like-minded Atlassian users at free events near you!Find a group
Connect with like-minded Atlassian users at free events near you!
Unfortunately there are no AUG chapters near you at the moment.Start an AUG
You're one step closer to meeting fellow Atlassian users at your local meet up. Learn more about AUGs
We're bringing product updates and pro tips on teamwork to ten cities around the world.Save your spot