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.
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