Seeking advices on dealing with tenants from previous owner

[Nelson Xu]

Hello everyone, first time posting in this forum. My case is a little complicated and I try to sort out the sequence of actions. Appreciated your time in advance on this post.

I have a property in Southern California with tenants from previous owner. The tenants has been living in the unit for quite some time. The rent is below market. They're paying rent mostly on time. However, I just learned today that one of the tenants has health issue and won't able to work. They want to terminate the lease (month-to-month lease) and use security deposit as last month of the rent.

Problem is there's no security deposit in the lease when I got this property from previous owner. There are 5 people living in the unit while only person name is on the lease.

Questions are

How should I deal with this tenant in current situation? 

What would the move out process look like? for any property damages.

Can I still demand full or partial rent paid back in the future?

Thank you!