I have a couple on a periodic joint agreement. Unfortunately one of them wishes to leave. Is a deed of assignment ok to use as there is no replacement tenant?

Your best option is do nothing then both of the couple remain each liable for ALL the rent, so if there are issues you can sue either or both for what is owed, even if one hasn't lived there for some time.

However, if either give valid notice to quit then that notice ends the tenancy for both of them, and you'd need to offer a new tenancy to the sole remaining - if you wanted to.

Because it is periodic. Any one of them can bring the tenancy to an end. If the other stays it would be a new agreement and they would be liable for all.

But to answer your question. No a deed of assignment is not necessarily best on a periodic. Just start a new agreement. with new deposits etc. this is clean and clear.

I have a family of four on a periodic, one of them has to leave as they do not have leave to remain in the UK and are unauthorised to rent. The remaining three are happy to maintain the tenancy as it is without adding a tenant.
The Home office have asked for proof that I have done everything required of me to show that this particular tenant has left the property, a Deed of asignment has been suggested.