Tenant is terminally ill - month 5 of 12 month lease in winter

[Dane C Allen]

A longstanding great tenant of mine is terminally ill and has moved out (it’s highly likely he won’t be back). He’s now living with family and receiving hospice care. He’s in very poor health and likely will not survive much longer (3-6 months). His lease is thru 7/31/19 and we’re approaching the dead of winter. I’ve been corresponding with his POA and have assured her I will make every effort to get the place rented ASAP . She has been understanding and appreciative of my making an effort to work with them during this difficult time. I already have a promising lead to fill his space by April 1st of 2019, though I’d like to get someone in there sooner. Not a situation I’ve dealt with before, makes me think, how would you handle? What implications are there for a situation like this? What recourse would I have if they decide to stop paying rent?