Hi, you have two viable options. In case you want to stay here just for a temporary period of time, you might want to apply for an extension to your visa due to health issues that prevented you from leaving Italy in time. You will need to get some stamps and letters from doctors for this. You can read more about Schengen visa extension here, I just replied about the same topic to someone else:
Schengen visa extensionhttp://www.maxi-pedia.com/forum/index.php?topic=1283.msg3152#msg3152In case you expect to stay in Italy for a relatively long time, you might be eligible for a permanent residence status (taking care of your relative - I am assuming he or she already has some legal long-term presence status).
Cheers.