Hi.
Liver abscesses are not a good thing to have but it is very rare for them to rupture and cause death, so be assured , they usually do not rupture because of nature's way to have a think lining around an abscess.
This looks to be secondary abscess, and if small can be treated by medicines alone. May need aspiration or surgery if large and is having symptoms like fever, chills, jaundice, pain or so.
Please post the recent reports and clinical findings for us to help you more.
Appendix too needs to be removed at an appropriate time to avoid further complications.

Such problem like liver abscesses always very dangerous. Of cause it depends from age of the patient, his present status, size of abscesses, how aggressive infection is there, locations in the liver and many other circumstances, which doctors will take into considerations. Treatment is also depends from above mentioned characteristics. But anyway will consist of antibiotic therapy and possible surgical drainage of abscesses. Surgical methods can be also varied. Recently often used minimally invasive techniques such as puncture and drainage of bacterial liver abscesses under ultrasound guidance, which can reduce numbers of operations, numbers of complications and shorten the hospital stay.
But this is just a sample, in each and every case the methods of treatment must define only your doctor.
The healing process takes a long time, mainly because of the difficulties of treatment and prolonged drainage of the abscesses.