Last Thursday Night My 88 Year Old Mother And I

Last Thursday night my 88 year old mother and I both started coughing a lot. The next day I woke up feeling like I had the flu with a 102 degree fever. My sister took my mother to stay with her and now today my sister has the flu. Mom is still coughing but hasn't had any flu symptoms. My sister's doctor said to give her Tamiflu just in case. How safe is that at her age and is it a good idea?