My 10 Month Old Began With A Fever Dec 10th.

My 10 month old began with a fever Dec 10th. And was off and on until a well later. Never a fever over 100.6. She was very congested and didn't have an appetite during this time. The doctor wanted to get blood work done since it had been so long since she had a fever. Blood work came back with neutrophils at 1200 and sed rate was 79. doctor gave her two shots in the leg off an antibiotic. doctor had us come back the next day for a recheck. Went to the lab for more blood work and the results were worse. Her neutrophils were now 180 and sed rate was 88. She received 4 more shots of antibiotics the next two days. But this time fever had stopped. Today, 20th, her neutrophils are 420 and sed rate is 70. So things are improving, but slowly. She has been on amoxicillin for the past three days. Do you have any insight on to what this could have possibly been and do you think she will always be neutropenic? Should she have been admitted? Thank you for any input.