dear user at first i would like to tell you that vomiting and diarrhea is very common problem in infants and you need not to worry about that but you should do simple steps.......vomiting and diarrhea may have more than one cause but the most common cause in that age 4.5 month is viral infection of the intestine ....viral infection causes severe watery diarrhea with vomiting and may lead to complication as dehydration......dehydration is the problem we worry about it because baby loss fluids in vomiting and diarrhea so you should give your baby oral rehydration therapy and fluids...oral rehydration sachets administered to protect the baby from dehydration and it is the most important step in management...if you do that your baby will be fine shortly and improve...other cause is allergy to certain milk formula that baby take and then you discuss with your doctor to change milk formula.......you are good mother....i wish your baby healthy life