Hello,
If it doesn't bother you when you leave it alone, then the infection cannot be very deep or very serious. Hydrogen peroxide is used in open wounds or infected points where there is pus which needs to be removed. It will not be absorbed through the skin. Ointments like Neosporin are absorbed through the skin and can act on somewhat deeper infections. You can apply the ointment inside the bellybutton as deep as you can and leave it to be absorbed.
Motrin 500mg twice daily will also help to decrease the inflammation. Once the inflammtion decreases, your natural body immunity should be able to handle the infection.
I hope this helps.
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Experienced in the management of Pregnancy,Infertility,Menopause and all problems in Pregnancy and Gynecology