If a bug bug bite is not made worse by scratching, it should resolve within a week. Bed bug bites usually itch though and you will be tempted to scratch them.
Everybody can react differently to bed bug bites. Some people may not show much more than a small bump. Others might have a more intense reaction. If the reaction is severe enough, your doctor may decide to prescribe an antihistamine, antibiotic or corticosteroid treatment.