Accidents from an illness, like vomit, can quickly soak in to your mattress and live a lingering smell. The best way to remove the smell is to first remove any lingering moisture.
Gently rub in a tablespoon of corn starch per hand size stain. Corn starch is a natural absorbent and deodorizer. Let sit for ten minutes, and then vacuum or brush away residue.
To prep for cleaning the stain, its a great idea to have a fan on standby. This will help your mattress dry and also circulate odors (and direct them towards the door or window).
After you have absorbed lingering moisture you will want to clean the stain. Never dump water or liquid on to a mattress; because they are sealed its an easy way to grow mildew when left to sit in water.
First, try a mild solution of a 1/4 cup of dish washing detergent (preferably granules) with 1 cup of warm water. Apply to a clean sponge or towel and lightly rub the stains and blot dry with a clean towel. Allow to dry. If smell is still present and stains are still there, your next step is to head for the kitchen cupboards for back up.
Grab baking soda, a spray bottle, and white vinegar. You will want to fill your spray bottle with enough vinegar so that it can produce a light mist.
Once your mattress is dry, sprinkle baking soda over stains. Then mist the areas with your vinegar, allow to bubble until fizzed out and then blot dry with a clean towel. Both vinegar and baking soda are natural deodorizers, and when combined they also produce a reaction that helps remove surface dirt. Again, sprinkle some corn starch on your stains and allow to sit and dry. Brush or vacuum up residue.
In most cases, a combination of these methods will help remove odors and stains from vomit. If the smell or stain persists, your next option is to purchase an enzyme based cleaner. Most enzyme based cleaners require a lengthier saturation time, so it is imperative you use this method early in the morning or at a time when the mattress can be left to dry overnight.
Professional mattress cleaning service is also available in most regions and often times done in conjunction with carpet cleaning.