You are offline, so I'll leave some helpful information.
I have seen a number of children with norovirus and similar intestinal viruses that ran no fever and seemed quite happy despite the vomiting and diarrhea. Such an infection may be the cause of your son's symptoms.
Prompt evaluation by the child's physician or in an emergency room is advised if you note any of the following in association with persistent vomiting/diarrhea: fever that is high (i.e., at or above 104-105 degrees Fahrenheit / 40-40.5 degrees Celsius) or lasts beyond 48-72 hours; blood (bright red, maroon, or black) or bile (light to dark green) in the vomit; blood (bright red or tarry black) or mucus (copious or bloody) in the bowel movements; diarrhea that lasts beyond 7 days; failure to urinate well at least once every eight hours; dry or sticky (not moist and slick) inner cheek lining when a clean finger is swiped against it; sunken eyes or failure to produce tears with crying; inability to take or hold down even one-half teaspoon sips of liquid; abdominal bloating with rigidity or tenderness to pressure; confusion, inconsolable irritability, or hard-to-rouse lethargy; intolerable pain or marked neck stiffness; coarse-sounding, too-rapid, or labored breathing.
If none of the above urgent criteria are met, then the cause of the vomiting/diarrhea may be viral (instead of bacterial, etc.), especially if there has been exposure to sick family or friends with similar symptoms. Please note that antibiotics do NOT work against viruses. I would also avoid anti-diarrheal meds unless advised by the child's physician (in a sense, these prevent the body from ridding itself of the viral infection). You may use Tylenol for ache or fever, but avoid ibuprofen (Motrin or Advil) as it may irritate the stomach.
To prevent dehydration and maintain nutrition, try this routine called "2 bites, 2 ounces."
Alternate every 10-20 minutes between offering:
½ - 2 ounce sips of clear liquids* (e.g., Pedialyte**; Pedialyte or other brand popsicles; white grape juice; flat non-caffeinated soda; clear bouillon or broth; Kool-Aid; Gatorade; ice chips or water** - must take other liquids besides ice chips/water).
1 - 2 bites of bland, starchy solids (e.g., plain yogurt***; bananas; applesauce; cooked rice; toast; Jello; Saltine or graham crackers; finely-chopped carrots; plain mashed white potatoes; pastas; consider non-fried lean meat if the starches are tolerated).
* Start with a measuring teaspoon--once the child has tolerated this a couple of times, go up to two teaspoons or a measuring Tablespoon. A measuring Tablespoon used for baking holds ½ ounce of fluid, two Tablespoons equals 1 ounce of fluid, and a measuring cup will help when giving larger volumes.
** In place of Pedialyte and similar over-the-counter oral rehydration solutions, you may use the following mixture of a level measuring half-teaspoon of salt and six level measuring teaspoons of sugar stirred until dissolved in 1 liter or 5 measuring cups (34-40 ounces) of clean drinking water.
*** At age 7-9 months and up, you may give plain yogurt labeled as having "active cultures" of Lactobacillus or Bifidobacterium. You may sprinkle a little sugar to improve flavor. Avoid yogurt with added fruit other than bananas.
If your child refuses solids, offer the liquids instead, gradually increasing the fluid volume up to 4 ounces as tolerated (stay at the fluid volume where no vomiting occurs). If he manages enough fluid intake to urinate well at least once every 8 hours, this will help to prevent dehydration. Starchy solids promote formed stools. Eating plain yogurt helps to restore normal intestinal function, while a more effective over-the-counter option similar to yogurt (and found helpful in my practice against viral- and antibiotic-associated diarrhea) is 1 capsule daily of "Culturelle for Kids" mixed into a snack of cool or room temperature applesauce, mashed bananas, or plain yogurt.
Avoid liquids and foods that may promote diarrhea (examples: prunes and pears; prune, pear, apple, or red grape juices; unpeeled fruits; high-fiber foods, such as bran- or whole-grain type cereal, bread, waffles, or muffins). Dairy products (e.g., milk, cheese, butter, and even milk-based formula) and fried/greasy foods may be difficult to digest or cause gas pains during an intestinal infection.
Viral sources of vomiting/diarrhea often resolve within 7-10 days. If there is only vomiting without clear cause for more than 48 hours, seek medical evaluation for possible uncommon causes (e.g., increased pressure inside the skull). Focus on hydration, and remember the urgent criteria at the start of this message.