Friend, as long as the children do not have contact with the feces they will be okay. Touching or playing with the puppy is safe. It sounds like your puppy has both Tapeworms and Roundworms. Drontal will treat both, Tapeworms and Roundworms. If he vomited the medication I would phone your vet to get another prescription and repeat it again. For the diarrhea this is what you can do:
1- Do not give things like Pepto Bismol or Imodium.
2- No food or water for 6-8 hours, need to give his gastrointestinal tract time to rest. Then, offer a small amount of water followed by a diet of rice (75%) and boiled boneless chicken or ground beef (25%) until the stool is back to normal then wean him back into his regular diet.
3- Supplement with a probiotic, these are friendly bacterias that help with regularity. You can buy this over the counter in any pet store.
4- If after doing the above you do not see improvement within 24 hours I would have him checked by your family veterinarian for a hands on examination.
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