If you've ruled out random virus, then....
....accidental poisoning, maybe? Could they have ingested or inhaled anything they shouldn't have? Berries or leaves from outside, any chemicals you might have around (laundry soap, perfume, toothpaste, dish soap, shampoo, lotion, conditioner, hand gel, bug spray, ant traps, paint thinner, fertilizer, etc....) , lead painted anything, carbon monoxide, medications (yours or theirs), vitamins, aspirin.... it might be worth asking your kids if they've gotten into something they shouldn't have. Also, check around inside and outside your home for anything out of place that they might have gotten into.
Don't assume they can't get to it just because you have it locked in a cabinet. Kids are smart, kids are curious, and kids don't understand the dire consequences of some of their actions. An open drawer is all my 3 year old needs to climb up onto my bathroom and counter tops (that seriously drives me crazy!!!). So check in all your cabinets, above and below!
Or a more likely scenario....
.... is that they started out with a 24 hour tummy bug, and just haven't fully gotten over it yet. If their tummies were upset, that can cause acids to build in their tummies, which can then turn into a mild case of reflux, which can cause the throat to burn (which feels like a sore throat), and the tummy to ache, which can lead to not wanting to eat, which can lead to more tummy aches, more acid build up, more sore throats, and more extreme fatigue....
I recommend putting your kids on an extremely bland diet for at least 48 hours. Saltines, dry white toast, rice, chicken soup.... no spices, nothing acidic, no dairy. Your kids need to eat a little bit every couple of hours, to keep their stomachs from getting completely empty (empty stomachs produce a LOT of acids!). If they don't want to eat.... well, how do you deal with them if they don't want to take medicine? Treat the food the same way as you would medicine and get them to eat.
If it's a tummy virus that just hasn't gotten better, the above treatment will help them improve fairly quickly. But don't stop the bland diet for at least 48 hours, and even after that, keep acidic and spicy foods out of their diet for at least a week after they are feeling better. Hope this helps!
