I think the cost could well be close. Here, if you call 911 for a medical emergency you will get a police patrol car, an ambulance, and a fire truck dispatched in most cases. That's at least three vehicles and at least half a dozen people.

If this is happening once a week then it really seems there ought to be a better way to handle it.

Edit: if you also assume that an ambulance dispatch is usually followed by an ER trip, the cost balloons even more. Seems really hard to think that a 911 dispatch followed by an ER trip once a week is the best way to handle a chronically ill person at home.

I don't know the exact rules but in the UK, when calling 999 you're asked which service you require. Fire service will only get police involved when they need it or automatically when more than 2 Fire Engines are dispatched. Ambulance calls can result in an Ambulance with a crew of 2 or a single paramedic in a sensible estate car (station wagon) with a big engine, they'll call the Air Ambulance if they need one and it's free. Dispatchers will send who they think you need as they see fit. That includes Lifeboat Service or Mountain Rescue. Or Police, of course. Each service will obviously ask for backup as required.

The BIG difference with what you've described though is the layout of the UK is much denser than USA. The resources available and the nature and frequency of call outs will obviously be quite different.

How much would they save if they bought the fire department a Prius for responding to medical calls?