Would complete elimination of the deer population eliminate the Lyme disease threat?

Has anyone really looked into the rise of Lyme disease, and if this is directly related to a rise in the deer population? I understand that the ticks can live on other hosts, such as rodents and birds. But it seems that deer is the preferred host.

I understand that Lyme disease is controversial in many ways, and it’s difficult to get solid info about the whole thing. There are many kids I know who supposedly have had (or still have?) Lyme disease. And it’s very common to return from a hike and have more than one tiny deer tick hiding somewhere (hair, skin, armpit, etc).

I consistently hear that a reduction in the deer population would be a guaranteed way of reducing the deer tick population, and therefore the occurrence of Lyme disease. If this is the case, would it be safe to assume that 0 deer = 0 deer ticks = 0 cases of Lyme disease? Note: I’m not proposing killing the deer. Just trying to figure this out.

Has anyone really done the work of researching this? If so,