I can't speak for English settlements elsewhere in North America, but in New England this was most definitely the case. The first Pilgrims arrived in Massachusetts in December 1619 and found the local population devastated by an epidemic (potentially leptospirosis) that had ravaged the coastal tribes for three years. Some historians propose a 90% mortality rate among certain groups.

I asked a similar question on this sub a while ago and received statistics and sources - it's worth checking out.

I apologize that I don't have the time right now to write a lengthy, complete answer. I hope to at least point you in the right direction, and highlight some previous posts that may be of interest.

The short answer to your question is that there is no one story of infectious disease spread after contact. Conditions varied based on time and place, dependent on the specific ecological conditions of the hosts, the pathogens, and the environment.

By the time English settlers arrived in number, and really just confined themselves to small tenuous beachheads on the Atlantic Coast of Massachusetts and Virginia, more than a century had elapsed since initial contact with Spanish explorers. A few decades ago scholars believed this initial interaction in North America prompted the spread of massive, uniform, catastrophic epidemics that decimated Native American populations from the Atlantic Coast deep into the interior of the continent. Our perception of disease and demography after contact has since shifted.

For example, current evidence suggests a period of relative demographic stasis in the years following contact, and places the first smallpox pandemic in the U.S. Southeast to 1696-1700 (Kelton). That is almost 80 years after the relative late arrival of English colonists, and almost two centuries after Spanish outposts and missions dotted the Florida landscape.

What changed in those 80 years?

Slavers operating out of Virginia and Carolina financed slaving raids that allowed for the spread of the pathogen where previous epidemics would have burned out. The social environment changed, and the poor health of human hosts fleeing the slavers likewise changed. Smallpox could now spread with great effect, where the pathogen had once been checked. I wrote a much longer account of the conditions needed to propagate a smallpox epidemic, the ecological changes accompanying the rise of the slave trade in the American South, and the 1696 smallpox epidemic here if you are interested. Basically, epidemics didn't just happen. The full force of the colonial cocktail (disease, famine, territorial restriction and displacement, warfare, social disruption, etc.) was needed to initiate and perpetuate the spread of infectious pathogens.

Finally, though we tend to think of disease as one massive tsunami surging ahead of colonists in North America, a more accurate depiction would be that infectious disease epidemics, like many of the other shocks of colonialism, act like a slowly rising tide. Northern Plains nations kept historical records in the form of Winter Counts, and allow us to time the arrival of disease on the Northern Great Plains. I wrote a longer write-up about these documents here, but the take home message is that epidemics arrived in a series of small waves, even before sustained contact with Europeans. On the Northern Plains an epidemic occurred roughly every 5.7 years for the entire population, but varied by band. The Mandan saw the recurrence of epidemics every 9.7 years, while the Yanktonai averaged an epidemic every 15.8 years. The longest epidemic free interval for any band was 45 years for the Southern Lakota, and the shortest was 14 years for the Mandan. There were years, and in the case of the Southern Lakota even decades, for population recovery of loses, provided other sources of mortality were limited. Unfortunately, when the colonial cocktail hits in full force, there is little time for recovery.

If you are interested, I also wrote a series on some of the common myths of conquest for another reddit community. You might find the posts on the myth of Death by Disease Alone and A Pristine Eden helpful for understanding the concepts I briefly touched on here.

Sources

Kelton Epidemics and Enslavement: Biological Catastrophe in the Native Southeast 1492-1715

Sundstrom (1997) “Smallpox Used Them Up: References to Epidemic Disease in Northern Plains Winter Counts, 1714-1920.”

While not answered in AskHistorians, these /r/badhistory posts seem to touch upon this idea in great detail, with a number of sources.

http://np.reddit.com/r/badhistory/comments/2u4d53/myths_of_conquest_part_seven_death_by_disease/

http://np.reddit.com/r/badhistory/comments/2v1mnf/myths_of_conquest_part_eight_a_pristine/