Hey there! Before I answer this question, I want you to do something for me. Take a look at your question and substitute "blacks" in place of "Native Americans." Do it again, this time substituting "whites." Would you ask either of those questions?

You're asking a titanic question that can't really be answered in a short amount of time. Native Americans can't be lumped together any more than whites, blacks, Asians ... really any ethnic group can. I know you didn't mean anything by it and that you're just trying to learn (as we all are), but sometimes we need to stop and look at our own inherent biases.

Now, with that standard disclaimer out of the way, let me give you your answer. :)

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I'm going to talk about disease among Alaska Natives, and there's two reasons for that. The first and simplest reason is that I have sources at hand that I think are relevant. The second is that disease and disease resistance spreads along lines of communication and trade. The more isolated a group is, the less likely they are to have been exposed to a disease before. Without exposure, that group cannot have developed resistance to a disease.

With those factors in mind, groups in isolated places -- deep in the jungle, say, or high in the mountains -- are going to be more vulnerable to diseases spread by people coming from more trafficked places. When a disease arrives in a place that has little or no resistance to it, we call that a virgin field epidemic.

Alaska is one of these isolated places, and it has had plenty of virgin field epidemics. Until the 1920s and the arrival of regular airplane service in Alaska, traveling into rural Alaska was an ordeal. You had to spend weeks -- if not months -- on the trail, traveling by ship or sled, and this chronological protection meant a sick individual could get better (or die) and thus was no longer a disease carrier by the time he or she got to his remote destination.

I'm sure there are places in the jungles of South and Central America that could make the case for having the last virgin field epidemic, but I don't have that information and can't talk about them. If you're interested in Alaska disease history, there's no better book than Robert Fortuine's Chills and Fever: Health and Disease in the Early History of Alaska. I'll rely on that book for some sections of this answer. For an easy link about what I talk about later, check out M.W. Johnson's paper, "Results of 20 years of tuberculosis control in Alaska," published in 1973 in Health Services Reports.

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Let's limit our discussion to the 20th and 21st centuries. We could talk all day about smallpox, plague and other such things, but you're asking for the last major epidemic. I have to tell you, though, the last "major outbreak" hasn't happened yet. The rates of sexually transmitted disease and alcohol-related diseases remain far higher among Alaska Natives and American Indians than among the general American population.

The causes for this disparity can be argued, and I won't go into that because of the 20-year rule for this subreddit. Fortuine devotes an entire chapter to STDs, which he calls "the Gift of Venus," and the Alaska Section of Epidemiology writes regular reports about what is literally an STD epidemic in Alaska.

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I have a sneaking suspicion that you might not be satisfied with that answer, however, so let's talk about something else -- the campaign against tuberculosis in Alaska. This was one of Alaska's great public health success stories, and it took place from the 1950s through the early 1960s.

Tuberculosis had been endemic in Alaska since the late 18th century, when it was brought to the territory by explorers from Europe, where tuberculosis was endemic. By the second decade of the 19th century, it was endemic across Alaska as well. There are ample accounts from almost every European who kept a diary and described the consumptive cough among the Tlingit in Southeast Alaska, the Koniag/Alutiiq in Kodiak, and the Aleuts of the Aleutians. Various remedies were attempted -- one in Southeast Alaska involved dissolving slugs in slime and drinking the resulting concoction.

Tuberculosis found an easy breeding ground in Alaska. Winter meant close confines indoors, and smoky sod dwellings made lungs unhealthy in any event, less able to resist tuberculosis' spread. Tuberculosis continued to be a scourge into the 20th century as well, even as Alaska and Alaskans modernized. In 1950, tuberculosis was the No. 1 health problem in Alaska: Among Alaska Natives, it had a death rate of 653 per 100,000 people -- almost 25 times the national average for all races. It had an incidence rate of 1,854 cases per 100,000 people in 1952. By 1970, this incidence rate had fallen to 154 per 100,000 people. Tuberculosis fell from the leading cause of death among Natives to an insignificant item.

The difference was the development of tuberculosis-treatment drugs and techniques, coupled with a massive public health campaign. After the end of WWII, military hospital beds were converted for use by tuberculosis patients. The Public Health Service converted WWII surplus ships into traveling hospitals that sailed up and down the coast of Alaska, treating remote villages. Inhabitants received regular X-rays to detect tuberculosis, they received vaccinations to stop the spread of the disease, and if they were already infected, they received treatment to save their lives.

Coincident with this campaign, there were mass vaccination efforts to stop the spread of such diseases as measles, mumps and polio. All rates of infectious disease dropped dramatically during the 1950s and 1960s.

Tuberculosis has by no means disappeared entirely; in recent years, it has returned and Alaska now has the highest rate of tuberculosis in the United States (albeit at a figure far lower than even the 1970 figure). The great fear is that drug-resistant tuberculosis may spread to Alaska, and if that happens, we could see a return to the bad old days.

I'll echo /u/The_Alaskan with the caveat that this is a titanic question that is rather loaded with supposition.

Previously unexposed Native American populations aren't any more susceptible to infectious organisms than other immunologically naive populations. They have fully-functioning immune systems that operate on the same as principles of innate and adaptive immunity like all humans, with perhaps the only difference being in their slight homogeneity in one specific immunological important loci called HLA (human leukocyte antigen). The Americas were home to a variety of nasty infectious organisms before European pathogens arrived. I'll quote a previous answer from another history-related community for the sake of time...

A wide variety of gastrointestinal parasites accompanied the original migrants on their journey to the New World and can be found in coprolites and mummies across the Americas (see Goncalves et al. 2003 for a review of archaeoparasitology). New World populations were likewise subject to Chagas, pinta, bejel, tick-borne pathogens like Lyme disease and Rocky Mountain Spotted Fever, syphilis and tuberculosis. Finally, like all humans who interact with wildlife, New World populations would have been subject to zoonotic diseases that jumped from a non-human animal to a human host. The most famous New World zoonotic disease from a wildlife source is cocoliztli, presumed to be a viral hemorrhagic disease like Hantavirus, that killed millions in a series of epidemics that burned through Mexico in the late 16th century.

So the idea that Native Americans ever lived in a disease-free paradise is simply not accurate. They faced immunological challenges and survived in a hostile parasite-infested world similar to what all humans encounter. Now, since we have dealt with several issues, lets dive into your questions...

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I'll echo /u/The_Alaskan and say that we are currently witnessing disease epidemics in Native American populations. Though this skirts the 20-year rule I hope my fellow mods will grant me room to mention a few key public health issues in modern Amerindian populations since your question does specifically address changing access to medical care and the modern setting.

First, as /u/The_Alaskan mentioned, STIs and alcohol-related diseases have a higher prevalence in Native American communities. I will also add elevated rates of adult-onset diabetes, as well as heart disease and obesity in First Nations and Native American populations. As far as infectious organisms, throughout the world indigenous populations tend to lack of access to vaccines, sufficient nutrition, and adequate medical care that all increase their susceptibility and mortality to pathogens. In this sense, yes, indigenous populations do still have problems with immunity today, but it is due to differential access to resources and not due to their ancestry. Indigenous/Aboriginal populations are disproportionately affected by poverty and neglected tropical diseases, ranging from high helminth infection rates in the Inuit to malaria epidemics among the Yanomami.

In the slightly less recent past there were also epidemics of infectious disease both in previously uncontacted groups, as well as settled. /u/The_Alaskan did a great job explaining relatively modern epidemics in Alaska. Similarly, in the middle 20th century, previously isolated Amazonian groups like the Ache faced increased mortality, mostly due to respiratory infections, during their transition to missions. Again, forgive me for quoting previous answers, but I'm in a bit of a rush...

When the Ache transitioned onto reservations infectious disease mortality spiked during the transition period. Interviews with survivors lead us to believe 38% of the population died from respiratory diseases (though that number includes people who were infected, returned to the jungle, and died away from possible medical care). However, the Ache rallied and are now a growing population. The key factor for population survival is limiting other sources of excess mortality.

Finally, going back even further in history, we have records of periodic waves of disease in Northern Plains groups, even before sustained contact with Europeans. Here is a look at the instances of disease mentioned in Ogala Winter Counts to give you an idea of this periodic nature of disease spread...

Year	Disease
1780	Cough; smallpox or measles
1781	Smallpox
1785	Measles
1798	Smallpox
1798	Many pregnant women died
1801	Smallpox
1813	Whooping Cough
1818	Measles or Smallpox
1844	Measles or Smallpox
1849	Asiatic Cholera
1850	Smallpox
1861	Disease among the children (maybe smallpox?)
1901	Smallpox

So, really, Native Americans never stopped "getting killed off by disease" because humans never stop getting killed off by diseases. Improving access to medical care and providing adequate nutrition help improve morbidity and mortality rates for all humans, and hopefully, the disastrous epidemics in our shared past will not be repeated.