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Eugenics + Poverty

It actually happened.

If you feel like hating humanity you can look up the practise of forced sterilization, which was quite widespread even 40 years ago. And you still see genocides in 3rd world countries of undesirables, it is not a far stretch to connect autism with undesirables.
All you need is a tiny bit of propaganda in a paranoid or stressed society and they would have no problem doing what you describe, real countries have done exactly that.

In 1933 the Nazi passed the ‘Law for the Prevention of Hereditarily Diseased Offspring’, allowing for the forced sterilisation of those regarded as "unfit". Only a few years later they included panels for deciding which disabled people should be "mercy killed" instead of just being sterilized. They kept it hushed up enough most families though the people were just being put in special care facilities, later they would be told they had died of some common disease. The first gas chambers were invented because letting disabled people starve to death in camps took to long. Most of the techniques they applied to later concentration camps actually started as techniques for "solving" the disabled problem.

If poverty is as common as you say people in your fictional country will likely welcome such measures for the financial burden they remove, poor farmers often could not afford to feed the disabled. Also keep in mind shame can easily keep people from checking on institutionalized family members so you may not even see much checking to begin with, so the lies only need to be paper.

For the same reason that autistic people often have trouble in most societies today: The tendency toward literal observation and very direct speech. This can cause all sorts of trouble (think about telling your boss exactly what you think of her management style), and just "refrain from commenting honestly" can be difficult enough, much less the polite lies that are the typical social lubricant.

Combine this emperor-has-no-clothes straightforwardness with a society where political correctness is Serious Business, such as the Soviet Union (where people were afraid to be the first to stop clapping for political speeches), and you have a situation where being autistic (or inconveniently honest in general) will get you disappeared.

In WWII concentration camps targeted "problem" DNA, things that "ruined" the human race like genetically passed defects, homosexuality, and (of highest priority) the Jewish race.

As awful as that is, it's a real event, and it only happened ~80 years ago. Your story could follow a similar thing, where the intention could seem noble to some (who have 0% compassion or humanity) to "purify" the country of autism.

NOTE: My notes are for the United States, though there will be many similarities with other industrialized countries. And non-industrialized countries.

This is not unique.

Capturing and persecuting disabled children (in other words, putting them in institutions) was a very common response to "difference" in most of the 20th century and also some times before.

I know people who were institutionalized (with the approval of their families) for having Down Syndrome or juvenile arthritis (which doesn't affect intellect, just walking and using your arms). Even children with something as simple as club foot ended up in institutions.

Kids who needed physical adaptations to their environment (places to roll their wheelchair...or the wheelchair in the first place) were institutionalized when they were too big to carry around. Changing the location of the kid was easier than changing the infrastructure.

Mental health issues also were something that frequently pegged people as "unfit" for society. This included women who spoke out against sexism, people of color who were too "uppity," people suffering from trauma (especially things one didn't talk about, like being raped), and people with actual mental illness.

Historically, there were not a lot of people with autism in institutions. Simply because autism wasn't very common. Now it's extremely common (plenty of studies show that only a part of it is due to changes in diagnostic criteria and the increase in diagnosis overall). But if we were still a society with large amounts of institutionalization directly due to disability, the institutions would be filled with autistic children and adults too.

So if your fictitious society targets autistic children, it may simply be because autism is perhaps the most common condition in children with disabilities that aren't overly physical (blindness, losing a limb, etc).

We still have group homes and the like but they are not institutions and they are not for small children. Those kids who can't be with their families mostly go to foster care. But, guess what? We still have capture and persecution of people with disabilities in the US. It's called prison. Huge percentages of modern day prisoners are people with mental illness (often masked by self-medication aka drug abuse) or people who don't "fit in" in some manner or people with other kinds of disability.

With American-style prisons for profit, there's an incentive to increase the prison population. So, if you're part of a marginalized community, it doesn't take much to get sentenced. Then you're put to work. You get paid almost nothing while the prisons take payment from the companies who benefit from the cheap labor.

I'm talking about real life here. I'm not making this stuff up.

So if your world has a lot of people with autism and a lot of oppression of people who are different, then people with autism will be targeted. The only thing you have to do differently from real life is the fact that autism was not common in the 70's and 80's and is not common in 3rd world countries.

Adding to my answer as at least one person finds insufficient evidence of persecution:

What I describe above was not about lack of resources but was a systematic attack on anyone who was disabled. All of this was part of the eugenics movement (which did not originate in Germany in the 1930's, though they took it to new levels). In Nazi Germany, people with disabilities were rounded up and murdered. These murders largely took place before the concentration camps were fully set up. Institutions were emptied as people were massacred. People with mental illness and visible disabilities were taken from their homes to be killed.

In the US and many other countries, women with disabilities (especially anyone with an intellectual impairment or mental illness) were sterilized without their consent. In many cases, without their knowledge; they sometimes found out after years of unsuccessfully trying to have children with their husbands. Women of color with disabilities were especially targeted.

This was straight up eugenics. Lots and lots of documentation about this. The leaders of this movement were not secretive about what they were doing or why.

Because it's contagious.

Seriously. Some old study, possibly done by/under a previous government that's now seen as The Best In History by the currently governing party, could have found some strange but not readily disprovable results leading that autism might be contagious. Add news networks, lack of education, illiteracy and a predisposition to believe whatever comes from the capital, and it's a perfectly reasonable scenario.

It would also explain why they persecute people with autism and not with physical disabilities, as other answers suggested.

Why not take inspiration from the current reasons/ways autistic people are persecuted?

The belief that the "normal" child has been somehow replaced by the autism is a recurrent theme and could easily be segued into belief that the child is somehow possessed, or that their soul has been removed (Thanks Jenny!). Particularly in those who have displayed so-called "regression" and the non-verbal or those with vocal and physical tics.

Those of us who have more what would be classed as HFA/Aspergers, while not quite as Hammer-horror possessed-seeming commonly display behaviors such as difficulty making appropriate eye contact which can make us appear untrustworthy, deceptive or aggressive and attempts to imitate neurotypical social cues can also do this.

If you don't want to take the religious angle there is also the common misconception about it being a disease - from there it's a short hop to people treating them as lepers.

Utilitarianism

The members of the society might think that undo effort would go into maintaining members of the population that are autistic. Socialized medicine in Iceland has either eliminated or nearly eliminated Down syndrome by enforcing public policy in medicine to require doctors to guide expecting mothers towards terminating pregnancies with Down syndrome due to the cost associated with the provision of national medical resources. Combine the frugality of a poor nation with the likely stress associated with those that have mental illness on the medical community, and all you need is a change in moral viewpoint on what is or isn't murder. Look no further than Nazi Germany for how the definition of murder can be changed for adults in a society.

Fear

This is a big one. A government or ruling class could convince the entire population that "the autistic" have no emotion like normal human beings and are thereby violent, unhinged and no capable of existing in society. Those with autism often have outbursts that without the correct understanding could be viewed in a very twisted light. Many autistic individuals lack the ability to emote like those without autism. A government could hype up the population around psychopathy and the idea of psychopaths with no emotion being "terrifying" (just look at the recent sociopath scare on YouTube). It would not be difficult to make a jump from psychopathy to autism with a population that didn't have the best education (poorer countries poorer education generally speaking).

Fear and Utilitarianism

It might serve you well to utilize class-ism to the extent of these two separate motivations. The rich or ruling class looking at the health care costs and loss of productivity in families that have an atypical person, and the poor masses being convinced of a crazed maniac caricature of the autistic much like what was done with Marijuana smokers with "reefer madness" or anyone not "Aryan" in Nazi Germany.

I'm assuming that you mean people with obvious autism rather than normal-functioning autism which even with modern diagnosis can be hard to detect. The basis would be misunderstanding autism. Just like people think that autism can be caused by vaccines or that having disabled children can be as a punishment from God/Allah/whoever.

Religious punishment is an easy scapegoat. You or your parents are being punished for something and if society gets to interact with you they might get punished as well. Or any other number of reasons such as their idea of how holy they are could get tarnished by interacting with autistic people.

Society can also find it hard to deal with them and will simply punish them for it. In some cultures we already see that people are ashamed of having children with a disability and they'll be kept inside most of their lives. While others put them in homes away from society or lock them up. We also see a lot of victimshaming and victim punishment such as people who have been subject to rape getting locked up rather than the rapist. It took more than a hundred years and several world wars before a large portion of (western) humanity actually started protecting people with disabilities.

I guess I would be curious as to why it is labeled “autism”in a third world country? It could labeled many other things for many other reasons which would better describe a justification for their persecution. Perhaps, “retarded” or just sick or something. I would add it probably is not a reflection of any intelligent society to “persecute and possibly kill” these people with autism because the evolution of humanity might depend on such minds and personalities. I could see a “demon hunt” on such people if they lack empathy or have been proven to show uncontrollable behavior that lacks explanation. So like I said, “autism” may or may not be a reason. But if this is an idea for a book you are writing, perhaps they (country) has decided evolution is less important than pragmatism. And pragmatism can easily use these “autistic” people as examples or use them as experiments or anything else. They could also persecute them into behavior modification and have them be the ones who hunt themselves down. Which would be a wonderful irony about society. Ha!

Because some of them see through established propaganda and group think and can therefore become very politically dangerous if they manage to make people realize they have been fooled. That would be the real reason. The scapegoat reason? Hmm I don't know what would be easiest. If society is religious, maybe you can claim their social and mental differences are because they are evil or don't have a soul or something.

One possible reason could be that people with autism have been scapegoated to blame for the mass poverty amongst its citizens. Whenever a society hits hard times it will always look for something to put the blame on even if that something had absolutely nothing to do with it.

Another possibility could be that autism is seen to cause more problems for the country's society and that people with autism are better to be gotten ride of. Many "modern" societies have similar trains of thought and many people often treat autism as a burden or something to be avoided with many encouraging people with autism to do their best to control and hide their autistic tendency's. Now this usually isn't so bad in well of societies like in the U.S. or Europe but in a country that is on hard times like the one you have described, people are going to be much less tolerant of such things and will go to extreme lengths to "rectify" the problem.