I have studied homeless mentally ill since around 1985.
My first job in the field involved trying to find veterans with schizophrenia among the mentally ill homeless in Downtown LA. The research director had a grant to study their genes and their brains. First thing he said was that people suffering from schizophrenia and related unclassified psychoses are among the most likely to be homeless, constituting at least 3% of the downtown population, housed or unhoused. All of our research put the incidence of schizophrenia among the homeless in DTLA at 3-5%. As a result of this type of research, we've built suburban housing with mental health units for homeless veterans in California. Where I live, I can see improvement in services (as one of these centers is about half a mile from me - it's beautiful, houses almost entirely mentally ill veterans, including those with dementia). We recently opened a great outpatient treatment center (right next to the police station). We still lack enough beds for everyone, but we recognize that some schizophrenics absolutely won't accept living indoors. There's a local park where LE has decided to manage rather than evict - that too is near my house. So I see many improvements locally but I also fear for the fate of our big cities - where it's not going so well.
Anyway, the reasons so many are homeless and in big cities are two fold.
First, their families either cannot or will not take care of them. As their parents age, the parents can't follow them around and mediate their encounters with the world. Second, big cities provide anonymity and some services unavailable in smaller towns (where a jail cell may be the only place to hold a ranting, floridly psychotic person). One or both parents may be schizophrenic or with some symptoms, as well.
One of the symptoms of schizophrenia (esp. the "negative" schizophrenics) is avoidance of humans and a desire to walk.
Walking schizophrenics are everywhere, but rarely in small towns. Most small towns have 1-2 of this type per 10,000 population, but big cities have way more. Some small towns/families give bus tickets to the big cities. But in general these schizophrenics find their own way into larger areas, where homelessness outpaces resources AND where there are soup kitchens, showers, shelters, meal vouchers, homeless outreach workers, thrift shops, help with getting any source of disability payment, etc, etc. For whatever reason, they walk and walk. I've tracked some people as they wander 20-30 miles in a day or two.
So they end up in big urban centers. For example, Santa Monica has a free clinic (and has had since I started this work) and rapidly became a mecca for homeless from all over the nation (and sometimes, the world). DTLA has similar services. This is the second factor (the pull factor of services, treatment and ways of feeding oneself).
So the push factor (families and small towns rejecting or guiding mentally ill people toward bigger centers) and the pull factor (ability to live unhoused but with access to certain services), result in way more mental illness in bigger places. My own town isn't that big, but we have a large green belt, with year round water. It's basically river beds (two of them) and the unhoused people live (and die) there. In these situations, schizophrenic people meet each other, form communities, form romantic attachments and reproduce.
Schizophrenia is a genetic disorder.
These problems are chronic, as many as 3-5% of any population will suffer from psychosis at some point in their lives or have chronic psychosis. It is higher in big cities, due to the factors above.
It is lower in small towns where, basically, many are driven out or sent away or ignore (by their own families in many cases and the families are often suffering from the exhaustion of compassion and coping - and have other children as well).
Our societal decision in the 80's and 90's to get rid of long term care facilities was a terrible decision.
The non-veteran mentally ill population does not have such services. Indeed, most are incapable of locating services and need to be guided to them over and over. It's in the nature of their illness that they will leave and walk and eat very little and drink whatever liquids they find, etc. They seem to have a hard time living indoors. We don't know for sure, because most of this type of schizophrenic can comprehend human language, but rarely speak it properly (and when they do, it may or may not be intelligible; some of them will write but are effectively mute). I've known several schizophrenics who were "impersonators." One was a family of schizophrenics (mom and dad both schizophrenic; 12 children, all schizophrenic; 4 were psychics; 3 were Elvis impersonators; the rest lived off of those efforts). They truly believed they were psychic and could read minds. They had a little sign in the yard of their ramshackle house (this was in New Mexico), stating palm reading and psychic services. They got occasional gigs as Elvis impersonators.
Neely may have had a belief at times that he actually *was* Michael Jackson (as these Elvis impersonators actually believed that they were Elvis). It's a common symptom of schizophrenia. It's an extraordinarily painful illness, as well. High rates of suicide (and I believe Neely's anger and shouting was part of an overall desire to die - the "I'll take a bullet!" is a classic wish (to die, but in a vague way that they can't operationalize unless someone like police take them down for truly threatening behavior - which is rare; or someone like Penny is involved; trained LE and mental health emergency teams use other methods, naturally). Mental pain is terrible pain, leads to dual diagnosis (drug addiction).
At any rate, such people are indeed scary to many. Where I live, LE is trained to recognize and deal with schizophrenics, I'm proud to say. It's been a big part of my life's work to help provide the training (which is all done internally in the Sheriff's division, nowadays - the goal was to train enough LEO's that they could find great and more experienced teachers with each new generation of officers - and so it has come to pass). But my city is smallish (200,000) and the feeder towns from where the schizophrenics come are mostly close by (a region with about 1 million who are connected by public transport, which the schizophrenic population does use). In my personal life, I have known schizophrenic physicians, engineers, college professors. A good friend (schizophrenic; mother was schizophrenic) was one of the psychiatrists who worked alongside me in the research projects. She ended up devoting her life to the schizophrenic populations of the Pacific Northwest, half of whom came from other parts of the country. She gave me a lot of insights into schizophrenia and was expert in helping medication-refusing patients. She traveled from rural clinic to clinic up until the time she died. Had hundreds of patients. We have a deficit of psychiatrists in the US. The older group is dying off and it is not a popular first choice for newly minted MD's.
New York *is* trying but as one of the world's largest cities, boy, it must seem almost hopeless. There are at least 5 schizophrenics that are from my region who are now in NY (I talk to their parents from time to time). No one knows exactly how they got there (and those are just the parents who *know* where their mentally ill children are; a good friend had a tragic event when his schizophrenic son moved away while in remission; long story, but he ended up being killed by police in Sacramento).
I'm just riffing off your mention of the Big City problem. It is a huge problem. And since not everyone lives in a big city, not everyone cares or wants to divert tax dollars from, say, the federal budget to this problem. I'm not sure it's getting worse, it's been a problem for a long time, but it's possible that population growth in the mega-cities has brought the entire system to where we are now. Which is not a good place.
No one can handle schizophrenia merely by urging "personal responsibility" to the sufferer. It's a disease and it's global. Its causes are known, there are treatments, but it is a painful, difficult illness to treat, everywhere.
IMO.