Xjyuk

Modern Earth; a series of connected events (natural disasters, mass migration, drug-resistant pandemic, collapse in biodiversity, armed conflict) have led to a significant reduction in human population over a single generation.

Does "civilisation" continue? Societies could certainly adapt to short-term disruptions—but if the complex, global, inter-connected supply chains upon which they currently depend do not recover sufficiently quickly, would they not collapse entirely?

For example, a number of contemporary industries (unfortunately) depend upon oil: not only is most transportation fuelled by petroleum; but most fertilisers, pesticides, pharmaceuticals and plastics are also derived from it. If oil extraction, refinery and distribution were to suddenly cease for a sustained period (before such dependent industries have identified and adapted to alternatives), oil reserves would deplete and those industries themselves could collapse. That in turn could lead to a collapse of further dependent industries, including the manufacture of machines used in virtually any given supply chain.

Do we have any idea how many people are required to keep "it" (the modern industrial world) functioning?

Please give due consideration to the fact that disaster-struck areas may be uninhabitable, with their populations migrating en-masse and seeking refuge in other areas—thereby placing further strains on the system.

Clarification

There have been some very probing and helpful comments, which have prompted me to spend some considerable time reflecting on exactly what my question is and this edit is an attempt to elaborate. My thanks to all those commenters who contributed to this discussion.

By "significant reduction in human population over a single generation", let's assume the reduction is unforeseen, uniform across all demographics and complete within 10 years. To be clear, it is an "unmitigated apocalypse".

By "maintain the modern industrial world", I meant maintain our current technology. It strikes me that a good measure of this is the smartphone—it requires mining and refining many minerals, including some that are so rare that viable ore has only been found in a few locations on the planet; even once refined, maufacture of some parts (e.g. 7nm-process semiconductor fabrication) is so specialist that the number of capable facilities can be counted on one hand; even once assembled and distributed, their function depends on an infrastructure of components (antennæ, switches, routers, servers, etc) that themselves require such parts and maintenance thereof.

I'm going to respond to this with a framing challenge, because as it stands I think you're asking the wrong question.

The issue is that the modern industrial world is a product of population DENSITY rather than total population. This may seem semantic but in a post-apocalyptic environment it's very meaningful. Modern industrialism depends on and benefits from massive economies of scale that accrue from having lots of people all close enough together to quickly and easily exchange goods and services.

In a post-apocalyptic scenario, if you're envisioning two million survivors scattered across the entire planet, they're all going to be spending pretty much all of their time just keeping themselves fed and sheltered, and it's straight back to a hunter/gatherer economy worldwide.

If, however, you're envisioning two million survivors concentrated in a specific area that (for example) quarantined itself before anybody could be infected with the zombie virus, then you'd potentially have enough critical mass of different kinds of expertise to allow cooperative maintenance and task specialization: e.g. you have a bunch of people whose entire job is providing the food for everybody. You have another bunch whose entire job is maintaining power/water/electric. You have another bunch who kill the zombies before they swim across the river.

This is basically how urban societies developed in the first place, and the same rules would apply in an apocalypse.

To more directly answer your question: The larger an area your population of survivors is spread across, the more people you need. The more widely distributed the resources you require are (food, water, factories, sources of raw materials), the fewer people you need.

The number could potentially be as low as a few hundred in a commune out in the middle of Nowhere, Idaho that already has its own self-sufficient sources of food, water, and power, and a smattering of skilled electricians, mechanics, and engineers.

I've used a personal analogy for this, which I call "The brake pad problem." I estimate (without any serious research) that maybe a few hundred people around the world really know what it takes to manufacture an automobile brake pad. If you kill them all off, we're in serious trouble at least until a new process (or a reconstruction of the original) can be completed.
The idea is that everyone is incredibly specialized, and our outputs are very interconnected. This makes the whole system unstable. My WAG, then, is less than 5% of the people in industrial countries/regions would cause a complete collapse.

You might want to read Lucifer's Hammer for more ideas.

This answer might sound mean, but most people can be killed off without affecting the industries, if you choose the right people. You only need the people directly employed in the industries, the people supporting the cities and governments where those people live, the people providing supplies to those. Currently that's basically the West, factory places in Asia and some others scattered around the world. Probably around a billion people needed to keep everything running as it is, but could be less.

Although it depends where and in how many places you want to see modern civilisation. In all the places it exists around the world? In a select few nations? Or is one city enough?

What we DON'T need

Let's start the answer in reverse. Who we can dismiss? Who we don't need to maintain modern technological levels. A good part of humanity if I says so. No offense for anyone.

• Employees of huge companies who provide service to the masses? They
  can be significantly down-scaled. If functionality, the maintaining
  of technology instead of profit the purpose, only their engineering
  team. You can scrap their HR, sales, and so on departments.


• Most services and its workers too - you don't need 20 kind of chips
  and cars and so on, to maintain modern technology levels.


• Any kind of Media related profession - SCRAP!


• Governmental bodies - Significantly downscaled, based on the
  situation and type of government.


• Homeless, unemployed, and so on... they don't contribute to modern
  technology, so they too can be dismissed.

Other factors: Human intellect, productivity, automation

• Automation - we are heading towards a heavily automated world, most
  mundane task can be or will be replaced by robots.


• Human intellect - Automation means, you can dismiss almost all the
  heavy-lifters. You will however need highly educated and intelligent
  people to operate and maintain the machinery.


• Productivity - much depends on the disposition of the people. Are
  they lazy? Hard-workers? Do you need police to manage big masses of
  them and so on. This too will affect the final number.

The final factor is, in this scenario, do all human capable of reaching the same intellectual level? Do they born equal in that regard? Or for example every hundredth people has high enough intellect?

The envisioned technology level of (modern) also changes to final number. Steam-engines are easier to understand than nuclear. A possibly future automation could reach fully automated factories with a few overseer required.

To calculate an accurate number without these factors, is beyond my capability.
At guess, a few hundred thousand or million, highly specialized human blessed with great intellectual capability could, in theory, all over the world operate and maintain the necessary machinery to keep it going.