Let's say that humans were biologically immortal. Once they reach adulthood, they effectively stop aging. Due to disease though, the average lifespan is 100 years. (In practice, your access to healthcare would have a huge effect on your lifespan, but we'll ignore that for the purpose of this question, since it would make it much more difficult to answer.) For ease of answering this question, we'll say that distribution of lifespans is exactly an exponential distribution with mean 100 years.

Additionally, humans have hit the carrying capacity of the Earth, and so only reproduce to replace those who died. In particular, we'll assume that some sort of steady-state has been reached.

My question is, given these constraints, what would be the distribution of ages be? Basically what I'm looking for is an equation or graph that tells you how many people are at a given age.

Note that I'm talking about the current age, not the lifespan.

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    "but we'll ignore that for the purpose of this question". You can't ignore it, since that's intrinsic to the answer. (Afghanistan and Monaco will have fundamentally different age-of-death distributions.) – RonJohn Nov 13 at 1:46
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    Just because you say to ignore a factor doesn't mean a reasonable answer can ignore that factor. (And "equal healthcare" doesn't say anything about the quality of the healthcare.) – RonJohn Nov 13 at 2:17
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    at any rate, the longer one is alive, the likelier one is to contract a serious disease, just on the basis of extended risk. but the risk on any particular day is no more than any other (in contrast to mortals whose susceptibility increases). quantifying that would be way to much work for me, though. its an interesting question. i do expect that upon maturity, the probability graph is linear. – theRiley Nov 13 at 4:25
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    Really interesting answers. But im not sure on how theres enough information to answer that question. Really curious. – user10645073 Nov 13 at 9:11
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    As it stands this question should be answered by a calculator or predictive modelling, not humans. You've declined or ruled out the factors that would make human interpretation useful. – nullpointer Nov 13 at 13:02
up vote 11 down vote accepted

Curve describing distribution of age of death is: $$ \frac{e^{-\frac{x}{100}}}{100} $$ Use whatever plotting program you want to have it visualised

It's easily found by normalising to $1$, finding average and using preset average of 100: $$ \int_{0}^{\infty}a*e^{-x*b}=\frac{a}{b}=1 $$ $$ \frac{\int_{0}^{\infty} a*x*e^{-x*b}}{\int_{0}^{\infty}a*e^{-x*b}} = \frac{\frac{a}{b^2}}{\frac{a}{b}}=\frac{1}{b}=100 $$ hence $a=b$ and $b=\frac{1}{100}$.

Assuming that your population is stable and has not experienced any advancements in medicine in last 2000 years (assuming population of 1 billion, age of 1612 is moment when expected number of people drops below 1, so we can safely assume that no one survives to age of 2000), distribution of ages in population will look exactly like age of death distribution, just multiplied by population. For 7 billion, it's: $$ 7*10^7 e^{-\frac{x}{100}} $$ This is because you made some very simplifying assumptions, which you forgot to take to logical end. Making distributions of lifespans exponential you have de facto introduced yearly death rate (about 1%, because $\frac{e^{-\frac{1}{100}}}{100} \approx 0.99$), independent to anything. Infant has same chance to die next year as 1500 years old great$^{70}$grandfather (or 1000000 years old). Every single person has same chance to die per year. Every generation will shrink by fraction equal to chance of death. It also means that total number of deaths is constant and equal to size of population times chance of death, and as infants born to replenish population age, they die off each year in exactly same exponential way, year younger bracket is smaller by chance of death fraction and year older group is larger by chance of death fraction. Because of that, age distribution is exact same as lifespan distribution.

If there have been any changes in medicine that might have impacted lifespan (or rather yearly chance of death, because that metric is more useful in your population) or any wars and catastrophes that might have killed abnormally many people, there is insufficient data to provide answer; you have to define those advancements and/or wars/catastrophes.

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    @PyRulez Please do not introduce errors when editing answers. You removed normalisation constant ($\frac{1}{100}$) in your edit. – M i ech Nov 13 at 14:06
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    Oh whoops, sorry. Was not intentional. – PyRulez Nov 13 at 14:07
  • Also, "If there have been any changes in medicine that might have impacted lifespan (or rather yearly chance of death, because that metric is more useful in your population) or any wars and catastrophes that might have killed abnormally many people, there is insufficient data to provide answer; you have to define those advancements and/or wars/catastrophes." At that point, I would just do a discrete simulation myself. I'm not going to make worldbuilding.se do all my math. – PyRulez Nov 13 at 14:08
  • Also, its kinda counterintuitive that they are essentially the same. You'd think that average age would be less than average lifespan, since the age of any living person is less than their lifespan. – PyRulez Nov 13 at 14:52
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    @Arthur: If everyone dies at exactly 100 then your chance of dying isn't uniform, which is one of the constraints of the question and the one that makes that point work. Think of it as a sampling exercise. If the 'dead' people are sampled from the population of 'alive' people with a uniform chance of sampling then the 'dead' sample will inevitably look like the 'alive' distribution, as long as you have enough people and the 'alive' distribution doesn't change over time. If you only sample people at age 100 then of course the two won't look the same. – Joe Bloggs Nov 13 at 16:38

Since you want an exact Exponential Distribution with a mean of 100, I just found an online calculator and put in it for you. Just multiply the Y Axis by your total population.

I know this might be viewed as a tongue in cheek answer but you did literally say we'll say that distribution of lifespans is exactly an exponential distribution with mean 100 years and In particular, we'll assume that some sort of steady-state has been reached.

Now if you factored in something different, like the fatality rate from ages 0-5 is 500% higher and the fatality rate from ages 100+ is 500% higher, you might get some more interesting looking curves. But you have basically told us the answer in your question. An exponential distribution with mean 100 years.

https://mathcracker.com/exponential-probability-calculator.php#results

## Heading ##

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    This. Exactly, in fact. – Joe Bloggs Nov 13 at 13:36
  • Sigh, the lifespan (i.e. age of death) follows an exponential distribution. I was asking about the current age. – PyRulez Nov 13 at 13:45
  • @PyRulez Your current rules specify an exponential distribution, and a mean age of 100 years. This predetermines the shape of the distribution to e^-(1/100). BUT: How many people are there in any given age group? This is not solveable unless you tell us how many people there are in total, which you did not do. You also did not specify whether this is steady-state or not (as the people stop aging in a fertile bracket, it would need some pretty good family planning to not be runaway...) – bukwyrm Nov 13 at 14:03
  • @bukwyrm The question said mean lifespan of 100 years, not mean age of 100 years. – PyRulez Nov 13 at 14:06
  • @PyRulez If, after a looong time, you ask everyone who lived how old they were when they died, and if this is a steady state, mean age is mean span. If this is not steady state, we need to know in what way - did immortality suddenly set on? at what historic population distribution & size? what was the breeding pattern afterwards? – bukwyrm Nov 13 at 14:17

Asymptotic curve

1% death rate

The average age of this population is 99. I did this in a way I could wrap my head around it, using excel.

Imagine that each year 1% of each birth cohort dies of disease or accidents. You have a population of 1,000,000. Your biggest cohort is of course infants: you have 10000 of them and they are 1% of your population. 1% die and your population of 1 year olds is 9900. 1% die and your population of 2 year olds is 9801.

Each year, the 1% that is subtracted from the population is a smaller amount. That gives you an asymptotic curve as in the diagram above.

I wish my math were good enough to provide a multiplier such that for any age you could figure the # of individuals of that age in the population of 1,000,000. But I did it up in excel to make the graph. Lacking elegant math, I would make the graph and then use the curve to approximate number of individuals of any given age.

In my model (with average age 99; I think because of rounding) there were 3660 100 year old individuals comprising 0.366% of the population. There were 9 700 year olds and, of course, 1 1000 year old.

  • 2
    I'm not sure the actual numbers being presented here are either correct or relevant but the structure of the resulting population almost certainly is. – Ash Nov 13 at 9:41
  • This is known as a geometric distribution, with the parameter $p$ as the chance of dying in any specific year. – A Simmons Nov 13 at 9:48

I don't think there's enough information available to give a concrete answer to this. Maybe someone else can, but I don't think it's possible.

So much depends on things we don't know, such as how long this has been going on, and when/how this immortality began.

Here's what I mean: If this technology/magic were instituted immediately it means that, in general, we'd expect each person to die, a hundred years from their DOB. (As an average) If this took place in the modern U.S. the generations would be locked into rolling waves.

For example, according to wiki 1.8% of the population is 85 or older. Based on your criteria, when that generation dies, it is replaced with exactly the same number of new births.

So what happens is that the age distribution numbers are almost the same, but switch places. That 1.8% drops to the 'bottom' of the age distribution. 80-85 years later, the wave 'peaks' and what used to be 15-19 year olds at ~7% of the population are now all about 100 years old.

It won't be exact, but if that's close to how one scenario would play out. You'd have to remove the percentages of deaths that are from most 'natural causes' while keeping deaths by accident, contagious disease, homicide and suicide, you'd need some noise, and the math would be pretty granular, but stick to the general 'wave' pattern.

Or

It takes effect by socioeconomic strata, or by state mandate, etc. And depending on how long this has been going on, given the admitted population controls, it could be literally anything you wanted it to be. A society could have it locked at 1% birth rate, every year, into eternity.

  • But the world is living longer, and having fewer children. That's already changing the distribution. – RonJohn Nov 13 at 3:04
  • @RonJohn which is entirely irrelevant, if the premise of this question takes place immediately. – user49466 Nov 13 at 3:51
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    why should we assume the change takes place immediately. why not assume a gaussian distribution, or even a flat distribution and let things equilibrate? i'm not a math wizard, though, so maybe i'm all wet here. that assumption just seems arbitrary. best of all would be an 'in the wild' human population in the pre-industrial age, or paleolithic age. – theRiley Nov 13 at 4:43
  • @Theriley I'm not assuming that. I stated that how the change takes place, how long it's been, etc, will effect the distribution, and used this as an example. It's up to the OP to set the ground rules. – user49466 Nov 13 at 4:46
  • i see. thanks for the clarification. – theRiley Nov 13 at 5:44

The first thing we have to look at is a population pyramid. Below is the 2015 pyramid from CIA world factbook data.

Population pyramid 2015 https://www.indexmundi.com/world/age_structure.html

We currently have a hint of population growth at the bottom, then dropping off through to an upper limit of around 100. Median age of 30.4. (You've asked for average age rather than median age, this is far harder to calculate.)

Since you have stated a static population, with good healthcare in peacetime, the pyramid should be less of a pyramid and more of a column, starting to taper off where aggregate accident rates and incurable medical conditions start to kill people off.

An aging population would be narrow at the bottom with a bulge higher up. A growing population is more a classic pyramid.

A quick and dirty answer would be to extend the above pyramid age ranges 3fold, but of course this pyramid is limited by natural age limitations hence would not be valid.

The shape you're looking for is going to be a cumulative normal distribution of death rates rather than a clean exponential decay as others have suggested.

Lets say there a thousand humans (a really small earth). If we distributed them roughly across ages so they balance out at 100. I could have a:

  • very dense demographic with lots of 100 year olds.
  • a split demographic with many young individuals say 10-20, and many older individuals 180-190.
  • or some kind of flat demographic ~5 people of each age between 0 and 200.

That is the distribution you've described. Take your pick.

If you want something more realistic... (and grossly oversimplified).

Then lets consider mortality. Without aging doing any killing by itself, you'll find that we die purely from disease and accidents. This probably won't change the mortality rate very much. I'll draw some figures from Australian adult deaths. So roughly 0.0468% to 0.0782% of the population dies due to something.

That's about 46-78 dead humans a year (in our thousand human population). So about that many births per year.

  • 78 one year olds
  • 73 two year olds
  • 65 three year olds
  • 63 four year olds
  • 20 five year olds
  • 15 six year olds
  • 80 six year olds
  • 45 seven year olds
  • 46 eight year olds
  • 48 nine year olds
  • 20 ten year olds
  • 72 eleven year olds
  • 12 twelve year olds
  • 22 thirteen year olds
  • 46 fourteen year olds
  • 23 fifteen year olds
  • 31 sixteen year olds
  • 87 seventeen year olds
  • 44 eighteen year olds
  • 38 nineteen year olds
  • 22 twenty year olds
  • 56 twenty-one year olds
  • 64 twenty-two year olds

and because of the small population cap, 51 people are executed at random. But if you raised it, to something more manageable, you'll see that this roughly makes a flat population that will slowly taper out (at the rate of a 5 hundredths of a percent).

Lets see what this population looks like, but with no ritualised forced deaths, and every member living till natural causes (like accidents) kill them. Lets also keep that median straight on top of 100 (roughly speaking).

  • A is the number of 100 year olds
  • B is the total number of years past 100 that the oldest individual (roughly) is.
  • C is half the total population.
  • D is the death rate.
  • E is the eldest member.
  • Y is the total number of new borns.
  • X is the difference between the total 100 year olds, and the total newborns.

  • X = 100 * D

  • Y = A + X
  • A = B * D
  • C = 100 * X / 2 + 100 * A
  • C = A * B / 2
  • E = 100 + B

This is represents two triangles and a rectangle like:

 *
 **
 * *
X*  *
 *   *
 +****+
 *    **
 *    * *
A*    *  *
 *    *   *
 *    *    *
 *****+******
  100    B

So assuming D = 0.0005

  • X = 0.05
  • A = B * 0.0005
  • B = (1/0.0005) * A
  • B = 2000 * A
  • C = A * B / 2
  • C = A * (2000 * A) / 2
  • C = 1000 * A * A

  • C = 100 * 0.05 / 2 + 100 * A

  • C = 100 * A + 2.5

  • C = C

  • 100 * A + 2.5 = 1000 * A * A
  • 0 = 1000 * A * A - 100 * A - 2.5
  • A ~= 0.12071...

  • A = B * D

  • B = A / D
  • B ~= 0.12071 / 0.0005
  • B ~= 241.42

  • Y = A + X

  • Y ~= 0.12071 + 0.05

  • E = 100 + B

  • E = 100 + 241.42
  • E = 341.42

So the oldest thing is roughly 341.42 years old. And their lives one 17th of a newborn. For a median age of 100. (Statistically speaking). That is a total average population of 341.42 * 0.17071 / 2 ~= 29.141 individuals.

29 people isn't a satisfying population. Its an almost extinct one. Lets take the approximately 7 billion people alive today, and give them a shot of immortality. and institute the replacement only birth rate, with the constant death rate of 0.0005. (Lets say immortality makes people more adventurous and take on risks, without learning much about avoiding them). Fast forward many years and allow the natural randomness to even out a little, (so that I can use the simple triangle model and ignore every other event).

 *
 **
 * *  C
A* D*
 *   *
 +****+
    B
  • D = 7,000,000,000
  • D = A * B / 2
  • C = A / B
  • C = 0.0005

  • A/B = 0.0005

  • A = 0.0005 * B
  • 7,000,000,000 = A * B / 2
  • 7,000,000,000 = 0.0005 * B * B / 2
  • 7,000,000,000 * 2 / 0.0005 = B * B
  • B * B = 28,000,000,000,000
  • B ~= 5,291,502.62
  • A = 0.0005 * B
  • A = 2645.75

So natural causes gives the oldest person at about 5.3 million years old. Most people would have "accidentally" died by now.

Also there are around 2645 new infants born every year. This means a mere 6 average-large sized schools of each level of education are sufficient for educating the entire population. Imagine that, overnight most of the worlds schools can just be shutdown. Of course there will now be a large upswing in golf-course requirements.

Hope this helps paint a very crude picture.

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