By Saeed Mirshekari
February 28, 2014
Abstract
Will Living People Ever Outnumber the Dead? Probably not! It has been shown that
only about 6.5 percent of all people ever
born were alive in 2011 \\\[1]. But the human population on Earth has been growing
very rapidly in the recent decades such that
one could imagine one day in which the number of alive people reaches the number of all
the dead people. Assuming the world's average life expectancy to remain constant and
equal to its current value (68 years) and the
world's annual growth rate to stay constant
but just a bit less than its current value (1%),
we show that living people will never outnumber the dead.
Growth of Living Population
The human population has started to grow extremely
fast only since the last few centuries, as you can see
in Figure 1. The rate of population growth has
obviously changed over time. But, because of all
the human improvements and developments in the
modern world it is not unlikely to reach a constant
rate of population growth and life expectancy in the
future.
The ''annual population growth rat'' (r) is the
rate at which the number of individuals in a population increases in a year as a fraction of the initial
population. It is very easy to show that for a constant population rate of r the living population will
be growing as \\\[eq.1]
 = N_0 (1 %2B r) \times t)
where 
is the living population at the initial time.
Population in the world is currently growing at a rate of about 1:14% per year \\\[2] and the current world's
average life expectancy is 67.88 years \\\[3].
Figure 1: The estimated size of human population from
10,000 BCE-2000 CE. \\\[3]
How Many People Are Dead?
Knowing only the constant values of current annual
population growth rate, r, and the world average life
expectancy, 
, there is a simple way to estimate
the number of dead people at time 
.
Suppose after 
the annual growth rate and
the average life expectancy reach their constant values. The number of living people at this time is
given by Equation (1) as )
. Now the question is that after a single period of the average life
expectancy, i.e. at 
, how many people
have died in average in this time period? The answer is simply Alive(
). In other words, in average, the number of people who have died since 
until
is same as the number of people who
were alive at . Without losing any generality
we can set 
and generalize this idea to the
future times in which 
and write \\\[eq.2]
 = \sum^{N-1}_{n=1} Alive(n [ L ]) %2B Dead (t_0))
where N is an integer number and Dead(t0) is the
number of all people who have died at any time in
the history before t = t0. Using Equation (1), the
above summation can get as simplified as \\\[eq.3]
 = N0\frac{(1 %2B r)^t - 1}{(1 %2B r)^{[ L ]} - 1} %2B Dead(t_0))
Critical Values of 
and 
Based on Equations (1), (3), it's not difficult to show
the living population ultimately outnumber the dead
if and only if:
 [ L ] > 2)
For example, with the current value of the world average life expectancy, i.e. 67.88 years, the living population can outnumber the dead if and only if the annual population growth rate is not less than the critical value of 1.02%. This number is only slightly smaller than the current value for the world i.e. 1.14%. Although we have considered r to be
constant, recent studies \\\[2] predict a small decrease in the rate of population growth to a value less than 1% in close future. This small decrease would be enough to conclude that the living population never outnumber the dead with the current world average
value of life expectancy.
On the other hand, as another example, if we fix the population growth rate at its current value, the world average life expectancy has to be greater than 61.12 years to ultimately allow the living population outnumber the dead. Any values of r and less than their critical values cause the dead population outnumber the living forever.
References
[1] Carl Haub, http://www.prb.org (2011)
[2] http://www.worldometers.info
- Thanks to M. Le Delliou for useful discussions.
- This calculations might be vey well known and we probably
just have rediscovered them here.
Saeed Mirshekari
Saeed is currently a Director of Data Science in Mastercard and the Founder & Director of OFallon Labs LLC. He is a former research scholar at LIGO team (Physics Nobel Prize of 2017).