Demographics

% World population from the World Bank (live), fitted with a logistic curve
d = webread('https://api.worldbank.org/v2/country/WLD/indicator/SP.POP.TOTL?format=json&per_page=100');
rows = d{2}; yr = []; pop = [];
for k = 1:numel(rows)
  if ~isempty(rows{k}.value), yr(end+1) = str2double(rows{k}.date); pop(end+1) = rows{k}.value / 1e9; end
end
[yr, idx] = sort(yr); pop = pop(idx);

% logistic model P(t) = K / (1 + exp(-r (t - t0))), fitted by least squares
model = @(p, t) p(1) ./ (1 + exp(-p(2)*(t - p(3))));
err = @(p) sum((model(p, yr) - pop).^2);
p = fminsearch(err, [11, 0.03, 1990]);
tt = 1960:2100;
plot(yr, pop, 'o', 'MarkerSize', 4), hold on
plot(tt, model(p, tt), 'LineWidth', 2), hold off
xlabel('year'), ylabel('billions')
legend('World Bank data', sprintf('logistic fit, ceiling %.1f bn', p(1)))
title(sprintf('World population: %.2f bn in %d', pop(end), yr(end)))
subtitle('A simple curve fit, not a projection: the UN uses age-structured models')

% Stylised age structures (illustrative shapes, not a specific country's data):
% a young, fast-growing population and an older, ageing one. Percent of population per 5-year group.
ages = 0:5:95; groups = numel(ages);
young = 14 * exp(-ages/22); young = 100 * young / sum(young);
old = (4 + 1.5*exp(-((ages - 55)/18).^2)) .* exp(-max(ages - 65, 0)/11); old = 100 * old / sum(old);
labels = arrayfun(@(a) sprintf('%d–%d', a, a + 4), ages, 'UniformOutput', false);

subplot(1, 2, 1)
barh(-young/2), hold on, barh(young/2), hold off
yticks(1:2:groups), yticklabels(labels(1:2:end)), xlim([-8 8])
title('Young population'), xlabel('% (men ← → women)')
subplot(1, 2, 2)
barh(-old/2), hold on, barh(old/2), hold off
yticks(1:2:groups), yticklabels(labels(1:2:end)), xlim([-8 8])
title('Ageing population'), xlabel('% (men ← → women)')
sgtitle('Population pyramids')

% Stylised four-stage model: death rates fall first, birth rates follow decades later
t = 1750:2050;
death = 12 + 23 ./ (1 + exp((t - 1880)/25));       % per 1000 people per year
birth = 12 + 26 ./ (1 + exp((t - 1950)/20));
growth = birth - death;
pop = cumprod([1, 1 + growth(1:end-1)/1000]);       % relative population size

subplot(2, 1, 1)
plot(t, birth, t, death, 'LineWidth', 2), hold on
fill([t fliplr(t)], [birth fliplr(death)], 'g', 'FaceAlpha', 0.15), hold off
legend('birth rate', 'death rate', 'natural growth'), ylabel('per 1000 per year')
title('Demographic transition (stylised)')
subplot(2, 1, 2)
plot(t, pop, 'LineWidth', 2), ylabel('population (× start)'), xlabel('year')