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  "body": "\n## Example\n\nConsider an age-structured population, and let $a \\ge 0$ denote age.\nLet $l(a)$ be the probability of surviving to age $a$, and $m(a)$ the birth rate of individuals of age $a$.\nAssume $l m$ is nonnegative, integrable on $[0, \\infty)$ and not almost everywhere zero.\nThe birth rate $B(t)$ at time $t \\ge 0$ satisfies a [[renewal equation|EWVVH3EKMR]]\n$$B(t) = G(t) + \\int_0^t B(t - a) \\, l(a) \\, m(a) \\, da,$$\nwhere $G(t)$ collects the births from individuals already alive at time $0$.\nThis is a [[Volterra equation of convolution type|SNFYTC1TTA]] with kernel $k = l m$.\n\nLooking for exponential solutions $B(t) = Q e^{rt}$ of the equation without the term $G$, with the integral taken over all ages $a \\ge 0$, gives the *Euler–Lotka equation*\n$$\\int_0^\\infty e^{-r a} \\, l(a) \\, m(a) \\, da = 1.$$\nIts left side is the [[Laplace transform|0V49JEZW06]] $\\hat k(r)$ of the kernel, so $r$ is a root of $1 - \\hat k(r) = 0$, the denominator of the Laplace transform method.\n\nCheck on an instance: let $l(a) m(a) = b \\, e^{-\\mu a}$ with constants $b \u003e 0$ and $\\mu \\ge 0$.\nFor $r \u003e -\\mu$ the left side is $\\int_0^\\infty b \\, e^{-(r + \\mu) a} \\, da = \\frac{b}{r + \\mu}$.\nIt equals $1$ exactly when $r = b - \\mu$, the birth rate minus the death rate.\n\n## Remarks\n\nThe left side is strictly decreasing in real $r$, where it converges, so the Euler–Lotka equation has at most one real root.\nThis real root is the intrinsic rate of increase of the population.\nThe net reproduction number is $R_0 = \\int_0^\\infty l(a) m(a) \\, da$, the value of the left side at $r = 0$.\nIf $R_0 \u003e 1$, the left side tends to $0$ as $r \\to \\infty$, so there is a real root and it is positive.\nIf $R_0 \\le 1$ and a real root exists, it is not positive.\n\n## Explanations\n\n### L1 human\n\nEach newborn will, over its life, produce offspring at various ages.\nIf the population grows like $e^{rt}$, then births long ago count less, by the factor $e^{-ra}$, compared with births today.\nThe Euler–Lotka equation says that, with this discount, each newborn exactly replaces itself.\nSolving it gives the long-run growth rate $r$ of the population.\n",
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