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The density of individuals at age $a$ evolves by transport with unit ageing speed and loss rate $\mu (a)$. New individuals enter at age zero through the integral birth boundary. Exponential separated profiles lead to the Euler-Lotka equation.

The mean lifetime number of offspring is the birth rate integrated against the probability of surviving to each age. For a continuous age model it is ${\int }_0^{\infty }b(a)\exp [-{\int }_0^a\mu (s)ds]da$. The Euler-Lotka equation compares this with one to locate positive population growth.

Substitution of the separated exponential profile into the age-structured population equation gives $r^{\prime }=-(\gamma +\mu (a))r$. ThereforeFor $C\ne 0$, the birth boundary condition cancels $C$ and becomes the Euler-Lotka equationThe survival probability to age $a$ is $e^{-\mu a}$ for constant death rate. Hence the expected lifetime offspring count, or net reproduction rate, isThe Gamma integral evaluates the Euler-Lotka equation as $Bp!/(\lambda +\mu +\gamma {)}^{p+1}=1$, requiring $\lambda +\mu +\gamma >0$. Its real solution and profile areThus $B^{\ast }=(\lambda +\mu {)}^{p+1}/p!$, and $B>B^{\ast }$ gives positive growth. The profile is a nonzero formal separated solution for all positive $B$; a finite total population additionally requires $\gamma +\mu >0$, equivalently $(Bp!{)}^{1/(p+1)}>\lambda$. A decaying-in-time mode need not have an integrable age profile.

Substitution into the renewal boundary condition and cancellation of $A_{0}T(t)$ gives the Euler-Lotka equationThis is the necessary condition selecting the growth exponent $r$.

If incidence grows exponentially, ${\Delta }_t=C{e}^{\rho t}$, substitution in the infectious disease renewal equation gives the discrete Euler-Lotka equationFor $X\sim \mathrm{Geometric}(p)$ on $\{0,1,\dots \}$, $g_i=\mathbb{P}(X=i-1)=p(1-p{)}^{i-1}$. The finite geometric series therefore yieldsand consequently