Euler-Lotka equation

The Euler-Lotka equation - sometimes just named after Euler or Lotka - is an equation that relates important quantities of the population dynamics in a population broken down by age. It is used, for example, in ecology and epidemiology and is named after Leonhard Euler (1760) and Alfred J. Lotka (1911).

Change of populations

The development of female populations over time is considered as a function of age. It is assumed that there are always enough male partners available for reproduction. Be

• ${\displaystyle w(a)}$the proportion of female individuals that an age reached ${\displaystyle a}$
• ${\displaystyle m(a)}$ the average number of female offspring of a female individual of age ${\displaystyle a}$
• ${\displaystyle r}$ Lotka's intrinsic rate of natural growth in the female population (also known as specific survival rate).

Then the Euler-Lotka equation reads:

${\displaystyle \int _{a=0}^{\infty }w(a)\cdot m(a)\cdot e^{-r\,a}\,da=1}$

It results from the following consideration. The number of births to mothers aged at the time equal to the number of births at the time (number of mothers, including those who do not to the time of survival) multiplied by the average number of births to mothers age and summed or integrated over For all ages of the mothers the integral equation results (a linear homogeneous integral equation of the Volterra type 2nd type): ${\displaystyle a}$ ${\displaystyle t}$ ${\displaystyle t-a}$ ${\displaystyle t}$ ${\displaystyle m(a)}$ ${\displaystyle a}$ ${\displaystyle a}$

${\displaystyle B(t)=\int _{0}^{\infty }B(t-a)\cdot w(a)\,m(a)\,d\,a}$

with the birth rate (births per unit of time) at the time . If you take an exponential approach for the population and thus also for the development of the birth rates: ${\displaystyle B(t)}$ ${\displaystyle t}$

${\displaystyle B(t)=B_{0}\cdot e^{r\,t}}$ and if this is used, the result is:

${\displaystyle B(t)=\int _{0}^{\infty }B(t)e^{-r\,a}\cdot w(a)\,m(a)\,d\,a}$

(Renewal equation for the birth process) and thus by shortening the Euler-Lotka equation. ${\displaystyle B(t)}$

Usually the discrete version is used (age in years):

${\displaystyle \sum _{a=0}^{\infty }w(a)\cdot m(a)\cdot e^{-ra}=1}$

That can also be written:

${\displaystyle \sum _{a=\alpha }^{\beta }w(a)\cdot m(a)\cdot e^{-ra}=1}$

where and indicate the time limits of fertility of a female individual. ${\displaystyle \alpha }$ ${\displaystyle \beta }$

The expression

${\displaystyle R=\sum _{a=\alpha }^{\beta }w(a)\cdot m(a)}$

or in the continuous version

${\displaystyle R=\int _{0}^{\infty }w(a)\,m(a)\,d\,a}$

is the average total number of female offspring of a female individual in the course of her life and is called the net reproduction rate . Population growth occurs when is greater than 1. ${\displaystyle R}$ ${\displaystyle R}$

The product of the probability of survival and the fertility rate is often considered:

${\displaystyle n(a)=w(a)\,m(a)}$

for which the Euler-Lotka equation is

${\displaystyle 1=\int _{0}^{\infty }n(a)\,e^{-ra}\,d\,a}$

If you normalize by dividing by , you get the distribution density of the generation time: ${\displaystyle n(a)}$ ${\displaystyle R=\int _{0}^{\infty }n(a)\,d\,a}$

${\displaystyle g(a)={\frac {n(a)}{R}}}$

and the Euler-Lotka equation can be written:

${\displaystyle {\frac {1}{R}}=\int _{0}^{\infty }g(a)\,e^{-ra}\,d\,a=Mg(-r)}$

where the Laplace transformation is the distribution density of the generation time g at the point . In statistics it is also referred to as the moment generating function . The equation can be interpreted in such a way that this equation between the reproduction rate and growth rate clearly gives the form of the distribution of the generation time and vice versa. ${\displaystyle Mg(-r)}$ ${\displaystyle -r}$ ${\displaystyle R}$ ${\displaystyle r}$

It can be shown that the Euler-Lotka equation, which has the form of a characteristic equation, has exactly one real root , since the function on the right-hand side of the Euler-Lotka equation is strictly monotonically decreasing. So that is must be positive . All other roots come in complex-conjugate pairs , wherein the real root dominates the complex roots: . Although they lead to fluctuations in the birth rate, they are asymptotically dominated by the real eigenvalue and the population asymptotically strives towards a stable age distribution. While the determination of the real eigenvalue with Lotka's method usually does not cause any problems, the treatment of the complex roots is cumbersome. Willy Feller provided a more elegant treatment of the problem than the elementary methods of Lotka in 1941 via the Laplace transformation. ${\displaystyle r_{1}}$ ${\displaystyle \psi (r)}$ ${\displaystyle 1=\psi (r)}$ ${\displaystyle r_{1}}$ ${\displaystyle R_{0}=\psi (0)>1}$ ${\displaystyle r_{j},r_{j}^{*}}$ ${\displaystyle r_{1}>Re\,r_{j}}$

Age structure analysis can also be handled in vector form with so-called Leslie matrices, with entries for fertility and survival rates for each age group.

Epidemiology

We are looking for an equation that connects the base reproduction number with the initial exponential growth of the epidemic (growth exponent ). ${\displaystyle R_{0}}$ ${\displaystyle r}$

For this purpose, an instantaneous infection rate is first expressed, the mean number of secondary infections of an individual in the time after the point in time at which they themselves were infected (age of the infection). ${\displaystyle R_{0}}$ ${\displaystyle n(a)}$ ${\displaystyle a}$

${\displaystyle R_{0}=\int _{0}^{\infty }n(a)\cdot d\,a}$

Normalized via the distribution density of the serial interval ${\displaystyle n(a)}$

${\displaystyle g(a)={\frac {n(a)}{R_{0}}}}$

so you get

${\displaystyle \int _{0}^{\infty }g(a)\cdot d\,a=1}$

${\displaystyle g(a)}$is variable over time and, for example, has a maximum of around 3 days in influenza and has already dropped sharply after 10 days.

Let the number of new infections at the moment for which an exponential approach was chosen at the beginning of the epidemic: ${\displaystyle B(t)}$ ${\displaystyle t}$

${\displaystyle B(t)=B_{0}\cdot e^{r\,t}}$

Then one can write the integral equation:

${\displaystyle B(t)=\int _{0}^{\infty }B(t-a)\cdot n(a)\,d\,a=\int _{0}^{\infty }B(t)e^{-r\,a}\cdot n(a)\,d\,a}$

and thus the Euler-Lotka equation for epidemiology:

${\displaystyle 1=\int _{0}^{\infty }e^{-r\,a}\cdot n(a)\,d\,a}$

If you insert one you get for the basic reproduction number: ${\displaystyle n(a)=g(a)\cdot R_{0}}$

${\displaystyle R_{0}={\frac {1}{\int _{0}^{\infty }e^{-r\,a}\cdot g(a)\,d\,a}}}$

The expression in the denominator has the form of a Laplace transform of so that one can write: ${\displaystyle Mg(-r)}$ ${\displaystyle g(a)}$

${\displaystyle R_{0}={\frac {1}{Mg(-r)}}}$

If one restricts the distribution of the serial interval (which is an approximation for the generation time ) to a fixed value for the generation time ( i.e. turns the distribution into a delta function ), one obtains ${\displaystyle T_{G}}$

${\displaystyle R_{0}=e^{r\,T_{G}}}$

Often this is approximated as what corresponds to the value from the SIR model (see the article Basic reproduction number ). In the SIR model, both the mean infectious period and the mean generation time can be used in this formula. Depending on the selection of the distribution of the serial interval, different formulas are obtained for the relationship between the number of reproductions and the growth rate : ${\displaystyle R_{0}=1+rT_{G}}$ ${\displaystyle R}$ ${\displaystyle r}$

In the literature there are also other Euler-Lotka equations for the relationship between the number of reproductions and the growth rate , but these are always given by specifying the distribution density of the generation time via its Laplace transformation. For example, assuming a normal distribution with mean and standard deviation (Dublin, Lotka 1925): ${\displaystyle R}$ ${\displaystyle r}$ ${\displaystyle g(a)}$ ${\displaystyle T_{g}}$ ${\displaystyle \sigma }$

${\displaystyle R=e^{rT_{g}-{\frac {1}{2}}\,r^{2}\,\sigma ^{2}}}$

One can also determine empirical distributions of the generation time in a histogram. Let the relative frequencies for the generation time between the limits for the age in the histogram be: ${\displaystyle a_{i},a:a_{i+1}}$ ${\displaystyle y_{i}}$

${\displaystyle R={\frac {r}{\sum _{i=1}^{n}y_{i}(e^{-ra_{i-1}}-e^{-ra_{i}})\cdot {\frac {1}{a_{i}-a_{i-1}}}}}}$

literature

• JC Frauenthal: Analysis of age-structure models , in: TG Hallam, SA Levin, Mathematical Ecology. An Introduction, Springer 1986, pp. 117-147
• N. Keyfitz: Introduction to the mathematics of population , Addison-Wesley 1968
• Mark Kot: Elements of mathematical ecology , Cambridge UP 2001

Individual evidence

1. ^ Euler equation, Euler-Lotka equation, spectrum lexicon of biology
2. ↑ ^{a b c d e f} Marc Lipsitch, J. Wallinga: How generation intervals shape the relationship between growth rates and reproductive numbers , Proc. Roy. Soc. B, Volume 274, 2007, pp. 500-604, online
3. ^ Kot, Elements of mathematical ecology, Cambridge UP 2001, chapter 20
4. ^ Feller, On the integral equation of renewal theory, Annals of Mathematical Statistics , Volume 12, 1941, pp. 243-267
5. ^ Kot, Elements of mathematical ecology, Cambridge UP 2001, chapter 22
6. ↑ The following illustration follows a lecture by Marc Lipsitch, Euler-Lotka for Epidemiologists , youtube , 2014
7. ↑ Dublin, Lotka, On the true rate of natural increase, as exemplified by the population of the United States , J. Am. Stat. Assoc., Vol. 150, 1925, pp. 305-339