Mathematical demographics

As Mathematical Demography is defined as the branch of demographics, the context and method of demography in mathematical analyzes and justified. Here come u. a. mathematical demographic models to use.

Objects of investigation

The math. Demographics include a. the creation and analysis of:

math. Population models,
Mortality and fertility rates,
Dependencies of the ratio of women to men,
different types of instabilities in a population.

Furthermore are examined u. a .:

Approaches to stabilize a population,
but also the demographic theory of relationships
Micro-demographics, e.g. B .: Birth rates - influenced by contraceptives, as well
the family demographics.

The focus here is on the investigation of the internal relationships of a population and less on the prediction of population development.

Example 1: Calculating the reproductive rate for an inpatient population

The aim is to calculate the approximate number of children per woman with which it is possible to maintain the size of the population in the form of a stationary population. The calculations are based on data from the USA, particularly from 1967.

The relative number of births at which girls are born is: 0.488. That means 48.8% of the total number of births are girls, based on data from the USA from 1967. In the case of three children per mother, the average is: 3 · 0.488 = 1.464 daughters per mother.
Not all daughters will survive until they can reproduce. US data from 1967 give the factor: 0.9665. In the case of a three-child family, this is an average of 0.9665 · 1.464 = 0.9665 · (3 · 0.488) = R ₀ = 1.415 daughters per woman who reach their fertility phase .
We assume that an unknown proportion of women cannot or do not want to have children: 10%. This then gives 1.415 (daughters per mother) · 0.9 = 1.273 fertile daughters with their own child, per woman.

For the possible case of an average of two children per mother, a value of 2.0 · 0.488 · 0.9665 · 0.9 = R _2.0 = 0.849 is obtained as the reproduction rate or replacement factor per generation (= daughters with child, per Mother ). Taking into account a generation duration of 26.14 years (US data for 1967), the 26.14th root (26.14 years) results in the same factor r per year: and you also get:) the replacement factor of r = 0 , 9938 per year for the case of an average of 2.0 children per mother. This corresponds to a decrease in the population of 0.62% per year. The biological half-life of the affected population is then 111 years. ${\ displaystyle r ^ {26.14} = R_ {2.0}}$ ${\ displaystyle r = {\ sqrt [{26.14}] {R_ {2.0}}}}$

For the case of three children per mother, the following overall equation can be written:

(1)

{\ displaystyle 3 \ cdot 0 {,} 488 \ cdot 0 {,} 9665 \ cdot 0 {,} 9 = R}

,

with R as the reproduction rate “fertile daughters with their own child, per mother”. In this case one obtains: R = 1.273 daughters with child, per mother. This in turn results in an annual population growth of 0.93% with an average generation duration of 26.14 years. This results in a factor of 16 for the new US population size in a little less than three hundred years.

In order to be able to keep the population as stationary , we set R = R _stationary = 1 (daughters with child, per mother). We are now looking for the mean value x ₁ children per woman in order to satisfy the following equation:

(2)

{\ displaystyle x_ {1} \ cdot 0 {,} 488 \ cdot 0 {,} 9665 \ cdot 0 {,} 9 = R _ {\ mathrm {station {\ ddot {a}} r}} = 1}

.

We get: x ₁ = 2.36 children per fertile woman, in this case (the mean final number): children per mother . The greatest uncertainty factor here is the assumption of 10% women who cannot or do not want to have children, see point 3 above.

With current US death rates and an average of three children per mother, three hundred years from now, we would have a US population of more than three billion people. What we need, however, is only an average of 2.36 children per mother, according to the author (see source, p. 426, approx. Line 16).

The often mentioned figure of 2.1 children per woman is related to this and can be calculated as follows.

From x ₂ · 0.488 · 0.9665 = R _stationary = 1, the value x ₂ = 2.12 is obtained.

This value of “2.12 children per woman” describes the mean final number of children per woman - including women who have no children of their own for life - taking into account US data from 1967.

These numbers show that the mean value needs fine tuning; Even small deviations can lead to intolerable deviations up or down over a long period of time, according to the author (p. 426, approx. line 26).

Example 2: Approaches to stabilize a population size

Frisch (1972) pointed to the theoretical possibility of an equilibrium mechanism for the size of a population. Malnourished women reach fertility later and menopause earlier than well-nourished women. Depending on the z. B. natural availability of food with this mechanism the possibility of increasing or decreasing birth rates. However, this approach is not suitable for compensating for short-term fluctuations in population size ( Keyfitz / Caswell , p. 427, line 10).

Another theory gives a further approach to a mechanism for stabilizing population size through a legal adjustment of the minimum age at marriage. The author of the source argues that this legal minimum age at marriage effectively eliminates the respective part of the fertility curve below the specified age (Keyfitz / Caswell, p. 504, line 4). In the example of India, this results in a total of 3.24 million births or 18% of the births of mothers under the age of 20 in 1961 with 18.14 million births . In order to assess the effectiveness of such an adjustment to the statutory minimum age at marriage, the author points out the theoretical possibility that the number of children that a family may wish to have a fixed number of children could not be implemented in the personal envisaged period, but with only a few years delay.

Whether there is already a stabilizing mechanism using such a fertility curve at a fixed minimum marriage age, in which, in practical terms, the person who distracts from family events (at least in Germany, particularly protected by Art. 6 Para. 1 of the Basic Law), also at the same time, for compensation, in permissibly sufficiently extensive on or about existing legally valid marriage laws and the like. a. (eg 18 years in Switzerland, Austria.: before mature 16-J and below (a minor) to the minimum age of marriage at the request , otherwise 18 J) indicates + in (a) greater or (b) on a smaller scale and enlightens , or enlighten Must not be found in the source so far. The scope of this information about the possibilities of a marriage (from e.g. 16 years, i.e. already during school time) would then be the possible control parameter for stabilizing a population size.

Example 3: Markov chains for individual life courses

The source describes a life cycle that an individual goes through, from birth to death, using a mathematical model . Such a life cycle is marked u. a. through various events, which can then also be reflected in the model. Some of these events, such as mating and reproduction, are optional (p. 245, lines 5-6), but marriage is also included. Others, like death, are inevitable. One approach to handling this information is to use matrix models or Markov chains . These can contain the states to be described in a formal mathematical representation, but also the transition probabilities between these states.

Life cycle graph for an age-classifying life cycle

Fig. 1: Simple model of a life cycle graph for an age-classifying life cycle. The mathematical states are represented in a simplified manner by '1', '2' and '3'. In this model, P ₁ and P _{2 describe} the transition probabilities to the respective subsequent state, and F ₂ and F _{3 describe} the corresponding reproduction probabilities .

A so-called life cycle graph is suitable for the construction of a matrix population model. A simplified model is mentioned here as an example: a life cycle graph for an age-classifying life cycle .

The transition probabilities between the states from the model can be captured with the following general equation (p. 69):

(3)

{\ displaystyle {\ vec {n}} (t + 1) = {\ boldsymbol {A}} {\ vec {n}} (t)}

.

Here is A , the population projection matrix represents and, in the case of altersklassifizierenden life cycle models as Leslie matrix referred to. The vector describes the state frequency distribution and is also called the population vector. The time difference is referred to as the projection interval. ${\ displaystyle {\ vec {n}} (t)}$ ${\ displaystyle t \ rightarrow t + 1}$

According to the life cycle graph from Fig. 1 , the elements of the matrix A and the elements of the population vector can be inserted into equation (3) as follows: ${\ displaystyle {\ vec {n}} (t)}$

(4)

{\ displaystyle {\ vec {n}} (t + 1) = {\ begin {pmatrix} 0 & F_ {2} & F_ {3} \\ P_ {1} & 0 & 0 \\ 0 & P_ {2} & 0 \ end {pmatrix}} {\ begin {pmatrix} n_ {1} \\ n_ {2} \\ n_ {3} \ end {pmatrix}} (t)}

.

The matrix A contains the scalar elements for the reproductive probabilities ( F of 'fertility') and for the probabilities of the transition to the next state. The population vector contains the state frequency distribution as a column vector with the scalar elements for which the following applies :, and which in the model named as an example with three states can be normalized to: ${\ displaystyle F_ {i} \ geq 0}$ ${\ displaystyle P_ {i} \ geq 0}$ ${\ displaystyle {\ vec {n}} (t)}$ ${\ displaystyle n_ {i}}$ ${\ displaystyle n_ {i} \ geq 0}$ ${\ displaystyle \ textstyle \ sum _ {i = 1} ^ {3} n_ {i} = 1.}$

After performing the matrix multiplication ( matrix times vector ), the following result for the population vector is obtained after the intended first projection interval: ${\ displaystyle t + 1}$

(5)

{\ displaystyle {\ vec {n}} (t + 1) = {\ begin {pmatrix} F_ {2} \ cdot n_ {2} + F_ {3} \ cdot n_ {3} \\ P_ {1} \ cdot n_ {1} \\ P_ {2} \ cdot n_ {2} \ end {pmatrix}} (t + 1)}

.

This initially formal result can be verified in Fig. 1 by the respective results `` at the end of the arrows '': the transition probability ( ) multiplied by the associated population size in the respective initial state 1, 2 or 3. By executing such a projection step multiple times, i.e. If the matrix multiplication is carried out multiple times and using specific numerical values, the development of the population vector can now be tracked or predicted via a large number of projection steps within the framework of the mathematical model . The population matrix A from equation (3) with its elements can be constant in the model, but it can also vary. If it varies, it can depend on external factors e.g. B. the weather, or also depend on internal conditions of the population itself. Furthermore, random or deterministic effects can occur and the behavior can be periodic or aperiodic. ${\ displaystyle F_ {2}, F_ {3}, P_ {1}, P_ {2}}$ ${\ displaystyle n_ {i}}$ ${\ displaystyle {\ vec {n}} (t)}$

Life cycle graph for a size-classifying life cycle

Fig. 2: Model of a life cycle graph for a size-classifying life cycle with three states. The transition probabilities G in the respective subsequent state, the residence probabilities P per state and the reproduction probabilities F are entered .

Another model can be used to mathematically describe a size-classifying life cycle graph . Fig. 2 shows such a graph with the states S ₁ , S ₂ and S ₃ . These states describe three size classes here, with the size for the subsequent state increasing in each case. An individual can move from the states S ₁ & S ₂ to the next state with the probability G _i and with the probabilities P _i also dwell in the respective state S ₁ , S ₂ or S ₃ . Reproduction is described by the probabilities F ₂ and F ₃ and produces new individuals in the state S ₁ . So-called 'absorbing states' are not explicitly drawn in this graph. Such an absorbing state (Keyfitz / Caswell, p. 246, lines 13-14) occurs as soon as an individual comes into a state but can never leave it. The state S ₃ becomes, for example, an absorbing state if the following applies: F ₃ = 0 and P ₃ = 1.

By appropriately inserting the probabilities F ₂ , F ₃ , G ₁ , G ₂ , P ₁ , P ₂ and P ₃ from Fig. 2 into the population projection matrix A from equation (3), the following population vector for the projection step is obtained : ${\ displaystyle t + 1}$

(6)

{\ displaystyle {\ vec {n}} (t + 1) = {\ begin {pmatrix} P_ {1} & F_ {2} & F_ {3} \\ G_ {1} & P_ {2} & 0 \\ 0 & G_ {2 } & P_ {3} \ end {pmatrix}} {\ begin {pmatrix} n_ {1} \\ n_ {2} \\ n_ {3} \ end {pmatrix}} (t)}

.

After multiplying the matrix times the population vector, one obtains for this first projection step in detail:

(7)

{\ displaystyle {\ vec {n}} (t + 1) = {\ begin {pmatrix} P_ {1} \ cdot n_ {1} + F_ {2} \ cdot n_ {2} + F_ {3} \ cdot n_ {3} \\ G_ {1} \ cdot n_ {1} + P_ {2} \ cdot n_ {2} \\ G_ {2} \ cdot n_ {2} + P_ {3} \ cdot n_ {3} \ end {pmatrix}} (t + 1)}

.

This behaves similarly for the second projection step; the following equation is obtained by repeated multiplication with the associated population projection matrix A :

(8th)

{\ displaystyle {\ vec {n}} (t + 2) = {\ begin {pmatrix} P_ {1} & F_ {2} & F_ {3} \\ G_ {1} & P_ {2} & 0 \\ 0 & G_ {2 } & P_ {3} \ end {pmatrix}} {\ begin {pmatrix} P_ {1} \ cdot n_ {1} + F_ {2} \ cdot n_ {2} + F_ {3} \ cdot n_ {3} \ \ G_ {1} \ cdot n_ {1} + P_ {2} \ cdot n_ {2} \\ G_ {2} \ cdot n_ {2} + P_ {3} \ cdot n_ {3} \ end {pmatrix} } (t + 1)}

and can again apply matrix multiplication ( matrix times vector ) to it to multiply .

For the case of m discrete projection steps using multiple multiplications of the matrix A for these projection steps and taking into account the brackets for products ( associativity of matrices ), equation (3) can be written as follows:

(9)

{\ displaystyle {\ vec {n}} (t + m) = {\ boldsymbol {A}} ^ {m} {\ vec {n}} (t)}

.

Here a multiple multiplication ( matrix times matrix for block matrices ) means the matrix A with itself - using exactly m matrix multiplications. This type of matrix exponentiation can already be found in PH Leslie (1945) in the form: 'matrix M ^t '. ${\ displaystyle {\ boldsymbol {A}} ^ {m}}$

In general, the Keyfitz / Caswell source notes that a matrix model contains a large amount of information ('contains a great deal of information', p. 245, line 7) about the events, their probabilities, and the associated sequences.

If necessary, Markov chains can also be evaluated as a mathematical Markov process in a continuous time domain instead of for discrete time steps . The source 'van Kampen' defines Markov chains as a class of Markov processes, whereby according to the source for Markov chains in particular discrete whole-number time steps apply. With the somewhat more complex so-called master equation (Pauli master equation or 'M equation'), Markov processes can now be evaluated in the continuous time domain .

Life cycle graph for going through an education system

Fig. 3: Graph for running through a training system. The projection interval is 4 years. The red marked state S ₆ is treated as an absorbing state in this model.

A somewhat more complex model is required to mathematically describe a graph as a representation of the progression through an education system up to the state of employment. The classification of the training levels for the model from Keyfitz / Caswell (p. 248) is, however, somewhat simplified here in only five "states" S ₁ to S ₅ . The classification in this still quite general system takes place in S ₁ : School classes 1-4, S ₂ : School classes 5-8, S ₃ : School classes 9-12, S ₄ : Specialized training ('Technical'), S ₅ : University (' College'). As the final state in this model, employment is described with state S ₆ . This state S ₆ is recorded in this described model as an absorbing state, i. H. From this state onwards, further states are no longer reached, neither through reproduction, nor through death, or through other transitions. The absorbing state S ₆ is therefore not a transition state either.

For a mathematical description it is advisable to split the population projection matrix A into:

(10)

{\ displaystyle {\ varvec {A}} = {\ varvec {T}} + {\ varvec {F}}}

with the dimensions of each individual matrix .

{\ displaystyle \ left ({\ begin {array} {c} s \ times s \ end {array}} \ right)}

The matrix T describes the transitions (without reproduction and without repetitions), the matrix F describes the reproductions, s is the number of transition states in the model without the absorbing state S ₆ and thus both the number of rows and the number of the columns of the respective block matrix. Depending on the focus of the examinations in the respective model, T (transition matrix) or rather F (fertility matrix) can now be examined more closely. In the present case of the model from Fig. 3 , the matrix T is examined more closely. With five transition states the dimension of this matrix is then s = 5. The absorbing state S ₆ (employment) in this model counts separately. The transition probabilities that this state S ₆ lead will be with the row vector m recovered and used in equation (11) to the matrix T to the larger matrix P to expand. As already mentioned, reproductions are not shown here. The elements of the associated matrix F in this case are 0 and are recorded as column vector 0 in equation (11). The following abbreviated mathematical representation for the transition matrix P is used to describe the Markov chain :

(11)

{\ displaystyle {\ varvec {P}} = \ left ({\ begin {array} {c | c} {\ varvec {T}} & {\ varvec {0}} \\\ hline {\ varvec {m} } & 1 \ end {array}} \ right)}

with the dimensions taking into account .

{\ displaystyle \ left ({\ begin {array} {c | c} s \ times s & s \ times 1 \\\ hline 1 \ times s & 1 \ times 1 \ end {array}} \ right)}

{\ displaystyle {\ text {number of lines}} \ times {\ text {number of columns}}}

The states S ₁ to S ₅ are considered to be 'transitions'; H. they each have an achievable subsequent state. They are with the transition matrix T acquired. The representation used in this formula (11) with marking lines between the individual elements of the matrix is used for identification in order to visualize that this is a correspondingly composed matrix. For the row vector m it must be taken into account that the total sum of the probabilities in equation (11) per column in the matrix P must be exactly 1: ${\ displaystyle \ left ({\ begin {array} {c | c} & \\\ hline & \ end {array}} \ right)}$

(12)

{\ displaystyle \ sum _ {i} {t_ {ij}} + m_ {j} = 1}

with m _j as elements of the row vector m and t _ij as elements of matrix T .

Furthermore, analogous to equation (3) , the following equation applies for a column vector in the present case , which is intended to reflect the probability distribution of the states from Fig. 3 : ${\ displaystyle {\ vec {x}} (t)}$

(13)

{\ displaystyle {\ vec {x}} (t + 1) = {\ boldsymbol {P}} {\ vec {x}} (t)}

.

To create the matrix P or the matrix T , proceed as follows as an example: The transition probability p ₃₂ in column 2 and line 3 for the equation (14) below applies to the transition from state S ₂ 'old' to state S. ₃ 'new' and is also intended as a transition according to Fig. 3 . If there is no transition in the life cycle graph for the point to be checked in the matrix, a zero ('0') is noted at the associated position in the matrix. All elements of the matrix must be checked accordingly. From the life cycle graph from Fig. 3 , including the transitions into the absorbing state S _6, the following equation is obtained for the transition matrix P :

(14)

{\ displaystyle {\ boldsymbol {P}} = \ left ({\ begin {array} {ccccc | c} 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & p_ {32} & 0 & 0 & 0 & 0 \\ 0 & 0 & p_ {43} & 0 & 0 & 0 & 0 & 0 & 0 & p_ { \\\ hline 0 & 1-p_ {32} & 1-p_ {43} -p_ {53} & 1 & 1 & 1 \ end {array}} \ right)}

with the overall dimension .

{\ displaystyle (s + 1) \ times (s + 1)}

From this it becomes clear that the transition matrix P is required for the calculation of the projection steps, but the somewhat smaller matrix T is sufficient for recording the probability values of the transitions in the Markov chain. The matrix P can then be calculated from T. You can also see from this example that 'branches' in a life cycle graph, e.g. B. from state S ₃ to S ₄ or to S ₅ can be mapped in a simple manner using a matrix model. ${\ displaystyle t \ rightarrow t + 1}$

In general, Quelle (D. Schulz, 2010) notes on Markow chains that they are a simple and descriptive model for mathematically mapping real-world processes.

Researchers at the Federal Institute for Population Research (BiB, Germany) report that there are “considerable differences between the professions with regard to childlessness. In 2009, childlessness was lowest among women farmers and women who were not gainfully employed (mostly housewives). " A high level of childlessness can be found among "female managers", where "the proportion of childless women was 45 percent." Other sources report: "Women with a university degree in Germany still give birth to significantly fewer children than women without a university degree" (2012 article) and the "number of female university graduates without young talent continues to rise in this country" (Germany, 2018). Similarly, differences in the number of children per woman depending on nationality are reported, e.g. B. "If Austrian women have an average of 1.27 children in their lifetime, the figure for foreign women (origin) is 1.84 children."

A careful investigation of different life path decisions using the mathematical model of the Markov chains, whether the determination of the exact causer and the sources of distraction from family events, with a subsequent, sufficiently extensive and legally permissible compensation by the causer exactly at these determined points, to one Adjustment of the different values of the number of children per woman z. B. in different professions, or different training or with different nationalities can not be found in source Keyfitz / Caswell.

Web links

MPIDR - Mathematical Demography - demogr.mpg.de , accessed on December 17, 2018.
Karl-Peter Hadeler, Hans Heesterbeek: Mathematical Demography and Epidemiology (springer.com), accessed on December 17, 2018.

Individual evidence

↑ ^a ^b ^c ^d ^e ^f Nathan Keyfitz , Hal Caswell : Applied Mathematical Demography. , 3rd edition Springer Science + Business Media, 2005, ISBN 0-387-22537-4 , p. 425f. (English)

^ RE Frisch: Weight at menarche: Similarity for well-nourished and undernourished girls at differing ages, and evidence for historical constancy. In: Pediatrice. Volume 50, 1972, pp. 445-450; Source taken from "Keyfitz / Caswell".

^ ^A ^b Nathan Keyfitz, Hal Caswell: Applied Mathematical Demography. 3. Edition. Springer Science + Business Media, 2005, ISBN 0-387-22537-4 , p. 504. (English)

^ Nathan Keyfitz, Wilhelm Flieger: World Population: An Analysis of Vital Data. The University Press, Chicago, Illinois, USA 1968, p. 659.

↑ Switzerland Art 94 ZGB, A. Marital status, zgb.gesetzestext.ch, accessed on January 10, 2018.

↑ Austria: formerly aged 16 and under (minors) on request , otherwise 18 years , help.gv.at, accessed on January 15, 2018.

^ ^A ^b ^c Nathan Keyfitz, Hal Caswell: Applied Mathematical Demography. 3. Edition. Springer Science + Business Media, 2005, ISBN 0-387-22537-4 , pp. 245ff. (English)

^ Nathan Keyfitz, Hal Caswell: Applied Mathematical Demography. 3. Edition. Springer Science + Business Media, 2005, ISBN 0-387-22537-4 , pp. 68f. (English)

^ PH Leslie: On the Use of Matrices in Certain Population Mathematics. In: Mathematical Demography. 2nd Edition. (selected papers), Springer, 2013, ISBN 978-3-642-35857-9 , p. 215 (tenth line from the top).

^ NG van Kampen : Stochastic Processes in Physics and Chemistry. 2., revised. Edition. Elsevier Science Publishers, 1992, ISBN 0-444-89349-0 , p. 89 (bottom).

^ NG van Kampen: Stochastic Processes in Physics and Chemistry. 2., revised. Edition. Elsevier Science Publishers, 1992, ISBN 0-444-89349-0 , p. 90 (above).

^ NG van Kampen: Stochastic Processes in Physics and Chemistry. 2., revised. Edition. Elsevier Science Publishers, 1992, ISBN 0-444-89349-0 , p. 97 (equations 1.4, 1.5 and 1.6)

^ NG van Kampen: Stochastic Processes in Physics and Chemistry. 2., revised. Edition. Elsevier Science Publishers, 1992, ISBN 0-444-89349-0 , p. 98 (fourth line from the top)

^ Nathan Keyfitz, Hal Caswell: Applied Mathematical Demography. 3. Edition. Springer Science + Business Media, 2005, ISBN 0-387-22537-4 , p. 246. (English)

^ Nathan Keyfitz, Hal Caswell: Applied Mathematical Demography. 3. Edition. Springer Science + Business Media, 2005, ISBN 0-387-22537-4 , p. 247. (English)

↑ D. Schulz: An introduction to time-discrete Markov chains . Research work. GRIN-Verlag, Munich 2010, ISBN 978-3-640-79708-0 , p. 3 (above).

↑ ^a ^b ^c Miriam Hollstein, September 19, 2012: Female academics have more children , in the section “A problem especially in West Germany”, welt.de, accessed on May 5, 2018.

↑ ^a ^b focus.de, September 20, 2012: Germany's birth rate is falling again , in the section: “Female academics are having more children again”, accessed on May 5, 2018.

↑ ^a ^b Dorothea Siems, March 7, 2018: The sad top position of German academics , 1st sentence in the article (html), welt.de, accessed on May 6, 2018.

↑ ^a ^b auslaender.at, January 4, 2012: The birth rate among foreign women, in the section: “The birth rate in Austria”, accessed on May 5, 2018.

[Keyfitz_ApplMathDemography-1] ↑ ^a ^b ^c ^d ^e ^f Nathan Keyfitz , Hal Caswell : Applied Mathematical Demography. , 3rd edition Springer Science + Business Media, 2005, ISBN 0-387-22537-4 , p. 425f. (English)

[2] RE Frisch: Weight at menarche: Similarity for well-nourished and undernourished girls at differing ages, and evidence for historical constancy. In: Pediatrice. Volume 50, 1972, pp. 445-450; Source taken from "Keyfitz / Caswell".

[KeyfitzCaswell_b-3] A ^b Nathan Keyfitz, Hal Caswell: Applied Mathematical Demography. 3. Edition. Springer Science + Business Media, 2005, ISBN 0-387-22537-4 , p. 504. (English)

[4] Nathan Keyfitz, Wilhelm Flieger: World Population: An Analysis of Vital Data. The University Press, Chicago, Illinois, USA 1968, p. 659.

[5] Switzerland Art 94 ZGB, A. Marital status, zgb.gesetzestext.ch, accessed on January 10, 2018.

[6] Austria: formerly aged 16 and under (minors) on request , otherwise 18 years , help.gv.at, accessed on January 15, 2018.

[KeyfitzCaswell_3rdEd_S245-7] A ^b ^c Nathan Keyfitz, Hal Caswell: Applied Mathematical Demography. 3. Edition. Springer Science + Business Media, 2005, ISBN 0-387-22537-4 , pp. 245ff. (English)

[KeyfitzCaswell_3rdEd_S68-8] Nathan Keyfitz, Hal Caswell: Applied Mathematical Demography. 3. Edition. Springer Science + Business Media, 2005, ISBN 0-387-22537-4 , pp. 68f. (English)

[9] PH Leslie: On the Use of Matrices in Certain Population Mathematics. In: Mathematical Demography. 2nd Edition. (selected papers), Springer, 2013, ISBN 978-3-642-35857-9 , p. 215 (tenth line from the top).

[10] NG van Kampen : Stochastic Processes in Physics and Chemistry. 2., revised. Edition. Elsevier Science Publishers, 1992, ISBN 0-444-89349-0 , p. 89 (bottom).

[11] NG van Kampen: Stochastic Processes in Physics and Chemistry. 2., revised. Edition. Elsevier Science Publishers, 1992, ISBN 0-444-89349-0 , p. 90 (above).

[12] NG van Kampen: Stochastic Processes in Physics and Chemistry. 2., revised. Edition. Elsevier Science Publishers, 1992, ISBN 0-444-89349-0 , p. 97 (equations 1.4, 1.5 and 1.6)

[13] NG van Kampen: Stochastic Processes in Physics and Chemistry. 2., revised. Edition. Elsevier Science Publishers, 1992, ISBN 0-444-89349-0 , p. 98 (fourth line from the top)

[KeyfitzCaswell_3rdEd_S246-14] Nathan Keyfitz, Hal Caswell: Applied Mathematical Demography. 3. Edition. Springer Science + Business Media, 2005, ISBN 0-387-22537-4 , p. 246. (English)

[KeyfitzCaswell_3rdEd_S247-15] Nathan Keyfitz, Hal Caswell: Applied Mathematical Demography. 3. Edition. Springer Science + Business Media, 2005, ISBN 0-387-22537-4 , p. 247. (English)

[16] D. Schulz: An introduction to time-discrete Markov chains . Research work. GRIN-Verlag, Munich 2010, ISBN 978-3-640-79708-0 , p. 3 (above).

[MiriamHollstein_2012-17] Miriam Hollstein, September 19, 2012: Female academics have more children , in the section “A problem especially in West Germany”, welt.de, accessed on May 5, 2018.