Time-evolution of age-dependent mortality patterns in ...€¦ · Time-evolution of age-dependent mortality patterns in mathematical model of heterogeneous human population Conclusion

Time-evolution of age-dependent mortality patterns in mathematical model of heterogeneous human population

Time-evolution of age-dependent mortalitypatterns in mathematical model of heterogeneous

human population

Severine Arnold (-Gaille), Universite de Lausanne(collaboration avec Demetris Avraam et Bakhti Vasiev, Universite de

Liverpool)

Longevity 117 September 2015

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Outline

Table of contents

Introduction

Model

ApplicationsAll ages excluding the extrinsic causes of deathAll ages, all causes of death

Conclusion

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Introduction

Figure: Mortality Rates, Sweden, 2000

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Introduction

Gompertz law of mortality

[Gompertz, 1825]:

mx = m0eβx

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Introduction

Gompertz law of mortality

0.001

0.01

0.1

1

40 50 60 70 80 90 100 110 120

Mortality rate

Age

A

0.00001

0.0001

0.001

0.01

0.1

1

40 60 80 100 120

Mortality rate

Age

B

Figure: Mortality Rates, Sweden, 1900 (Panel A) and 2000 (Panel B)

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Introduction

Compensation effect

0.00001

0.0001

0.001

0.01

0.1

1

0 20 40 60 80 100

Mortality rate

Age

Figure: Mortality Rates, Sweden, 1900 (black) and 2000 (red)

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Introduction

Compensation effect

0

0.0001

0.0002

0.0003

1900 1920 1940 1960 1980 2000

m0

Year

A

0.07

0.08

0.09

0.10

0.11

1900 1920 1940 1960 1980 2000

β

Year

B

Figure: Evolution of the exponential parameters, Sweden, age 40+

→ Strehler and Mildvan correlation [Strehler and Mildvan, 1960]:

ln(m0) = ln(M)− βX

m0 = Me−βX

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Introduction

Compensation effect

1.E-5

1.E-4

1.E-3

0.076 0.084 0.092 0.1 0.108

m0

β

A

trendline: 𝑚0 = 0.791(±0.463)𝑒−103.6(±6.1)𝛽

1.E-4

1.E-2

1.E+0

40 60 80 100

Mortality rate

Age

Year 1900

Year 1925

Year 1975

Year 2000

B

𝑋 = 103.6 ± 6.1,𝑀 = 0.791 ± 0.463

Figure: Compensation effect in 40+ mortality dynamics

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Introduction

Aim of this work

Populations are heterogeneous.

→ Each subpopulation obeys the exponential law.

→ Can we model the mortality of the entire population as amixture of weighted exponential terms?

→ If yes, do we observe the compensation effect in eachsubpopulation?

[Avraam et al., 2014]

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Model

Gompertz and an extension

Gompertz law of mortality [Gompertz, 1825]:

mx = m0eβx

An extension of [Avraam et al., 2013]

mx =n∑

j=1

ρjxmjx =n∑

j=1

ρjxmj0eβjx

[Booth and Tickle, 2008]

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Applications

All ages excluding the extrinsic causes of death


0.00001

0.0001

0.001

0.01

0.1

1

10

0 20 40 60 80 100

Mortality rate

Age

A

0.00001

0.0001

0.001

0.01

0.1

1

10

0 20 40 60 80 100

Mortality rate

Age

B

Figure: Heterogeneous model, Sweden, 1951 (Panel A) and 2010 (PanelB)

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Applications



Figure: Evolution of the exponential parameters of a two-subpopulationmodel, Sweden

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Applications



1.E-5

1.E-4

0.086 0.096 0.106

m20

β2

𝑋 = 100.4 𝑀 = 0.5854

A

1.E-5

1.E-4

1.E-3

1.E-2

1.E-1

1.E+0

0 20 40 60 80 100

m2x

Age

1951

1970

1990

2010

B

ܯ ,100.4= = 0.5854

Figure: Compensation effect for the second subpopulation, Sweden

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Applications



Figure: Evolution of the fractions of a two-subpopulation model, Sweden

→ homogenization of the population

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Applications

All ages, all causes of death


0.00001

0.0001

0.001

0.01

0.1

1

0 20 40 60 80 100 120

Mortality rate

Age

A

0.000001

0.00001

0.0001

0.001

0.01

0.1

1

0 20 40 60 80 100 120

Mortality rate

Age

B

Figure: Heterogeneous model, Sweden, 1900 (Panel A) and 2000 (PanelB)

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Applications



0.5

1

1.5

2

2.5

1900 1920 1940 1960 1980 2000

Year

m10

0

0.2

0.4

0.6

0.8

1

1.2

1900 1920 1940 1960 1980 2000

Year

m20

0

0.004

0.008

0.012

0.016

1900 1920 1940 1960 1980 2000

Year

m30

0

0.00004

0.00008

0.00012

0.00016

1900 1920 1940 1960 1980 2000

Year

m40

0

0.5

1

1.5

2

2.5

3

1900 1920 1940 1960 1980 2000

Year

β1

0

0.2

0.4

0.6

0.8

1900 1920 1940 1960 1980 2000

Year

β2

0

0.1

0.2

0.3

1900 1920 1940 1960 1980 2000

Year

β3

0.083

0.088

0.093

0.098

0.103

1900 1920 1940 1960 1980 2000

Year

β4

0

0.04

0.08

0.12

0.16

1900 1920 1940 1960 1980 2000

Year

ρ10

0

0.02

0.04

0.06

0.08

1900 1920 1940 1960 1980 2000

Year

ρ20

0

0.1

0.2

0.3

1900 1920 1940 1960 1980 2000

Year

ρ30

0.5

0.6

0.7

0.8

0.9

1

1900 1920 1940 1960 1980 2000

Year

ρ40

Figure: Evolution of the parameters of a four-subpopulation model,Sweden

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Applications



1.E-5

1.E-4

1.E-3

0.085 0.09 0.095 0.1 0.105

m40

β4

𝑋 = 106.8 𝑀 = 1.1591

A

1.E-5

1.E-4

1.E-3

1.E-2

1.E-1

1.E+0

0 20 40 60 80 100

m4x

Age

1900

1930

1970

2000

B

(𝑋=106.8, 𝑀=1.1591)

Figure: Compensation effect for the fourth subpopulation, Sweden

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Applications



What part of past mortality decrease is due

to the homogenization of the population and

what part is due to a real mortality decrease?

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Applications



0.00001

0.0001

0.001

0.01

0.1

1

0 20 40 60 80 100

Mortality rate

Age

Period 1900

Period 2000

Figure: Reduction in Swedish mortality rates within one century

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Conclusion

Concluding remarks

We consider a model of heterogeneous population composed ofseveral subpopulations having different mortality dynamics.

→ Each subpopulation follows the Gompertz law of mortality

Two main findings:

→ The compensation law of mortality is confirmed at thesubpopulation level;

→ Homogenization of the population over time.

Further steps:

→ Model giving potential insights for mortality at extreme oldages;

→ Any links with genes and natural selection?

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Conclusion

Bibliography I

Avraam, D., Arnold, S., Jones, D., and Vasiev, B. (2014).Time-evolution of age-dependent mortality patterns inmathematical model of heterogeneous human population.Experimental Gerontology, 60:18–30.

Avraam, D., de Magalhaes, J., and Vasiev, B. (2013).A mathematical model of mortality dynamics across thelifespan combining heterogeneity and stochastic effects.Experimental Gerontology, 48:801–811.

Booth, H. and Tickle, L. (2008).Mortality modelling and forecasting: A review of methods.Annals of Actuarial Science, 3:3–43.

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Conclusion

Bibliography II

Gompertz, B. (1825).On the nature of the function expressive of the law of humanmortality and on a new mode of determining life contingencies.

Philosophical Transactions of the Royal Society of London,115:513–585.

Strehler, B. L. and Mildvan, A. S. (1960).General theory of mortality and aging (a stochastic modelrelates observations on aging, physiologic decline, mortalityand radiation).Science, 132:14–21.

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Conclusion

Thank you very much for yourattention!

[email protected]

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Time-evolution of age-dependent mortality patterns in ...€¦ · Time-evolution of age-dependent mortality patterns in mathematical model of heterogeneous human population Conclusion