Life Table and Population Projection Using the Leslie Matrix

A PROJECT SUBMITTED TO THE FACULTY OF THE GRADUATE SCHOOL OF THE UNIVERSITY OF MINNESOTA BY

Amber Koslucher

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE

Dr. Richard Green

July, 2016 Acknowledgements

I would like to give a special thanks to my advisor, Dr. Richard Green. His help and knowledge have been a tremendous help to me during my undergraduate and graduate career at the University of Minnesota Duluth. All of his time and help have been greatly appreciated. I would also like to thank my parents for supporting me throughout my academic career. Their constant support and encouragement have helped me along the way.

i Abstract

A life table is a table that shows different values pertaining to the mortality rate for humans. The table is broken up by age and includes the following values: the probability that a person of age x will die within the next year, the number of people surviving from the original cohort to age x, the number of people of age x that die within the year, and the expected lifetime remaining for a person of age x. These values are often used to compare the health of different countries. This can be done by comparing their life expectancy at birth, the age-adjusted mortality rates, or by looking at their population pyramids. An age-distribution is the number of people alive for each age x from the original cohort. Using mortality rates along with fertility rates, future age-distributions can be determined. The Leslie Matrix is a tool used to make such projections. The Leslie Matrix uses fertility and survival rates to project the age- distribution to the following year. The Leslie Matrix will be used to project the United States population from the year 2010 to the years 2011, 2016, and 2050.

ii Contents

Acknowledgements i

Abstract ii

List of Tables v

List of Figures vii

1 Life Table 1 1.1 Definitions and Assumptions ...... 1 1.2 History of Life Table ...... 2 1.3 Life Table Calculations ...... 2 1.4 Life Table Applications ...... 4 1.5 Life Table Example ...... 5

2 Life Expectancy 10 2.1 Life Expectancy ...... 10 2.2 Importance of Life Expectancy ...... 10 2.3 Comparison ...... 11

3 Mortality Rates 13 3.1 Crude Mortality Rate ...... 13 3.2 Age-Specific Mortality Rate ...... 14 3.3 Age-Adjusted Mortality Rate ...... 16 3.4 Comparison ...... 19

iii 4 Life Table Functions 21 4.1 Density Function ...... 21 4.2 Cumulative Distribution Function ...... 22 4.3 Survival Function ...... 23 4.4 Hazard Function ...... 26

5 Population Pyramids 27 5.1 Definition ...... 27 5.2 Types of PopulationPyramids ...... 27

6 Population Projections 32 6.1 Population Projection Definition ...... 32 6.2 Leslie Matrix ...... 32 6.3 Stable Age-Distribution ...... 34

7 My Findings and Projections 37 7.1 Data Being Used ...... 37 7.2 United States Projections ...... 43 7.3 United States Projections Using Different Mortality and Fertility Rates 50 7.4 Stable Age Vector ...... 56 7.5 Conclusion ...... 58

References 60

iv List of Tables

1.1 Graunt’s Life Table ...... 3 1.2 United States Life Table 2010 ...... 5 1.2 United States Life Table 2010 ...... 6 1.2 United States Life Table 2010 ...... 7 1.2 United States Life Table 2010 ...... 8 1.2 United States Life Table 2010 ...... 9 2.1 Comparing Life Expectancy, 2010 - 2014 ...... 11 3.1 Comparing Crude Mortality Rates 2010 - 2014 ...... 14 3.2 United States 2000 and 1940 Standard Population ...... 18 3.3 Comparing Crude and Age-Specific Mortality Rates 2010 - 2014 . . . . . 19 7.1 Birth Rates by Age of Mother. United States, 2010 ...... 38 7.2 Survival and Fertility Rates for Women in the United States, 2010 . . . 39 7.2 Survival and Fertility Rates for Women in the United States, 2010 . . . 40 7.2 Survival and Fertility Rates for Women in the United States, 2010 . . . 41 7.2 Survival and Fertility Rates for Women in the United States, 2010 . . . 42 7.3 Comparing Female Age-Distributions of the United States: 2010, 2011, 2016 and 2050 ...... 44 7.3 Comparing Female Age-Distributions of the United States: 2010, 2011, 2016 and 2050 ...... 45 7.3 Comparing Female Age-Distributions of the United States: 2010, 2011, 2016 and 2050 ...... 46 7.3 Comparing Female Age-Distributions of the United States: 2010, 2011, 2016 and 2050 ...... 47

v 7.4 Total Number of Women and People for United States: 2010, 2011, 2016, and 2050 ...... 49 7.5 Total Number of Women and People for United States: 2010, 2011, 2016, and 2050, with Changes in Mortality Rates ...... 53 7.6 Total Number of Women and People for United States: 2010, 2011, 2016, and 2050, with Changes in Fertility Rates ...... 56 7.7 Rate of Population Change for Leslie Matrices Used ...... 57

vi List of Figures

3.1 Age-Specific Mortality Rates for the United States, 2010 ...... 15 3.2 Log(Age-Specific Mortality Rates)+5 for the United States, 2010 . . . . 16 3.3 Comparing Crude and Age-Adjusted Mortality, United States ...... 19 4.1 Density Function for the United States, 2010 ...... 22 4.2 Cumulative Function for the United States, 2010 ...... 23 4.3 Three Types of Survivorship Curves ...... 24 4.4 Survival Function for the United States, 2010 ...... 25 5.1 Three Types of Population Pyramids ...... 28 5.2 Population Pyramid of United States ...... 29 5.3 Population Pyramid of Nigeria ...... 29 5.4 Population Pyramid of Japan ...... 30 5.5 Population Pyramid of the World ...... 30 7.1 Birth Rates for the United States 2010 ...... 38 7.2 United States Age-Distributions: 2010, 2011, 2016, 2050 ...... 48 7.3 United States Projection to 2016 with Differing Mortality Rates . . . . 51 7.4 United States Projection to 2050 with Differing Mortality Rates . . . . 51 7.5 United States Projection to 2016 with Differing Fertility Rates . . . . . 54 7.6 United States Projection to 2050 with Differing Fertility Rates . . . . . 54 7.7 Stable Age-Distributions with Different Mortality and Fertility Rates . . 58

vii Chapter 1

Life Table

1.1 Definitions and Assumptions

A life table is a table that shows, for each age, what the probability is that a person will die before his or her next birthday, the number of the original cohort that survived, the number of people from the cohort dying, and the life expectancy remaining. There are two types of life tables, the cohort life table and the period life table. The cohort life table, also called the generation life table, presents the mortality experience of a particular cohort from the moment of birth until death. Therefore the cohort life table will follow the original cohort until no one from the cohort remains. Humans tend to live for a long time, and thus it will take a long time to complete a cohort life table. Since it generally takes longer than 100+ years to complete a cohort life table, this form of a life table is not very practical and therefore is not often used. The period life table, also called the current life table, presents what would happen to a hypothetical cohort if it experienced throughout its entire lifetime the mortality conditions that are current for each age group. Newborns experience mortality rates currently present for newborns. Then next year the newborns that survived their first year of life will be a year old and they will experience mortality rates that one-year olds are currently experiencing, and so on. Therefore the period life table is complete immediately, and this makes this form of the life table much more applicable. The period life table may thus be thought of as rendering a snapshot of current mortality experiences[1]. From here on we will be referring to the period life table when mentioning

1 2 life table. There are some assumptions that are made in constructing a life table. It is assumed that the cohort being considered is closed to migration. Therefore no one is entering or leaving the population. This assumption of course holds when looking at the World population, but not when looking at the United States population. We assume that everyone in the cohort experiences the same mortality rates. Thus there is no inequality in health; higher class people experience the same rates as lower class people. This assumption also does not hold in the United States. Lastly, it is assumed that deaths are evenly distributed throughout the year[2]. This is not always the case. For example, newborns are much more likely to die right after birth than they are months later.

1.2 History of Life Table

Data on fertility and mortality rates have been around for a long time. John Graunt is credited with producing the first life table in 1662. The life table that he made closely resembles the life tables used today. His observations were based on the bills of mortality of the city of London. They were used to create a system to warn of the onset and spread of bubonic plague in the city. His work resulted in the first statistically-based estimation of the population of London[3].

1.3 Life Table Calculations

A life table starts with a cohort of some given number of newborns. The cohort can be of any size. In the United States it is common to use 100,000 newborns when con- structing a life table. The World Health Organization, an agency of the United Nations that is concerned with public health, also uses 100,000 newborns when constructing a life table for the World. To construct a life table, we start with number of deaths that each age will experience for a given time interval. The most common time intervals to use are one- or five-year periods. In the United States, one-year time intervals are used.

We define dx to be the number of people that are age x at the beginning of the year and that will die within the next year. The Vital Statistics Cooperative Program 3 Age Interval Proportion of Deaths in Interval Proportion of Surviving to Interval 0-6 0.36 1.00 7-16 0.24 0.64 17-26 0.15 0.40 27-36 0.09 0.25 37-46 0.06 0.16 47-56 0.04 0.10 57-66 0.03 0.06 67-76 0.02 0.03 77-86 0.01 0.01

Table 1.1: Graunt’s Life Table of the Center for Disease Control and Prevention’s National Center for Health Statistics analyzes the death certificates collected by funeral directors, physicians, medical exam- iners, and coroners. This information is used to estimate the number of deaths by age each year. From this it is possible to calculate the remaining values found on the life table[1].

We define lx to be the number of people that are of age x at the beginning of the year. We know that l0 = 100, 000 since we assumed that there are 100, 000 newborns to start with. To calculate the number of one-year olds there are, we take the original 100, 000 newborns and subtract the number of newborns that died throughout their

first year. Thus we get that l1 = l0 − d0. Therefore the number of people of age x + 1 is equal to the number of people that were alive at age x minus those who did not survive that year. Then, in general, we have that

lx+1 = lx − dx.

We define qx to be the mortality rate that a person of age x will experience. This can be found by taking the number of people that are of age x that die within the year, dx, and dividing by the number of people that are of age x at the beginning of the year, lx. Therefore we have that dx qx = . lx 4

We define Lx to be the number of person years lived within the age interval x to x + 1, or the number of person years lived within the time interval of one year. This can be broken down into two different parts: those who survive the whole year, they each contribute one year, and those that do not make it throughout the whole year, they contribute the portion of the year that they survived. Since it is assumed that deaths are evenly distributed throughout the year, it follows that each person to die in that time interval contributes half a year. Thus we get that 1 L = l − d . x x 2 x

We define Tx to be the total number of person years lived from age x to age ω, where ω denotes the maximum age possible for the population. Therefore the total number of person years lived can be thought of as the total number of years left to be lived by the total number of people that are of age x at the beginning of the year. In the United States we let ω = 100. Thus the total number of person years lived can be represented as ω X Tx = Li. i=x

One of the most important values in the life table is the life expectancy. The remaining life expectancy for a person aged x is denoted by ex. This can be thought of as taking the average number of years left to be lived of those who survived to age x. Therefore it is the total number of person years yet to be lived, Tx, divided by the number of people that are alive at time x, lx. The remaining life expectancy is thus calculated by the following equation

Tx ex = . lx

1.4 Life Table Applications

There are several applications of the life table. One example would be calculating the probability that a person aged x will survive to age x + n. This will be given by

lx+n  = (1 − qx) · (1 − qx+1) ... (1 − qx+n−1). lx 5 Another application would be given that a man and woman get married when the man is of age x and the woman is of age y, what is the probability that they will live together for the next year n years given that divorce is not an option. This will be given by ∗ lx+n ly+n  ∗ ∗ ∗ = (1 − qx) ... (1 − qx+n−1) · (1 − qy) ... (1 − qy+n−1). lx ly The asterisk denotes the difference in the woman’s and man’s age-distributions. This is needed since men and women have different survival functions.

1.5 Life Table Example

The United States started a census in 1790 to enumerate the population of the country. The census has taken place every 10 years since. The United States Census Bureau uses two methods to count everyone. They have questionnaires mailed out to every home in the United States, and they have workers traveling door-to-door. Of course this can not be completely accurate, so the Bureau uses statistics and other population information to estimate the number of people that have not been accounted for. The United States Census Bureau believes that the lastest census, in 2010, has been the most accurate one to date[6]. Table 1.2 is the 2010 United States life table made by United States Department of Health and Human Services Centers for Disease Control and Prevention. This table is for all people in the United States and thus includes all races and sexes. This is from the 2010 final mortality statistics collected and the 2010 population estimates based on the census[1].

Table 1.2: United States Life Table 2010

Age qx lx dx Lx Tx ex 0-1 0.006667 100,000 667 99,419 7,619,510 76.2 1-2 0.000449 99,333 45 99,311 7,520,090 75.7 2-3 0.000322 99,289 32 99,273 7,420,779 74.7 3-4 0.000247 99,257 25 99,245 7,321,507 73.8 4-5 0.000178 99,232 18 99,223 7,222,262 72.8 6 Table 1.2: United States Life Table 2010

Age qx lx dx Lx Tx ex 5-6 0.000166 99,215 17 99,206 7,123,039 71.8 6-7 0.000147 99,198 15 99,191 7,023,832 70.8 7-8 0.000129 99,183 13 99,177 6,924,642 69.8 8-9 0.000109 99,171 11 99,165 6,825,465 68.8 9-10 0.000087 99,160 9 99,156 6,726,299 67.8 10-11 0.000072 99,151 7 99,148 6,627,144 66.8 11-12 0.000078 99,144 8 99,140 6,527,996 65.8 12-13 0.000121 99,136 12 99,130 6,428,856 64.8 13-14 0.000209 99,124 21 99,114 6,329,726 63.9 14-15 0.000328 99,103 32 99,087 6,230,612 62.9 15-16 0.000451 99,071 45 99,049 6,131,525 61.9 16-17 0.000569 99,026 56 98,998 6,032,476 60.9 17-18 0.000690 98,970 68 98,936 5,933,478 60.0 18-19 0.000817 98,902 81 98,861 5,834,542 59.0 19-20 0.000945 98,821 93 98,774 5,735,681 58.0 20-21 0.001084 98,727 107 98,674 5,636,907 57.1 21-22 0.001216 98,620 120 98,560 5,538,233 56.2 22-23 0.001311 98,501 129 98,436 5,439,672 55.2 23-24 0.001354 98,371 133 98,305 5,341,237 54.3 24-25 0.001358 98,238 133 98,171 5,242,932 53.4 25-26 0.001348 98,105 132 98,039 5,144,760 52.4 26-27 0.001344 97,973 132 97,907 5,046,722 51.5 27-28 0.001345 97,841 132 97,775 4,948,815 50.6 28-29 0.001359 97,709 133 97,643 4,851,040 49.6 29-30 0.001384 97,576 135 97,509 4,753,397 48.7 30-31 0.001414 97,441 138 97,373 4,655,888 47.8 31-32 0.001444 97,304 140 97,233 4,558,516 46.8 32-33 0.001475 97,163 143 97,091 4,461,282 45.9 7 Table 1.2: United States Life Table 2010

Age qx lx dx Lx Tx ex 33-34 0.001506 97,020 146 96,947 4,364,191 45.0 34-35 0.001542 96,874 149 96,799 4,267,244 44.0 35-36 0.001592 96,724 154 96,647 4,170,445 43.1 36-37 0.001659 96,570 160 96,490 4,073,798 42.2 37-38 0.001738 96,410 168 96,326 3,977,307 41.3 38-39 0.001830 96,243 176 96,154 3,880,981 40.3 39-40 0.001941 96,066 186 95,973 3,784,827 39.4 40-41 0.002064 95,880 198 95,781 3,688,853 38.5 41-42 0.002217 95,682 212 95,576 3,593,072 37.6 42-43 0.002421 95,470 231 95,354 3,497,496 36.6 43-44 0.002684 95,239 256 95,111 3,402,142 35.7 44-45 0.002987 94,983 284 94,841 3,307,031 34.8 45-46 0.003303 94,699 313 94,543 3,212,190 33.9 46-47 0.003624 94,387 342 94,216 3,117,647 33.0 47-48 0.003968 94,045 373 93,858 3,023,431 32.1 48-49 0.004342 93,671 407 93,468 2,929,573 31.3 49-50 0.004746 93,265 443 93,043 2,836,105 30.4 50-51 0.005172 92,822 480 92,582 2,743,061 29.6 51-52 0.005617 92,342 519 92,083 2,650,479 28.7 52-53 0.006093 91,823 559 91,544 2,558,397 27.9 53-54 0.006611 91,264 603 90,962 2,466,853 27.0 54-55 0.007174 90,660 650 90,335 2,375,891 26.2 55-56 0.007792 90,010 701 89,659 2,285,556 25.4 56-57 0.008451 89,309 755 88,931 2,195,897 24.6 57-58 0.009121 88,554 808 88,150 2,106,965 23.8 58-59 0.009775 87,746 858 87,317 2,018,815 23.0 59-60 0.010415 86,889 905 86,436 1,931,498 22.2 60-61 0.011075 85,984 952 85,507 1,845,062 21.5 8 Table 1.2: United States Life Table 2010

Age qx lx dx Lx Tx ex 61-62 0.011791 85,031 1,003 84,530 1,759,554 20.7 62-63 0.012577 84,029 1,057 83,500 1,675,024 19.9 63-64 0.013484 82,972 1,119 82,412 1,591,524 19.2 64-65 0.014542 81,853 1,190 81,258 1,509,111 18.4 65-66 0.015783 80,663 1,273 80,026 1,427,853 17.7 66-67 0.017195 79,390 1,365 78,707 1,347,827 17.0 67-68 0.018699 78,025 1,459 77,295 1,269,120 16.3 68-69 0.020247 76,566 1,550 75,790 1,191,825 15.6 69-70 0.021917 75,015 1,644 74,193 1,116,035 14.9 70-71 0.023725 73,371 1,741 72,501 1,041,841 14.2 71-72 0.025734 71,631 1,843 70,709 969,340 13.5 72-73 0.028077 69,787 1,959 68,808 898,631 12.9 73-74 0.030750 67,828 2,086 66,785 829,824 12.2 74-75 0.033815 65,742 2,223 64,631 763,039 11.6 75-76 0.037090 63,519 2,356 62,341 698,408 11.0 76-77 0.040540 61,163 2,480 59,923 636,067 10.4 77-78 0.044677 58,684 2,622 57,373 576,144 9.8 78-79 0.049227 56,062 2,760 54,682 518,771 9.3 79-80 0.054348 53,302 2,897 51,854 464,089 8.7 80-81 0.060110 50,405 3,030 48,890 412,236 8.2 81-82 0.066576 47,375 3,154 45,798 363,346 7.7 82-83 0.073449 44,221 3,248 42,597 317,547 7.2 83-84 0.080709 40,973 3,307 39,320 274,950 6.7 84-85 0.090777 37,666 3,419 35,957 235,630 6.3 85-86 0.101080 34,247 3,462 32,516 199,674 5.8 86-87 0.112324 30,785 3,458 29,056 167,157 5.4 87-88 0.124544 27,327 3,403 25,626 138,101 5.1 88-89 0.137762 23,924 3,296 22,276 112,475 4.7 9 Table 1.2: United States Life Table 2010

Age qx lx dx Lx Tx ex 89-90 0.151991 20,628 3,135 19,061 90,199 4.4 90-91 0.167224 17,493 2,925 16,030 71,139 4.1 91-92 0.183440 14,568 2,672 13,232 55,108 3.8 92-93 0.200596 11,895 2,386 10,702 41,877 3.5 93-94 0.218632 9,509 2,079 8,470 31,175 3.3 94-95 0.237462 7,430 1,764 6,548 22,705 3.1 95-96 0.256985 5,666 1,456 4,938 16,157 2.9 96-97 0.277076 4,210 1,166 3,627 11,219 2.7 97-98 0.297597 3,043 906 2,591 7,593 2.5 98-99 0.318395 2,138 681 1,797 5,002 2.3 99-100 0.339311 1,457 494 1,210 3,205 2.2 100 and over 1.000000 963 963 1,995 1,995 2.1 Chapter 2

Life Expectancy

2.1 Life Expectancy

As mentioned above, the life expectancy remaining for a person aged x is denoted by ex and is defined to be the average number of years left to be lived for those who are currently age x. This would be the total number of person years yet to be lived after the start of that interval and dividing by the number of people alive at the beginning of that time interval. Therefore we have that

Tx ex = . lx The remaining life expectancy can be calculated for any age x; however, it is common to do life expectancy for newborns and people of age 40. Of course it makes sense to do it for newborns since this will be the overall amount of time that people born during that year can expect to live. Life expectancy at age 40 is often used when looking at inequality in health. Typically the higher income a person has the higher their life expectancy will be. When looking at the amount of income a person has it makes sense to look at people age of 40[15].

2.2 Importance of Life Expectancy

Life expectancy is one of the most commonly used values on the life table. It is used to calculate several important values such as life insurance premiums. Actuaries

10 11 need life expectancy values to calculate life insurance premiums because they need to know how long people can be expected to live. It is used to plan for retirement since people need to know how long they can expect to live after they retire. Life expectancy is used to plan for food, water, and shelter. If life expectancy is increasing there will be a change in the population. This can result in more people being alive in a given region and therefore increases the need for supplies such as food, water, and shelter. Life expectancy is often used to compare different regions. For this we use life expectancy at birth, that is, how long newborns can expect to live. This is used to compare the overall health of two regions. Of course the area with the higher life expectancy is considered to have the better health conditions.

2.3 Comparison

As mentioned above, the most common use of life expectancy values is to com- pare different populations. Using data from the World Bank Group, an international financial institution that focuses on human development in developing countries, the life expectancy from birth for the United States of America, Nigeria, Japan, and the World is given by the following table[7]:

Life Expectancy at Birth 2010 2011 2012 2013 2014

United States of America : 78.5 78.6 78.7 78.8 78.9 Nigeria: 51.3 51.7 52.1 52.4 52.8 Japan: 82.8 82.6 83.1 83.3 83.6 World: 70.5 70.8 71 71.2 71.5

Table 2.1: Comparing Life Expectancy, 2010 - 2014

Throughout my paper I am using United States data, so of course it was to be in- cluded in the table of life expectancy values. I chose to look at Nigeria’s life expectancy because it is substantially lower than that of the United States, in fact Nigeria’s life expectancy is one of the lowest in the World. On the other hand, Japan has one of the highest life expectancies in the World, and thus that is why I chose to use Japan in my data set. Lastly, I chose to include the World population as well to tie all the data 12 together. From this table it can be seen that the life expectancy is gradually increasing in all of the given populations. We can see that Nigeria has a much lower life expectancy at birth than the United States, Japan, and the World. From this we might conclude that the healthcare in Nigeria is not as good as the other regions. However, Nigeria’s life expectancy is increasing much more rapidly than all other regions. This would sug- gest that Nigeria is still a developing country with improving healthcare conditions. By just looking at the life expectancy from birth, the areas would be rated Japan, United States, World, and then Nigeria in terms of the best healthcare. Chapter 3

Mortality Rates

A mortality rate, also called death rate, is a measure of the number of deaths in a particular population that is then scaled to the size of that population. Typically given in units of deaths per 1,000, or 100,000 people. This definition is vague and there are different types of mortality rates. These different rates are defined and used differently.

3.1 Crude Mortality Rate

The crude mortality rate is defined to be the number of deaths throughout a time period divided by the average number of people in that time period. Usually when the term ”mortality rate” is used, what is meant is the crude mortality rate. The most common time period to use is one year. Since it is an average, the population size is measured in the middle of the time period. Therefore it is taken on July 1st if the period is the calendar year. This form of mortality rate measures the overall mortality of the entire population. total # of deaths crude rate = total # of people The problem with this measure of mortality arises when comparing regions with different age distributions. This is because populations with a higher number of older people will have a higher crude mortality rate since the older people will have a higher probability of dying. An example of this would be when comparing developed countries to underdeveloped countries. Typically developed countries have a higher proportion of older people, whereas the underdeveloped countries have a higher number

13 14 of younger people. The crude death rate would most likely be higher for the developed country. Table 3.1 uses the regions that were used in the life expectancy comparison to compare their crude mortality rates for the years 2010 to 2014. These years were chosen because these are the most current rates published by the World Bank[9]. Here the crude mortality rate is the number of deaths per 1,000 people from the population.

Crude Mortality Rates 2010 2011 2012 2013 2014

United States: 8 8.1 8.1 8.1 8.1 Nigeria: 14 13.7 13.4 13.2 12.9 Japan: 9.5 9.9 10 10.1 10 World: 7.9 7.9 7.8 7.8 7.7

Table 3.1: Comparing Crude Mortality Rates 2010 - 2014

When comparing these populations using the crude mortality rate, we see that the United States’ crude rate is remaining almost constant, Nigeria’s crude rate is decreasing rather rapidly, Japan’s crude rate is increasing slowly, and the World’s crude rate is decreasing slowly. Using the crude mortality rates alone, one would assume that the United States’ healthcare is remaining approximately the same, Nigeria’s healthcare is improving rather quickly, Japan’s healthcare is worsening, and the World’s overall healthcare is improving. The rankings based on crude mortality rates alone would be the World, United States, Japan and Nigeria. However, these assumptions are based on crude mortality rates alone. To get a better estimate on the healthcare of each region, other forms of mortality rates should be used.

3.2 Age-Specific Mortality Rate

The age-specific mortality rate is a death rate for a specific age group. Typically these intervals are either one or five years in length. In the United States the time period is one year. The age-specific mortality rates are values given in the life table. This value is calculated by taking the number of deaths throughout the given time period and dividing by the number of people aged x in that time period. Therefore the 15 equation becomes dx qx = . lx Figure 3.1 is a graph for the age-specific mortality rates for the United States from the year 2010[1].

Figure 3.1: Age-Specific Mortality Rates for the United States, 2010

By looking at this graph we can see that mortality rates are extremely low until about the age of 60. After the age of 60 the mortality rates start increasing rapidly. By age 100 the mortality rates are around 30%. These data seem reasonable when thinking about the human lifetime. Since the mortality rates are so low early in life, this graph is not very informa- tive. To get a better look at the age-specific mortality rates, the log of these rates are examined. Figure 3.2 is a graph of the log of the age-specific mortality rates that were given in Figure 3.1[1]. 16

Figure 3.2: Log(Age-Specific Mortality Rates)+5 for the United States, 2010

When looking at this graph, we can see that this graph is approximately linear from age 40 − 100. This is Gompertz’s Law. In 1825, Benjamin Gompertz proposed the idea that there is an exponential increase in mortality rates with age. This can easily be seen when looking at the log of the mortality rates[2]. For my graph, I took the Log of the age-specific mortality rates and then chose to add 5 so that the values were greater than zero.

3.3 Age-Adjusted Mortality Rate

As described above, when comparing mortality rates of two different populations, the age-distribution of each population plays a role. Therefore the crude death rate is not necessarily the most informative way to compare populations. This was realized in the middle of the nineteenth century by health practitioners in England when the age- distributions were changing rapidly in the populations being compared. F.G.P. Neison, an actuary, was the first to come up with the idea of using a standard population as weights to construct an age-adjusted mortality rate in 1844. This idea was first used 17 in 1883 in England and Wales using the standard population of the 1881 population. The standard population was changed every time there was a new census, that is, every ten years. This became cumbersome, so the population in 1901 became the standard population[4]. Age-adjusted mortality rates are rates that would have existed if the population under study had the same age-distribution as the standard population. Populations are always changing and thus the age-distributions are always changing. This makes it hard to compare regions if their age-distributions differ. To compensate for the change in the age-distribution, the age-specific mortality rates are multiplied by the proportion of the standard population that is of that age. The standard population will be used in calculating the age-adjusted mortality rate for all of the regions being compared. The standard population, sometimes called the standard millions, is the age-distribution used as a weight to make the age-adjusted mortality rate. This calculation then becomes

ω X age-adjusted rate = dxpx i=0 again, ω denotes the maximum age possible for the population, 100 in the United States, and px denotes the proportion of the standard population that is of age x. The United States wanted to compare mortality rates with England, so the 1901 England population was used as the standard population. This remained until the 1940s, when it was determined that there was a significant difference between the standard population and the United States population in the 1940s. Therefore the National Center for Health Statistics decided that the 1940 census population would be used as the new standard population. In the mid-1990s the National Center for Health Statistics decided that a new standard population was again needed. It was recommended that the 2000 population age-distribution be used. Since then the United States has used the 2000 United States population age-distribution as the standard population[4]. The 1940 and the 2000 standard populations are given in Table 3.2. These values were given by the National Cancer Institute[14]. These values are given as the standard million, thus they are calculated such that the total population, all age groups combined, equals one million. 18 Table 3.2: United States 2000 and 1940 Standard Population

Age 2000 Standard Million 1940 Standard Million 0 13,818 15,343 1-4 55,317 64,718 5-9 72,533 81,147 10-14 73,032 89,208 15-19 72,169 93,670 20-24 66,478 88,007 25-29 64,529 84,277 30-34 71,044 77,789 35-39 80,762 72,495 40-44 81,851 66,742 45-49 72,118 62,697 50-54 62,716 55,114 55-59 48,454 44,383 60-64 38,793 35,911 65-69 34,264 28,911 70-74 31,773 19,515 75-79 26,999 11,422 80-84 17,842 5,881 85+ 15,508 2,770

When comparing the United States population from 1940 and 2000, we see a lot of changes in the age-distribution. In 1940 there were more newborns, children, and young adults than there are in 2000. There are more adults and older people in 2000 than there in 1940. Therefore it made sense to change the standard population from 1940 to 2000 for the United States. 19 3.4 Comparison

As mentioned above, there is a difference between the crude mortality rate and the age-adjusted mortality rate. This can be shown in the table and graph below. The crude and age-adjusted mortality rates are from the National Vital Statistics Reports from the years 2010, 2011, 2012, and 2013[8]. The standard population used in the calculation of all of the age-specific mortality rate is the United States age-distribution from 2000. Both of these rates are given per 100,000 people.

Mortality Rate 2010 2011 2012 2013

Crude: 799.5 806.5 810.2 821.5 Age-Specific: 747 740.6 732.8 731.9

Table 3.3: Comparing Crude and Age-Specific Mortality Rates 2010 - 2014

Figure 3.3: Comparing Crude and Age-Adjusted Mortality, United States

By looking at the crude mortality rate it would appear that there is an increase in mortality in the United States throughout the four year period. This can really be seen when looking at Figure 3.3. We see that the crude death rate is remaining approximately constant, whereas the age-adjusted rate is decreasing. When looking at the age-adjusted mortality rate alone, we see that it is decreasing. However, when combining the two rates, we can conclude that the United States population is increasing in age. This 20 conclusion is reached because the crude death rate is remaining constant, thus there are the same number of deaths in the total population year after year, but the age-adjusted mortality rate is decreasing with time. With a population growing older, there will be an increase in the number of deaths for older people and a decrease in the number of deaths for younger people. Therefore approximately the same number of deaths occur each year, and thus the crude rate remains constant. By examining Figure 3.3, we see that it is not always the case that the crude rate is above the age-adjusted rate. In fact before 2000, the age-adjusted rate was higher than the crude rate. The two rates are the same for the time period around 2000. This is because the 2000 standard population is used for the calculation of the age-adjusted rate. Of course then it is appropriate that the two rates are equal in 2000. It also seems appropriate that the rates surrounding 2000 are almost equivalent as the age- distribution does not change drastically those few years. From the example above, it is clear that the age-adjusted mortality rate is best used when comparing two regions with different age-distributions, or when comparing the same region at different times. It was mentioned that the age-adjusted mortality rates are better used to compare populations. I had originally planned to use age-adjusted mortality rates to compare my four populations, however, I could not find these data. Therefore the crude mortality rate was used above to compare the populations. The age-adjusted mortality rates would have been more informative. Chapter 4

Life Table Functions

4.1 Density Function

The density function of lifetime, sometimes called the event density, is defined to be the proportion of people dying at age t. This is denoted by f(t). The time of death will be denoted by T. This function can be found by using some of the values found on the life table. The density function is equivalent to the number of deaths per age, t, and dividing by the size of the original cohort.

f(t) = P (T = t)

d f(t) = t l0 ω X f(t) = 1 t=0 The density function for the United States for the year 2010 is given in Figure 4.1 below[1].

21 22

Figure 4.1: Density Function for the United States, 2010

4.2 Cumulative Distribution Function

The cumulative distribution function of lifetime is the proportion of people dying by age t. This is denoted by F (t).

F (t) = P (T ≤ t) Z t (4.1) = f(x)dx 0 0 ≤ F (T ) ≤ 1

F (t) ≤ F (t + 1)

Figure 4.2 illustrates the cumulative distribution function for the Unites States population using data from 2010[1]. These values were calculated by adding up the number of people that die at each age t, and dividing by 100, 000 to make this value be a rate. Therefore it is given by the following equation

100, 000 − l Pt−1 d F (t) = t = i=0 i . 100, 000 100, 000 23

Figure 4.2: Cumulative Function for the United States, 2010

By looking at the graph we see that a very small proportion of people die before the age of 40, and then the proportion of deaths increases slightly to the age of 60 and increases more rapidly from the age of 60 to 100. Since the United States life table only goes to age 100, the proportion of deaths by age 100 is 1.

F (100) = 1

4.3 Survival Function

The survival function is the proportion of people that survive to age t. This is denoted by S(t). S(t) = P (T > t) = 1 − F (t) = P [(T > 0) ∩ (T > 1) ∩ ... ∩ (T > t)] = P (T > 0) · P (T > 1|T > 0) · ... · P (T > t|[(T > 0) ∩ (T > 1) ∩ ... ∩ (T > t − 1)]) = P (T > 0) · P (T > 1|T > 2) · ... · P (T > t|T > t − 1)

= P1 · P2 · ... · Pn (4.2) 24

Where Pi = P (T > i|T > i − 1). The following assumptions apply to the survival function:

S(0) = 1

S(t + 1) ≤ S(t)

lim S(t) → 0. t→ω Therefore it is assumed that all newborns survive birth, the survival function is a non-increasing function, and that all people must die eventually. All of which make sense when thinking about the survival of humans. A common thing to do with the survival function is to look at the survivorship curve. The survivorship curve is a graph showing the number of people surviving to different times t. Figure 4.3 illustrates three very different shapes that the survival function can take[10].

Figure 4.3: Three Types of Survivorship Curves

There are three types of survivorship curves: Type I - This type of curve has a high survival rate in early and middle life, and then low survival rate after some point. A good example of this type would be humans. Humans are not likely to die as children and young adults, but around the age of 60 the 25 mortality rates increase rapidly. Type II - This type of curve has constant survival rate regardless of age if the log of the number of survivors is used. Therefore the organism experiences the same survival/mortality rate throughout its entire lifetime. Type III - The animals of this curve experience the greatest mortality rate early in life. They then have a much lower mortality rate for the remainder of their life. An example of this would be frogs. Frogs do not often make it past the tadpole stage, but if they do then can live for several years. As mentioned above, humans have a type 1 survivorship curve since we are not likely to die young, but have a higher chance of dying as we age. This can be displayed in the graph below of the human survivorship curve for the United States for the year 2010[1]. This graph is for all people in the United States and therefore includes all sexes and all races.

Figure 4.4: Survival Function for the United States, 2010

From this graph, we can see that almost all children and young adults survive. At the age of approximately 60, the survival rates start to plummet, and thus there are fewer people surviving. We see that over 50% of the population is alive at the age of 80. 26 From the graph of f(t), we can see that the modal age will be approximately 87. From the graph of F (t), we can see that the median age will be approximately 84.5. The mean age was calculated when calculating the life expectancy from birth. Therefore the mean age is 78.6.

4.4 Hazard Function

The hazard function is the instantaneous rate of failure. It will be denoted by h(t). The hazard function can be thought of as the probability of dying instantly given that the person has survived to age t. This value is given in the life table under the column qt, the probability that a person age t will die before the age t + 1. The hazard function will be given by the following equations:

P (t ≤ T < t + ∆t) h(t) = lim ∆x→0 ∆tS(t) (4.3) f(t) = . S(t)

The hazard function has the following property:

h(t) ≥ 0 ∀t ≥ 0. Chapter 5

Population Pyramids

5.1 Definition

A population pyramid, sometimes called an age pyramid, is a graph that illustrates the distribution of age groups by sex in a given population. It takes the shape of a pyramid because half of the pyramid is for the male age-distribution, typically the left half, and the other half is for the female age-distribution, typically the right half. Thus it can be thought of as two different histograms put back to back. One of the histograms depicts the age-distribution of females, and the other an age-distribution of males. The population pyramid tells how many people of each age and sex live in that country.

5.2 Types of PopulationPyramids

Population pyramids are used to understand the overall age-distribution of a pop- ulation. Population pyramids can take on different shapes, and these different shapes represent different age-distributions. These different age-distributions reveal a lot about the populations themselves. Thus the different shapes of population pyramids can be analyzed. Three main types of population pyramids are: expansive, constrictive, and stationary. Figure 5.1 shows the different shape that each population pyramid takes[11].

27 28

Figure 5.1: Three Types of Population Pyramids

The expansive pyramid has a very wide base and narrows as the age increases. This indicates that there is a high birth rate, since the base is very wide. The high birth rate causes a lot of young people in the population and therefore the wide base. This implies that the population is increasing rapidly. The death rate can be either high or low. If the death rate is high, then the pyramid would be narrowing much quicker and the maximum age of the population would be low. If the death rate is low, then the pyramid would be narrowing at a slower rate and the maximum age of the population would be higher. This type of populations typically represents developing nations with lower than average life expectancies. The constrictive pyramid is constricted at some particular age. Therefore at some point birth rates decreased. This will cause that area to have less people.This type of population typically represents a recently developed nation. The stationary pyramid has a somewhat equal proportion of people in each age- interval. Of course the values near the top of the pyramid will be smaller proportions since all people in the population must die eventually. The birth and death rates are low. Low birth rates will make the base of the pyramid small and low death rates will make the narrowing of the pyramid take a long time. The maximum age of the popu- lation will be the high. These indicate that the overall quality of life is high. This type 29 of population typically represents a well developed nation. Below are the population pyramids for the regions that have been under consid- eration. They have been made by the United Nations, Department of Economics and Social Affairs, Population Division[12].

Figure 5.2: Population Pyramid of United States

Figure 5.3: Population Pyramid of Nigeria 30

Figure 5.4: Population Pyramid of Japan

Figure 5.5: Population Pyramid of the World

From these population pyramids, it is clear that there are different shapes that a population pyramid can take. We see that the United States has a constrictive type of population pyramid, and therefore has many middle-aged people. Nigeria has an expansive population pyramid, and thus has a large portion of young people in the population. Japan also has a constrictive population pyramid. And the World also has 31 an expansive type of population pyramid. From this we can conclude that the United States and Japan are more developed countries than Nigeria. Of course we know this to be true. Chapter 6

Population Projections

6.1 Population Projection Definition

A population projection is the calculation of the expected number of people of each age that will be alive at future dates. This is calculated using the given age-distribution now and age-specific mortality rates and age-specific fertility rates. Population projec- tion can be understood by looking at two things: the people already alive at time t that survive to time t + n, and the newborns that were born in that given time interval.

6.2 Leslie Matrix

The Leslie Matrix is used in a discrete, age-structured model of population growth. It is a matrix that includes age-specific fertility rates along with age-specific survival rates. The Leslie Matrix will be denoted L from here on. The following assumptions are made when using the Leslie Matrix. It is assumed that only one sex will be considered, the female. This is because the Leslie Matrix includes fertility rates, and of course this is clearer when only looking at females. It is assumed that the population is closed to migration. Therefore there are no people coming into or going out of the region under study. Of course this is not the case in the United States population, but it is the case for the World population. We assume that the mortality and fertility rates remain constant throughout the time interval under study. Again, this will not be the case since these rates are constantly changing. Lastly, it is assumed that there is no maximum in

32 33 the size that the population can reach. Thus there is population increase or decrease in an unlimited environment[13]. This assumption is unrealistic because there will always be a limited supply of food, water, and shelter that are needed for survival. The Leslie Matrix is used to project a population in the following equation:

nˆt+1 = Lnˆt wheren ˆt+1 denotes the age-distribution vector at time t + 1 andn ˆt denotes the age- distribution vector at time t. Of course the Leslie Matrix is denoted by L. As mentioned above, the Leslie Matrix will consist of fertility and mortality rates. We will be using fertility rates to forecast the number of newborns that there will be the following year. This will be done by taking the age-specific fertility rates and multiplying by the age-distribution vector for females. The number of newborns next year will be denoted by n0t+1 . This will take the form

ω X n0t+1 = Filnt i=0 where Fi denotes the fertility rate of a mother aged i. We define the age-specific fertility rates to be the observed number of births to mothers age x and dividing by the average number of women in that given age group x. As stated before, the average number of women in the age group will be calculated in the middle of the year, on July 1st. Since only the females are considered in the matrix, the fertility rates are multiplied by some constant C. This constant C is the proportion of newborns that are female. We will be using mortality rates to forecast the number of people of all ages, other than newborns, that there are for the next year. This will be done by taking the age- specific survival rates. The age-specific survival rates are the probabilities that a person age x at the beginning of the year will survive the whole year. Therefore it is equivalent to 1 − dx. These rates will be denoted by Sx. Therefore we get that the age-specific survival rates that we will be using will be found by using the following equation

Sx = 1 − qx.

From these age-specific survival rates we can find the number of people of age x that there will be at time t + 1. For example, the number of one-year olds there will 34 be at time t + 1 is dependent on the number of newborns there were at time t and the probability of surviving the first year of life. This is because the only way to be a one-year old this year is by being a newborn last year and surviving the year. This will be given by the following equation

n1t+1 = S0n0t.

Then in general we have

nxt+1 = Sx−1nx−1t .

Putting all of these pieces back together, we are now ready for the Leslie Matrix. The matrix takes the following form   F0 F1 ... Fω−1 Fω   S 0 ... 0 0   0     0 S1 ... 0 0       .     .       .    0 0 ... Sω−1 0

Then the age-structured model becomes the following equation      n0 F0 F1 ... Fω−1 Fω n0      n  S 0 ... 0 0 n   1   0  1       n2   0 S1 ... 0 0 n2             .  =  .  .        .   .  .             .   .  .       nω 0 0 ... Sω−1 0 nω t+1 t.

6.3 Stable Age-Distribution

Using this Leslie Matrix we can find the stable population and the rate of change of the population. We define the stable population to be the age-distribution to which 35 a closed population would evolve to if its age-specific rates of fertility and mortality were to continue indefinitely. The finite rate of increase, λ, is defined as the factor that the stable population is multiplied by each year. Each age interval will be changing by this same constant. To find the stable population we need to find the eigenvalues and eigenvectors of the matrix L. This is done by solving the following equation

Lnˆ = λnˆ where λ is the eigenvalue andn ˆ is the associated eigenvector. To solve for λ, we need to solve the characteristic equation

f(λ) = det(L − λnˆ) = 0.

Assuming that ω = 100, for the United States this is the maximum age possible on the life table, we get the following determinant.

F0 − λ F1 ... F99 F100

S −λ ... 0 0 0

.

. = 0

.

0 0 ... −λ 0

0 0 ... S99 −λ By simplifying the determinant and expanding along the first column we get

101 98 λ − S0S1F2λ − ... − S0S1...S99F100 = 0.

According to Descartes’ Rule, the number of positive roots of a polynomial cannot be greater than the number of changes of sign in its coefficients. Thus when looking at the characteristic equation, there will be at most one positive root. The Secant Method will be used to find this positive root. The Secant Method is a root-finding algorithm that uses a succession of roots of secant lines to better approximate a root of a function f. It is defined by the following recurrence relation

x1 − x0 x = x1 − f(x1) f(x1) − f(x0) 36 where x0 and x1 are approximations to the root of the function x. For the characteristic equation given, let λ0 and λ00 be two approximations to the root of f(λ). The approximation for λ becomes

00 0 00 00 λ − λ λ = λ − f(λ ) . f(λ00 ) − f(λ0 )

As mentioned above, we define λ to be the limiting positive rate of population change. Thus during each time interval the number of individuals in each age class will be multiplied by λ. If λ < 1, then the population is decreasing in size. If λ > 1, then the population is increasing. If λ = 1, then the limiting population is stationary and thus remains unchanged year after year. After finding the rate of change, it is possible to find the stable age-distribution vector. Let λ∗ denote the unique positive eigenvalue of the Leslie Matrix under consid- eration. The associated stable age-distribution vector, denoted by nˆ∗, is given by the following equations Lnˆ∗ = λ∗nˆ∗

Lnˆ∗ − λ∗nˆ∗ = 0ˆ

(L − λ∗I)nˆ∗ = 0ˆ.

Therefore to find the stable age-distribution vector we need to find the kernel of L − λ∗I. Chapter 7

My Findings and Projections

7.1 Data Being Used

I am using the Leslie Matrix to project the United States 2010 age-distribution to the years 2011, 2016, and 2050. I am using age-specific fertility and mortality rates published in the National Vital Statistics Reports by the Centers for Disease Control and Prevention. I chose to use the year 2010 because this is when the last census was done, and therefore the most information can be found for this year. I also chose to use 2010 because the largest current life table published by the Center for Disease Control and Prevention is from the year 2011. By using the data from 2010 I am able to project the age-distribution for the year 2011 by using the Leslie Matrix and I will be able to compare with the known results for 2011. This will give an idea on how well the Leslie Matrix works for projecting the age-distribution a short amount of time, since it’s only one year. The age-specific mortality rates came directly from the National Vital Statistics United States Life Tables, 2010 article. As shown in Table 1.1, the mortality rates are given in the qx column for each age x. The age-specific fertility rates were found in the National Vital Statistics Report Births: Final Data for 2010. However; these rates are given in ranges of 2 or 5 year time intervals. The data was given as such:

37 38 Age 10-14 15-17 18-19 20-24 25-29 30-34 35-39 40-44 45-49 Fertility Rate: 0.4 17.3 58.2 90 108.3 96.5 45.9 10.2 0.7

Table 7.1: Birth Rates by Age of Mother. United States, 2010

Of course all other values not included in table are zero. Children under the age of 10 and women over the age of 50 seldom have children themselves. To get the age-specific fertility rates for each age, I chose to take the middle of each time interval and set the fertility rate to be the fertility rate for that age group. I then used Microsoft Excel to plot those data points and smooth out the curve.. From the graph I was able to read the age-specific fertility rates for each age. Of course these rates are approxi- mations.

Figure 7.1: Birth Rates for the United States 2010 39 Using the Leslie Matrix assumes that the fertility and mortality rates remain con- stant throughout the projections. To see what would happen if these values were to suddenly change in 2010, I chose to vary the fertility and mortality rates. I did a pro- jection with half of the fertility rate, double the fertility rate, half the mortality rate, and half the fertility rate. All of these rates are given in the table below.

Table 7.2: Survival and Fertility Rates for Women in the United States, 2010

1 1 Age Sx Sx = 1 − 2 · qx Sx = 1 − 2 · qx Fx 2 · Fx 2 · Fx 0-1 0.994447 0.9972235 0.988894 0 0 0 1-2 0.999593 0.9997965 0.999186 0 0 0 2-3 0.999774 0.999887 0.999548 0 0 0 3-4 0.999828 0.999914 0.999656 0 0 0 4-5 0.999864 0.999932 0.999728 0 0 0 5-6 0.999877 0.9999385 0.999754 0 0 0 6-7 0.999892 0.999946 0.999784 0 0 0 7-8 0.999903 0.9999515 0.999806 0 0 0 8-9 0.99991 0.999955 0.99982 0 0 0 9-10 0.999914 0.999957 0.999828 0 0 0 10-11 0.999913 0.9999565 0.999826 0 0 0 11-12 0.999906 0.999953 0.999812 0 0 0 12-13 0.999889 0.9999445 0.999778 .002 .001 .004 13-14 0.999862 0.999931 0.999724 .0035 .00175 .007 14-15 0.999828 0.999914 0.999656 .005 .0025 .01 15-16 0.999791 0.9998955 0.999582 .0085 .00425 .017 16-17 0.999754 0.999877 0.999508 .0145 .00725 .029 17-18 0.999718 0.999859 0.999436 .0245 .0125 .049 18-19 0.999683 0.9998415 0.999366 .032 .016 .064 19-20 0.999649 0.9998245 0.999298 .0375 .01875 .075 20-21 0.999613 0.9998065 0.999226 .0415 .02075 .083 21-22 0.999576 0.999788 0.999152 .045 .0225 .09 40 Table 7.2: Survival and Fertility Rates for Women in the United States, 2010

1 1 Age Sx Sx = 1 − 2 · qx Sx = 1 − 2 · qx Fx 2 · Fx 2 · Fx 22-23 0.999544 0.999772 0.999088 .0475 .02375 .095 23-24 0.999521 0.9997605 0.999042 .05 .025 .1 24-25 0.999504 0.999752 0.999008 .052 .026 .104 25-26 0.999487 0.9997435 0.998974 .053 .0265 .106 26-27 0.999467 0.9997335 0.998934 .054 .027 .108 27-28 0.999445 0.9997225 0.99889 .0535 .02675 .107 28-29 0.999421 0.9997105 0.998842 .053 .0265 .106 29-30 0.999392 0.999696 0.998784 .0525 .02625 .105 30-31 0.999359 0.9996795 0.998718 .0505 .02525 .101 31-32 0.999321 0.9996605 0.998642 .0485 .02425 .097 32-33 0.999278 0.999639 0.998556 .0445 .02225 .089 33-34 0.999232 0.999616 0.998464 .04 .02 .08 34-35 0.999181 0.9995905 0.998362 .0335 .1675 .067 35-36 0.99912 0.99956 0.99824 .0275 .01375 .055 36-37 0.999051 0.9995255 0.998102 .023 .0115 .046 37-38 0.998979 0.9994895 0.997958 .0185 .00925 .037 38-39 0.998905 0.9994525 0.99781 .0145 .00725 .029 39-40 0.998823 0.9994115 0 .997646 .0105 .00525 .021 40-41 0.998734 0.999367 0.997468 .0075 .00375 .015 41-42 0.998629 0.9993145 0.997258 .005 .0025 .01 42-43 0.998493 0.9992465 0.996986 .0035 .00175 .007 43-44 0.998324 0.999162 0.996648 .0025 .00125 .005 44-45 0.998133 0.9990665 0.996266 .0015 .00075 .003 45-46 0.99794 0.99897 0.99588 .001 .0005 .003 46-47 0.997747 0.9988735 0.995494 0 0 0 47-48 0.997538 0.998769 0.995076 0 0 0 48-49 0.997308 0.998654 0.994616 0 0 0 41 Table 7.2: Survival and Fertility Rates for Women in the United States, 2010

1 1 Age Sx Sx = 1 − 2 · qx Sx = 1 − 2 · qx Fx 2 · Fx 2 · Fx 49-50 0.997059 0.9985295 0.994118 0 0 0 50-51 0.996788 0.998394 0.993576 0 0 0 51-52 0.996511 0.9982555 0.993022 0 0 0 52-53 0.996244 0.998122 0.992488 0 0 0 53-54 0.995991 0.9979955 0.991982 0 0 0 54-55 0.995739 0.9978695 0.991478 0 0 0 55-56 0.995473 0.9977365 0.990946 0 0 0 56-57 0.995167 0.9975835 0.990334 0 0 0 57-58 0.994809 0.9974045 0.989618 0 0 0 58-59 0.994386 0.997193 0.988772 0 0 0 59-60 0.993908 0.996954 0.987816 0 0 0 60-61 0.993387 0.9966935 0.986774 0 0 0 61-62 0.992831 0.9964155 0.985662 0 0 0 62-63 0.992227 0.9961135 0.984454 0 0 0 63-64 0.991556 0.995778 0.983112 0 0 0 64-65 0.990792 0.995396 0.981584 0 0 0 65-66 0.98989 0.994945 0.97978 0 0 0 66-67 0.988852 0.994426 0.977704 0 0 0 67-68 0.98775 0.993875 0.9755 0 0 0 68-69 0.986624 0.993312 0.973248 0 0 0 69-70 0.985431 0.9927155 0.970862 0 0 0 70-71 0.984088 0.992044 0.968176 0 0 0 71-72 0.982533 0.9912665 0.965066 0 0 0 72-73 0.980817 0.9904085 0.961634 0 0 0 73-74 0.978862 0.989431 0.957724 0 0 0 74-75 0.976699 0.9883495 0.953398 0 0 0 75-76 0.974281 0.9871405 0.948562 0 0 0 42 Table 7.2: Survival and Fertility Rates for Women in the United States, 2010

1 1 Age Sx Sx = 1 − 2 · qx Sx = 1 − 2 · qx Fx 2 · Fx 2 · Fx 76-77 0.97163 0.985815 0.94326 0 0 0 77-78 0.968554 0.984277 0.937108 0 0 0 78-79 0.96504 0.98252 0.93008 0 0 0 79-80 0.961235 0.9806175 0.92247 0 0 0 80-81 0.957046 0.978523 0.914092 0 0 0 81-82 0.952331 0.9761655 0.904662 0 0 0 82-83 0.946956 0.973478 0.893912 0 0 0 83-84 0.940668 0.970334 0.881336 0 0 0 84-85 0.933037 0.9665185 0.866074 0 0 0 85-86 0.924439 0.9622195 0.848878 0 0 0 86-87 0.915224 0.957612 0.830448 0 0 0 87-88 0.905066 0.952533 0.810132 0 0 0 88-89 0.893912 0.946956 0.787824 0 0 0 89-90 0.881719 0.9408595 0.763438 0 0 0 90-91 0.868453 0.9342265 0.736906 0 0 0 91-92 0.854096 0.927048 0.708192 0 0 0 92-93 0.838644 0.919322 0.677288 0 0 0 93-94 0.822114 0.911057 0.644228 0 0 0 94-95 0.804546 0.902273 0.609092 0 0 0 95-96 0.786003 0.8930015 0.572006 0 0 0 96-97 0.766572 0.883286 0.533144 0 0 0 97-98 0.746365 0.8731825 0.49273 0 0 0 98-99 0.725517 0.8627585 0.451034 0 0 0 99-100 0.704183 0.8520915 0.408366 0 0 0 43 7.2 United States Projections

Using the fertility and mortality rates for the United States 2010, I put these values into the Leslie Matrix. The matrix takes the form:   0 0 0 0 0 0 ... 0 0 0 0   .994447 0 0 0 0 0 ... 0 0 0 0      0 .999593 0 0 0 0 ... 0 0 0 0      0 0 .999593 0 0 0 ... 0 0 0 0    0 0 0 .999774 0 0 ... 0 0 0 0      0 0 0 0 .999828 0 ... 0 0 0 0      0 0 0 0 0 .999864 ... 0 0 0 0    ......   ......     0 0 0 0 0 0 ....746365 0 0 0      0 0 0 0 0 0 ... 0 .725517 0 0   0 0 0 0 0 0 ... 0 0 .704813 0

Of course this a small version of the actual matrix. The Leslie Matrix that I am using is 101 × 101 in size. The fertility rates are all zero until the age of 12 and then are zero after the age of 46. This explains all of the zeros in the first row. The survival rates are given in the off diagonal. The age-distribution for the United States 2010 was given in 5−year age intervals. To get the age-specific distribution, I used a method similar to the one I used to find the fertility rates. I took the middle of the age for each of the age groups and assigned the value given for that age group to that specific age. I then plotted these points in Excel and connected the points using the smooth curve option. From this I was able to approximate the age-distribution by year for 2010. Multiplying the Leslie Matrix by the age-distribution from 2010, the age-distribution for 2011 is produced. Raising the Leslie Matrix to higher powers, we can get the age- distributions for future dates. This can be shown by the following equation:

nˆt+1 = Lnˆt

nˆt+2 = Lnˆt+1

nˆt+2 = L(Lnˆt) 44 2 nˆt+2 = L nˆt

n nˆt+n = L nˆt.

The age-distributions for United States females for the years 2010, 2011, 2016, and 2050 are given in Table 7.3.

Table 7.3: Comparing Female Age-Distributions of the United States: 2010, 2011, 2016 and 2050

Age 2010 2011 2016 2050

0-1 1968390 1990166 2001469 1868669 1-2 1971525 1957459 1990126 1862112 2-3 1973877 1970722 1987829 1864842 3-4 1977013 1973430 1984886 1867540 4-5 1979364 1976672 1981226 1869948 5-6 1982500 1979094 1977252 1872011 6-7 1985635 1982256 1955377 1873742 7-8 1985635 1985420 1969215 1875238 8-9 1994258 1985442 1972176 1876527 9-10 1998962 1994078 1975578 1877713 10-11 1983284 1998790 1978097 1878815 11-12 1983284 1983111 1981328 1879948 12-13 2016208 1983097 1984519 1881167 13-14 2042077 2015984 1984513 1882574 14-15 2057755 2041795 1993049 1884297 15-16 2104789 2057401 1997587 1886388 16-17 2128307 2104349 1981676 1888947 17-18 2144769 2127783 1981360 1892096 18-19 2147904 2144164 2013874 1895938 19-20 2143985 2147223 2039293 1900493 20-21 2124387 2143232 2054512 1905761 21-22 2119684 2123564 2101020 1911703 45 Table 7.3: Comparing Female Age-Distributions of the United States: 2010, 2011, 2016 and 2050

Age 2010 2011 2016 2050 22-23 2111844 2118785 2124039 1918162 23-24 2104005 2110880 2140018 1924941 24-25 2100870 2102997 2142724 1931693 25-26 2096950 2099827 2138431 1938269 26-27 2095382 2095874 2118541 1944458 27-28 2093815 2094265 2113542 1950197 28-29 2081272 2092652 2105448 1955351 29-30 2045996 2080066 2097375 1959755 30-31 1998962 2044752 2093980 1963338 31-32 1995042 1997680 2089769 1965996 32-33 1987987 1993687 2087860 1967706 33-34 1986419 1986551 2085904 1968431 34-35 1987203 1984893 2072966 1968001 35-36 2002881 1985575 2037342 1966163 36-37 2014640 2001118 1989965 1962964 37-38 2024831 2012728 1985451 1958641 38-39 2034238 2022763 1977752 1953369 39-40 2038941 2032010 1975455 1947582 40-41 2055403 2036541 1975426 1924004 41-42 2061674 2052800 1990120 1935376 42-43 2093031 2058847 2000820 1935816 43-44 2155743 2089876 2009818 1936407 44-45 2179260 2152129 2017832 1935794 45-46 2241973 2175191 2020934 1935504 46-47 2285088 2237354 2035449 1934809 47-48 2300766 2279939 2039642 1930659 48-49 2316444 2295101 2068401 1934457 46 Table 7.3: Comparing Female Age-Distributions of the United States: 2010, 2011, 2016 and 2050

Age 2010 2011 2016 2050 49-50 2300766 2310208 2127847 1933974 50-51 2300766 2293999 2148334 1913327 51-52 2316444 2293375 2207179 1907347 52-53 2270350 2308361 2246404 1932427 53-54 2230214 2262049 2258636 1950282 54-55 2191019 2221273 2270500 1957647 55-56 2120467 2181683 2251585 1994203 56-57 2073433 2110867 2248003 2007777 57-58 2025615 2063412 2259641 2014023 58-59 1970741 2015100 2210895 2007063 59-60 1912732 1959677 2167542 1992786 60-61 1869617 1901079 2124995 1963230 61-62 1799066 1857253 2051711 1946683 62-63 1746074 1786168 2000878 1926395 63-64 1657963 1732501 1948958 1905195 64-65 1626606 1643963 1889960 1887197 65-66 1528618 1611628 1827699 1867257 66-67 1403193 1513163 1779279 1847945 67-68 1329506 1387550 1704321 1826951 68-69 1246412 1313219 1645654 1794757 69-70 1152343 1229739 1553787 1741746 70-71 1116283 1135554 1514983 1677934 71-72 1026918 1098520 1414086 1649053 72-73 1003400 1008980 1288411 1615617 73-74 959502 984151 1210833 1584519 74-75 925010 939220 1124941 1552830 75-76 888950 903456 1029578 1529866 47 Table 7.3: Comparing Female Age-Distributions of the United States: 2010, 2011, 2016 and 2050

Age 2010 2011 2016 2050 76-77 857594 866087 986074 1500591 77-78 826238 833264 895649 1466786 78-79 791746 800255 862686 1428719 79-80 768228 764066 811674 1383473 80-81 744711 738447 768405 1342160 81-82 721194 712722 723591 1290061 82-83 688270 686815 682341 1248963 83-84 646723 651761 640698 1219988 84-85 599689 608351 596276 1162071 85-86 556574 559532 559378 1117543 86-87 509539 514518 521496 1055142 87-88 468619 466342 482959 974512 88-89 407631 424131 438035 890199 89-90 360597 364386 388538 792508 90-91 297885 317945 337703 700830 91-92 219494 258699 291729 614760 92-93 195977 187468 246753 516418 93-94 172459 164354 207948 426995 94-95 145023 141780 164306 346257 95-96 109747 116677 130817 270763 96-97 82310 86261 96335 209046 97-98 58009 63096 62656 157314 98-99 54873 43295 48887 114829 99-100 50954 39811 37217 81314 100 + 47034 35880 26807 56312 48

Figure 7.2: United States Age-Distributions: 2010, 2011, 2016, 2050 49 Looking at the table and graph for the age-distributions, we notice some changes in the population. There are more newborns in 2011 and 2016 than there are in 2010. However, there are fewer newborns in 2050. This can be explained by looking at the age-distribution for mothers. Fertility rates are given for women aged 12 − 46, and when looking at their age-distributions throughout the years we see that during the peak fertility ages, 20 − 30, that the number of women has increased from 2010 for 2011 and 2016, but decreased for 2050. Therefore there are more women of child bearing ages in 2011 and 2016 than there are in 2010, but not as many in 2050. When looking at the graph of the age-distribution we see that for the 2011 and 2016 age-distributions that they appear to be the age-distribution from 2010, but shifted over a bit. However, the 2050 age-distribution looks drastically different from any other age-distribution. There are fewer people up to the age of around 60. The number of people over the age of 60 is much higher for the 2050 projection. The Leslie Matrix is useful for projections because it projects the age-distribution from one time period to another. This is useful when one wants to know how many people of a certain age there will be. For example, it should be known the number of students entering college, people aged 18, so that colleges can prepare for an increase or decrease in the number of students. However, it is also useful to know the total number of people that will be alive at a given time. This helps us know how much material will be needed to keep people alive, such as water, food, and shelter. The total population for the years 2010, 2011, 2016, and 2050 are given in the table below.

Year Total Number of Women Total Number of People = 2·Women 2010 156,677,378 313,354,756 2011 157,412,204 314,824,308 2016 160,936,564 321,873,128 2050 164,239,897 328,579,794

Table 7.4: Total Number of Women and People for United States: 2010, 2011, 2016, and 2050 50 When comparing the total number of people, although this is only women, through- out of the years, we see that there is an increase in the total number of people that there are as time goes on. Therefore if mortality and fertility rates were to remain the same, there would be a considerable increase in the size of the population for the United States. While some changes would be seen immediately, they would be more drastic as time advances.

7.3 United States Projections Using Different Mortality and Fertility Rates

In Section 7.2 the age-distribution for the United States was projected to the years 2011, 2016, and 2050 using the Leslie Matrix and data from 2010. Now we will be comparing these same projections but with different rates for mortality and fertility. We will first look at the differences in mortality rates. The age-specific mortality rates are given in the life table of the United States, 2010. In the Leslie Matrix we use the survival rates, Sx = 1 − qx. To get the survival rates that would occur if mortality was suddenly to double we would take Sx = 1 − 2 · qx. To get the survival rates that 1 would occur if the mortality rate was to be cut in half, we would take Sx = 1 − 2 · qx. The age-distributions for 2016 and 2050 were found using the different mortality rates: actual mortality rate, half mortality rate, and double mortality rate. The age- distributions for the two years are given in the two graphs below. 51

Figure 7.3: United States Projection to 2016 with Differing Mortality Rates

Figure 7.4: United States Projection to 2050 with Differing Mortality Rates 52 The first graph shows the differences in the age-distributions for the 2016 pro- jection. I decided to leave out the 2011 projection since there would be little change in the age-distributions. The number of newborns would remain unchanged. This makes sense because the fertility rates will remain constant and therefore there will be the same number of newborns in 2011. The number of newborns will change in 2016 since the number of women of child bearing ages will change. We do not see much of a change in the age-distributions until around the age of 50. At this point we see that the line for half of the mortality is slightly above the line for double the mortality. Of course this makes sense because before the age of 50 the mortality rates are so small that doubling or halving the number will make little difference. However, when the mortality rates start to increase, around the age of 50, we can start to notice the difference in the age-distributions. The age-distribution is highest with the half mortality rate. The second graph shows the differences in the age-distributions for the 2050 pro- jection. This time the number of newborns differs. The fertility rate remains constant for all projections, but the mortality rates are changing. Since the 2050 projection is a projection of 40 years, we will see a lot more changes. The changes in mortality rates will cause changes in the number of women that are of child bearing age. Of course with the half mortality rate there will be a higher number of women that are of child bearing age, and therefore there will be a higher number of newborns for the mortality rate. We see that half the mortality rate results in more people in each age class, followed by the actual mortality rate, and lastly by the double the mortality rate. This was to be expected. Around the age of 60 is when large changes begin to appear. From varying the mortality rates we can see that halving the mortality rate for every age will make a big difference in the age-distribution down the road. The changes will not be seen immediately however. The same goes for doubling the mortality rate. The changes in mortality rates will change the life expectancy as well. The life expectancy for females in the United States in 2010 was 81.2. When the mortality rate was cut in half for every age group the new life expectancy became 88.0. When the mortality rate was doubled for every age group the new life expectancy became 73.5. Therefore doubling or halving the mortality rate only changes the life expectancy by 7 or 8 years, or about 8 or 9%. We will now look at the change in the total population caused by the change in 53 mortality rates. These numbers will be given in the table below.

Year Change in Mortality Total Number of Women Total Number of People 2010 Mortality Rate 156,677,378 313,354,756 2011 Mortality Rate 157,412,204 314,824,408 1 2011 2 · Mortality Rate 158,018,300 316,036,600 2011 2· Mortality Rate 156,203,540 312,407,080 2016 Mortality Rate 160,936,564 321,873,128 1 2016 2 · Mortality Rate 164,300,678 328,601,356 2016 2· Mortality Rate 155,544,976 311,089,952 2050 Mortality Rate 164,239,897 328,479,794 1 2050 2 Mortality Rate 178,885,601 357,771,202 2050 2· Mortality Rate 147,442,421 294,884,842

Table 7.5: Total Number of Women and People for United States: 2010, 2011, 2016, and 2050, with Changes in Mortality Rates

For 2011 we see that there is not much of a change in the total number of people for the different mortality rates. This would be because there has not been enough time to see the changes. However, we notice these changes more with the 2016 and 2050 data. We see that there are the most people when the mortality rate is halved, then comes the actual mortality rate, and lastly is the number of people there are when the mortality rate is doubled. While the change in mortality rates will eventually lead to changes in the total number of people, the changes are not extreme. We will now look at the change in fertility rates. The fertility rates were halved, doubled, or remained the same. Below are graphs with the change in fertility rates. 54

Figure 7.5: United States Projection to 2016 with Differing Fertility Rates

Figure 7.6: United States Projection to 2050 with Differing Fertility Rates 55 The first graph shows the differences in age-distributions for the 2016 projec- tions with varying fertility rates. We see that the only difference in the age-distributions is the number of newborns to five year-olds that there are. This makes sense because we projected the age-distribution for six years, and the mortality rates remain constant. We see that the number of newborns for double the fertility rate is approximately 400,000, the number of newborns for the actual fertility rate is approximately 200,000, and the number of newborns for half the fertility rate is approximately 100,000. The second graph shows the differences in age-distributions for the 2050 projec- tions with differing fertility rates. We see differences in the age-distribution until the age of 40. This fits since the projection is for 40 years, and after age 40 it is all based on mortality rates, which remain constant throughout these projections. We see that the number of newborns in 2050 for double the fertility rate is approximately 750,000 and approximately 50,000 for half the fertility rate. These numbers vary greatly and this can be explained by the increase in the number of women who are of child bearing age in the double fertility rate projection. Life expectancy is calculated using mortality rates found on the life table. The change in fertility rates will not cause a change in the mortality rates. Therefore the life expectancy will not be changed with the differing fertility rates and all life expectancies will remain the same. 56 Year Change in Fertility Total Number of Women Total Number of People 2010 Fertility Rate 156,677,378 313,354,756 2011 Fertility Rate 157,412,204 314,824,408 1 2011 2 · Fertility Rate 156,417,121 312,834,242 2011 2· Fertility Rate 159,402,370 318,804,740 2016 Fertility Rate 160,936,564 321,873,128 1 2016 2 · Fertility Rate 154,975,169 309,950,338 2016 2· Fertility Rate 172,859,357 345,718,714 2050 Fertility Rate 164,239,897 328,479,794 1 2050 2 Fertility Rate 120,220,849 240,441,698 2050 2· Fertility Rate 287,231,244 574,462,488

Table 7.6: Total Number of Women and People for United States: 2010, 2011, 2016, and 2050, with Changes in Fertility Rates

We can easily see that the change in the fertility rates causes a change in the total number of people alive. When the fertility rate is cut in half, the total number of people starts to decrease immediately and the results can really be seen in 2050. When the fertility rate is doubled, the total number of people starts to increase immediately. After 40 years the population has almost doubled in size. Therefore doubling the fertility causes a large increase in the number of people.

7.4 Stable Age Vector

It was mentioned in Chapter 6 that from the Leslie Matrix we can find the rate of change in the population. This value is the only positive eigenvalue. The rates of population change, λ, are given in the table below for all of the Leslie Matrices used in this paper. 57 Matrix Rate of Population Change λ − 1 Standard Leslie Matrix 0.998039 -0.001961 1 2 ·Fertility Leslie Matrix 0.9740457 -0.0259543 2· Fertility Leslie Matrix 1.023703 0.023703 1 2 · Mortality Leslie Matrix 0.998386 -0.001614 2· Mortality Leslie Matrix 0.9976736 -0.0023264

Table 7.7: Rate of Population Change for Leslie Matrices Used

We see that the rate of population change for the standard Leslie Matrix used is 0.998039. Since λ < 1, we can conclude that the population is decreasing in size. We see that the change in fertility rates has a rather big impact on the rate of population change. When the fertility rate is halved, λ decreases by 0.024 and when the fertility rate is doubled, λ increases by 0.026. When the mortality rate is halved, λ increases by only 0.0003, and the mortality is doubled, λ decreases by only 0.0003. All of the rates are less than one, except for when the fertility rate doubles. If the fertility rate were to double there would be an increase in the size of the population indefinitely. We see that all values are close to 1, the point at which the population is stationary. We can also find the stable age distribution by finding the corresponding eigenvec- tor. Using R I was able to find these values. I then standardized the values by making the total age-distribution sum to one. These distributions are graphed below. 58

Figure 7.7: Stable Age-Distributions with Different Mortality and Fertility Rates

7.5 Conclusion

Life expectancy and mortality rates were used to compare the United States, Japan, Nigeria and the World. The regions were ranked from highest to lowest life expectancy at birth, and were ranked from Japan, United States, World, to Nigeria. The regions were again ranked according to their crude mortality rates from lowest to highest, and were ranked from the World, United States, Japan, to Nigeria. Based on these two comparisons, we can conclude that Nigeria has the worst healthcare of the regions being compared. However, not much else can be concluded using the data given. The crude mortality rate does not accurately depict the mortality conditions of a region and it was decided in Chapter 3 that the age-adjusted mortality rates are better suited for comparisons. The data could not be found for these regions, so life expectancy rates were used instead. 59 Using data from the United States life table for women for 2010 and the construc- tion of a Leslie Matrix, the United States age-distribution was projected to the years 2011, 2016, and 2050. During this time period we noticed that there wasn’t much no- ticeable change for 2011 and 2016, but by the year 2050 changes in the age-distributions started to appear. There were fewer children and young adults, but there were more older people. We saw the total number of people increasing each year. When looking at the finite rate of increase change we saw that this value was slightly less than 1. This suggests that the population will eventually decrease in size. Therefore one can reach the conclusion that the population is increasing in size until a point in the future where it will eventually reach stability, this can be after a period of decreasing in size, after the stability point the population will be decreasing at the constant rate λ. Fertility and mortality rates were changed to see how each would affect the pop- ulation at a later date. We saw the most change by changing the fertility rate. When the fertility rate was halved the population decreased substantially and the rate of pop- ulation change was less than 1. When the fertility rate was doubled the population increased rapidly and the finite rate of increase was greater than 1. Therefore if all of the sudden women were having twice the number of babies than they were before, the population would be growing and would continue to grow indefinitely. Since life ex- pectancy is calculated by using mortality rates, which we assumed to remain constant, there was no change in the life expectancy. Therefore people lived for the same length of time, but there are more people. There was less change in growth rate when changing the mortality rates. Since mortality rates are generally pretty low, doubling the mortality rate will make a small change. Therefore there were not such large changes in the age-distributions. Of course when the mortality rate was halved people tended to live longer and when the rate was doubled people lived for a shorter period of time. These changes would cause a change in the life expectancy at birth; however, these rates aren’t as drastic as one might expect. There was a change of approximately ±7 years from what the life expectancy would be if there was no change in mortality. Both of the mortality rates resulted in the rate of population change being less than 1. References

[1] Arias, Elizabeth. ”United States Life Table, 2010.” National Vital Statistics Reports. no. 63 (2014).

[2] Keyfitz, Nathan. Introduction to the Mathematics of Population. Reading, Mas- sachusetts: Addison-Wesley, 1968.

[3] Lancaster, H.O. Expectations of Life. A Study in the Demography, Statistics, and History of World Mortality. New York, New York: Springer-Verlag, 1990.

[4] Keiding, Niels. ”The Method of Expected Number of Deaths, 1786-1986”. Interna- tional Statistical Review, no. 5 (1987): 1-20.

[5] Ahmad, Omar, Cynthia Boschi-Pinto, Alan Lopez, Christopher Murray, Rafael Lozano, and Mie Inoue. Age Standardization of Rates: A New WHO Standard. World Health Organization, 2001. Web. 1 March 2016.

[6] ”Census Bureau Releases Estimates of Undercount and Overcount in the 2010 Cen- sus”. United State Census Bureau 22 May 2012. Web. 3 June 2016.

[7] ”Life Expectancy at Birth, Total(Years)”. The World Bank. World Bank, n.d. Web. 17 May 2016.

[8] Xu, Jiaquan, Sherry Murphy, Kenneth Kochanek, and Brigham Bastian. ”Deaths: Final Data for 2013”. National Vital Statistics Reports. 64(2016). Web 27 May 2016.

[9] ”Death Rate, Crude(Per 1,000 People)”. The World Bank. World Bank, n.d. Web 17 May 2016.

60 61 [10] Pinder, John, James Wiener and Michael Smith. ”The Weibull Distribution: A New Method for Summarizing Survivorship Data”.Ecology 59 (1978):175-179.

[11] ”Types of Population Pyramids”. Understanding Society . n.d. Web. 20 May 2016.

[12] ”Population Pyramids of the World from 1950 to 2100.” United Nations, Depart- ment of Economics and Social Affairs, Population Division. Web. 23 May 2016.

[13] Leslie, P.H. ”Some Further Notes on the Use of Matrices in Population Mathemat- ics”. Biometrika. 35 (1948): 213-245.

[14] ”Standard Populations - 19 Age Groups.” National Cancer Institute: Surveillance, Epidemiology, and End Results Program. National Cancer Institute. n.d. Web. 8 June 2016.

[15] Chetty, Raj, Michael Stepner, Sarah Abraham, Shelby Lin, Benjamin Scuderi, Nicholas Turner, Augustin Bergeron, and David Cutler. ”The Association Between Income and Life Expectancy in the United States, 2001-2014”. JAMA. 315(2016): 1750-1766.