# Gompertz–Makeham law of mortality

The Gompertz–Makeham law states that the human death rate is the sum of an age-independent component (the Makeham term, named after William Makeham)[1] and an age-dependent component (the Gompertz function, named after Benjamin Gompertz),[2] which increases exponentially with age.[3] In a protected environment where external causes of death are rare (laboratory conditions, low mortality countries, etc.), the age-independent mortality component is often negligible. In this case the formula simplifies to a Gompertz law of mortality. In 1825, Benjamin Gompertz proposed an exponential increase in death rates with age.

The Gompertz–Makeham law of mortality describes the age dynamics of human mortality rather accurately in the age window from about 30 to 80 years of age. At more advanced ages, some studies have found that death rates increase more slowly – a phenomenon known as the late-life mortality deceleration[3] – but more recent studies disagree.[4]

Estimated probability of a person dying at each age, for the U.S. in 2003 [1]. Mortality rates increase exponentially with age after age 30.

The decline in the human mortality rate before the 1950s was mostly due to a decrease in the age-independent (Makeham) mortality component, while the age-dependent (Gompertz) mortality component was surprisingly stable.[3][5] Since the 1950s, a new mortality trend has started in the form of an unexpected decline in mortality rates at advanced ages and "rectangularization" of the survival curve.[6][7]

The hazard function for the Gompertz-Makeham distribution is most often characterised as ${\displaystyle h(x)=\alpha e^{\beta x}+\lambda }$. The empirical magnitude of the beta-parameter is about .085, implying a doubling of mortality every .69/.085 = 8 years (Denmark, 2006).

The quantile function can be expressed in a closed-form expression using the Lambert W function:[8]

${\displaystyle Q(u)={\frac {\alpha }{\beta \lambda }}-{\frac {1}{\lambda }}\ln(1-u)-{\frac {1}{\beta }}W_{0}\left({\frac {\alpha e^{\alpha /\lambda }(1-u)^{-(\beta /\lambda )}}{\lambda }}\right)}$

The Gompertz law is the same as a Fisher–Tippett distribution for the negative of age, restricted to negative values for the random variable (positive values for age).

Gompertz–Makeham
Parameters${\displaystyle \alpha >0}$ (real)
${\displaystyle \beta >0}$ (real)
${\displaystyle \lambda >0}$ (real)
Support${\displaystyle x\in \mathbb {R} ^{+}}$
PDF${\displaystyle \left(\alpha e^{\beta x}+\lambda \right)\cdot \exp \left(-\lambda x-{\frac {\alpha }{\beta }}\left(e^{\beta x}-1\right)\right)}$
CDF${\displaystyle 1-\exp \left(-\lambda x-{\frac {\alpha }{\beta }}\left(e^{\beta x}-1\right)\right)}$

## References

1. ^ Makeham, W. M. (1860). "On the Law of Mortality and the Construction of Annuity Tables". J. Inst. Actuaries and Assur. Mag. 8: 301–310.
2. ^ Gompertz, B. (1825). "On the Nature of the Function Expressive of the Law of Human Mortality, and on a New Mode of Determining the Value of Life Contingencies". Philosophical Transactions of the Royal Society. 115: 513–585. doi:10.1098/rstl.1825.0026.
3. ^ a b c Leonid A. Gavrilov & Natalia S. Gavrilova (1991) The Biology of Life Span: A Quantitative Approach. New York: Harwood Academic Publisher, ISBN 3-7186-4983-7
4. ^ Gavrilov, Leonid A.; Gavrilova, Natalia S. (2011). "Mortality Measurement at Advanced Ages: A Study of the Social Security Administration Death Master File" (PDF). North American Actuarial Journal: 432–447.
5. ^ Gavrilov, L. A.; Gavrilova, N. S.; Nosov, V. N. (1983). "Human life span stopped increasing: Why?". Gerontology. 29 (3): 176–180. doi:10.1159/000213111.
6. ^ Gavrilov, L. A.; Nosov, V. N. (1985). "A new trend in human mortality decline: derectangularization of the survival curve". Age. 8 (3): 93.
7. ^ Gavrilova, N. S.; Gavrilov, L. A. (2011). "Stárnutí a dlouhovekost: Zákony a prognózy úmrtnosti pro stárnoucí populace" [Ageing and Longevity: Mortality Laws and Mortality Forecasts for Ageing Populations]. Demografie (in Czech). 53 (2): 109–128.
8. ^ Jodrá, P. (2009). "A closed-form expression for the quantile function of the Gompertz–Makeham distribution". Mathematics and Computers in Simulation. 79 (10): 3069–3075. doi:10.1016/j.matcom.2009.02.002.
Algor mortis

Algor mortis (Latin: algor—coldness; mortis—of death), the second stage of death, is the change in body temperature post mortem, until the ambient temperature is matched. This is generally a steady decline, although if the ambient temperature is above the body temperature (such as in a hot desert), the change in temperature will be positive, as the (relatively) cooler body acclimates to the warmer environment. External factors can have a significant influence.

The term was first used by Dowler in 1849. The first published measurements of the intervals of temperature after death were done by Dr John Davey in 1839.

Benjamin Gompertz

Benjamin Gompertz (5 March 1779 – 14 July 1865) was a British self-educated mathematician and actuary, who became a Fellow of the Royal Society. Gompertz is now best known for his Gompertz law of mortality, a demographic model published in 1825.

Biodemography of human longevity

Biodemography is a multidisciplinary approach, integrating biological knowledge (studies on human biology and animal models) with demographic research on human longevity and survival. Biodemographic studies are important for understanding the driving forces of the current longevity revolution (dramatic increase in human life expectancy), forecasting the future of human longevity, and identification of new strategies for further increase in healthy and productive life span.

Cervical dislocation

Cervical dislocation is a common method of animal euthanasia. It refers to a technique used in physical euthanasia of small animals by applying pressure to the neck and dislocating the spinal column from the skull or brain. The aim is to quickly separate the spinal cord from the brain so as to provide the animal with a fast and painless death.

Dead on arrival (DOA), also dead in the field and brought in dead (BID), indicates that a patient was found to be already clinically dead upon the arrival of professional medical assistance, often in the form of first responders such as emergency medical technicians, paramedics, or police.

In some jurisdictions, first responders must consult verbally with a physician before officially pronouncing a patient deceased, but once cardiopulmonary resuscitation is initiated, it must be continued until a physician can pronounce the patient dead.

Death messenger

Death messengers, in former times, were those who were dispatched to spread the news that an inhabitant of their city or village had died. They were to wear unadorned black and go door to door with the message, "You are asked to attend the funeral of the departed __________ at (time, date, and place)." This was all they were allowed to say, and were to move on to the next house immediately after uttering the announcement. This tradition persisted in some areas to as late as the mid-19th century.

Death rattle

Terminal respiratory secretions (or simply terminal secretions), known colloquially as a death rattle, are sounds often produced by someone who is near death as a result of fluids such as saliva and bronchial secretions accumulating in the throat and upper chest. Those who are dying may lose their ability to swallow and may have increased production of bronchial secretions, resulting in such an accumulation. Usually, two or three days earlier, the symptoms of approaching death can be observed as saliva accumulates in the throat, making it very difficult to take even a spoonful of water. Related symptoms can include shortness of breath and rapid chest movement. While death rattle is a strong indication that someone is near death, it can also be produced by other problems that cause interference with the swallowing reflex, such as brain injuries.It is sometimes misinterpreted as the sound of the person choking to death, or alternatively, that they are gargling.

Dignified death

Dignified death is a somewhat elusive concept often related to suicide. One factor that has been cited as a core component of dignified death is maintaining a sense of control. Another view is that a truly dignified death is an extension of a dignified life. There is some concern that assisted suicide does not guarantee a dignified death, since some patients may experience complications such as nausea and vomiting. There is some concern that age discrimination denies the elderly a dignified death.

Gompertz distribution

In probability and statistics, the Gompertz distribution is a continuous probability distribution, named after Benjamin Gompertz. The Gompertz distribution is often applied to describe the distribution of adult lifespans by demographers and actuaries. Related fields of science such as biology and gerontology also considered the Gompertz distribution for the analysis of survival. More recently, computer scientists have also started to model the failure rates of computer codes by the Gompertz distribution. In Marketing Science, it has been used as an individual-level simulation for customer lifetime value modeling. In network theory, particularly the Erdős–Rényi model, the walk length of a random self-avoiding walk (SAW) is distributed according to the Gompertz distribution.

Lazarus sign

The Lazarus sign or Lazarus reflex is a reflex movement in brain-dead or brainstem failure patients, which causes them to briefly raise their arms and drop them crossed on their chests (in a position similar to some Egyptian mummies). The phenomenon is named after the Biblical figure Lazarus of Bethany, whom Jesus Christ raised from the dead in the Gospel of John.

List of actuaries

An actuary is a business professional who deals with a financial situation of risk and uncertainty. This is a list of notable actuaries and others who have influenced the profession.

Megadeath (or megacorpse) is one million human deaths, usually caused by a nuclear explosion. The term was used by scientists and thinkers who strategized likely outcomes of all-out nuclear warfare.

Necronym

A necronym (from the Greek words νεκρός, nekros, "dead" and ὄνομα ónoma, "name") is a reference to, or name of, a person who has died. Many cultures have taboos and traditions associated with referring to such a person. These vary from the extreme of never again speaking the person's real name, often using some circumlocution instead, to the opposite extreme of commemorating it incessantly by naming other things or people after the deceased.

For instance, in some cultures it is common for a newborn child to receive the name (a necronym) of a relative who has recently died, while in others to reuse such a name would be considered extremely inappropriate or even forbidden. While this varies from culture to culture, the use of necronyms is quite common.

Necrophobia

Necrophobia is a specific phobia which is the irrational fear of dead things (e.g., corpses) as well as things associated with death (e.g., coffins, tombstones, funerals, cemeteries). With all types of emotions, obsession with death becomes evident in both fascination and objectification. In a cultural sense, necrophobia may also be used to mean a fear of the dead by a cultural group, e.g., a belief that the spirits of the dead will return to haunt the living.Symptoms include: shortness of breath, rapid breathing, irregular heartbeat, sweating, dry mouth and shaking, feeling sick and uneasy, psychological instability, and an altogether feeling of dread and trepidation. The sufferer may feel this phobia all the time. The sufferer may also experience this sensation when something triggers the fear, like a close encounter with a dead animal or the funeral of a loved one or friend. The fear may have developed when a person witnessed a death, or was forced to attend a funeral as a child. Some people experience this after viewing frightening media.The fear can manifest itself as a serious condition. Treatment options include medication and therapy.The word necrophobia is derived from the Greek nekros (νεκρός) for "corpse" and the Greek phobos (φόβος) for "fear".

Negligible senescence

Negligible senescence is a term coined by biogerontologist Caleb Finch to denote organisms that do not exhibit evidence of senescence (biological aging), such as measurable reductions in their reproductive capability, measurable functional decline, or rising death rates with age.There are many species where scientists have seen no increase in mortality after maturity. This may mean that the lifespan of the organism is so long that researchers' subjects have not yet lived up to the time when a measure of the species' longevity can be made. Turtles, for example, were once thought to lack senescence, but more extensive observations have found evidence of decreasing fitness with age.Study of negligibly senescent animals may provide clues that lead to better understanding of the aging process and influence theories of aging. The phenomenon of negligible senescence in some animals is a traditional argument for attempting to achieve similar negligible senescence in humans by technological means.

There is "non-senescence" in the genus Hydra.There are also organisms that exhibit negative senescence, whereby mortality chronologically decreases as the organism ages, for all or part of the life cycle, in disagreement with the Gompertz–Makeham law of mortality (see also Late-life mortality deceleration). Furthermore, there are species that have been observed to regress to a larval state and regrow into adults multiple times, such as Turritopsis dohrnii.Recent studies have indicated a connection between phenomena related to negligible senescence and the general stability of an organism's genome, specifically transcription processes, over its lifetime.

Pallor mortis

Pallor mortis (Latin: pallor "paleness", mortis "of death"), the first stage of death, is an after-death paleness that occurs in those with light/white skin.

Post-mortem interval

Post-mortem interval (PMI) is the time that has elapsed since a person has died. If the time in question is not known, a number of medical/scientific techniques are used to determine it. This also can refer to the stage of decomposition of the body.

Skeletonization

Skeletonization refers to the final stage of decomposition, during which the last vestiges of the soft tissues of a corpse or carcass have decayed or dried to the point that the skeleton is exposed. By the end of the skeletonization process, all soft tissue will have been eliminated, leaving only disarticulated bones. In a temperate climate, it usually requires three weeks to several years for a body to completely decompose into a skeleton, depending on factors such as temperature, humidity, presence of insects, and submergence in a substrate such as water. In tropical climates, skeletonization can occur in weeks, while in tundra areas, skeletonization may take years or may never occur, if subzero temperatures persist. Natural embalming processes in peat bogs or salt deserts can delay the process indefinitely, sometimes resulting in natural mummification.The rate of skeletonization and the present condition of a corpse or carcass can be used to determine the time of death.After skeletonization, if scavenging animals do not destroy or remove the bones, acids in many fertile soils take about 20 years to completely dissolve the skeleton of mid- to large-size mammals, such as humans, leaving no trace of the organism. In neutral-pH soil or sand, the skeleton can persist for hundreds of years before it finally disintegrates. Alternately, especially in very fine, dry, salty, anoxic, or mildly alkaline soils, bones may undergo fossilization, converting into minerals that may persist indefinitely.

William Makeham

William Matthew Makeham (11 September 1826 – 17 November 1891) was an English actuary and mathematician. He had one wife, Hepzibah Reed, and seven children, William, Amy, Elizabeth, Thomas, Frederick, Emily, and George.

Makeham was responsible for proposing the age-independent Makeham term in the Gompertz–Makeham law of mortality that, together with the exponentially age-dependent Gompertz term, was one of the most effective theories to describe human mortality.Makeham was responsible for two important studies on human mortality:

"On the Law of Mortality and the Construction of Annuity Tables." J. Inst. Actuaries and Assur. Mag. 8, 301–310, 1860.

"On an Application of the Theory of the Composition of Decremental Forces." J. Inst. Actuaries and Assur. Mag. 18, 317–322, 1874.

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Continuous univariate
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