Введение
Математическое уравнение, связанное с уровнем смертности человека. Закон Гомперца — Макехама утверждает, что уровень смертности человека является суммой компоненты, зависящей от возраста (функция Гомперца, названная в честь Бенджамина Гомперца), которая экспоненциально возрастает с возрастом, и компоненты, не зависящей от возраста (термин Макехама, названный в честь Уильяма Макехама). В защищенной среде, где внешние причины смерти редки (лабораторные условия, страны с низкой смертностью и т. д.), компонента смертности, не зависящая от возраста, часто пренебрежимо мала. В этом случае формула упрощается до закона смертности Гомперца. В 1825 году Бенджамин Гомперц предположил экспоненциальное увеличение смертности с возрастом. Закон смертности Гомперца — Макехама довольно точно описывает возрастную динамику смертности человека в диапазоне от примерно 30 до 80 лет. В более старшем возрасте некоторые исследования показали, что смертность увеличивается медленнее — явление, известное как замедление смертности в пожилом возрасте.
The Gompertz–Makeham law states that the human death rate is the sum of an age dependent component (the Gompertz function, named after Benjamin Gompertz), which increases exponentially with age and an age independent component (the Makeham term, named after William Makeham). 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
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. 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. The hazard function for the Gompertz Makeham distribution is most often characterised as 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:
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).
Снижение смертности человека до 1950-х годов в основном было связано с уменьшением компоненты смертности, не зависящей от возраста (Макехама), в то время как компонента смертности, зависящая от возраста (Гомперца), была на удивление стабильной. С 1950-х годов наблюдается новая тенденция в смертности, выражающаяся в неожиданном снижении смертности в старшем возрасте и «прямоуголизации» кривой выживаемости. Эмпирическое значение бета-параметра составляет около 0,085, что подразумевает удвоение смертности каждые 0,69 / 0,085 = 8 лет (Дания, 2006). Функция квантилей может быть выражена в виде аналитической формулы с использованием функции Ламберта W:
The Gompertz–Makeham law states that the human death rate is the sum of an age dependent component (the Gompertz function, named after Benjamin Gompertz), which increases exponentially with age and an age independent component (the Makeham term, named after William Makeham). 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
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. 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. The hazard function for the Gompertz Makeham distribution is most often characterised as 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:
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).
Закон Гомперца эквивалентен распределению Фишера — Типпетта для отрицательного возраста, ограниченному отрицательными значениями случайной переменной (положительными значениями возраста).
The Gompertz–Makeham law states that the human death rate is the sum of an age dependent component (the Gompertz function, named after Benjamin Gompertz), which increases exponentially with age and an age independent component (the Makeham term, named after William Makeham). 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
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. 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. The hazard function for the Gompertz Makeham distribution is most often characterised as 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:
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).
Будущее долголетия человека
Когорты людей, родившихся после 1950 года, первыми должны ощутить заметное замедление исторической тенденции увеличения смертности, согласно исследованию 2023 года, которое делает этот вывод, опираясь исключительно на математические расчеты. Исследование предсказывает будущее, в котором рекорды долголетия будут регулярно побиваться после 2073 года, а некоторые прогнозы указывают на возможность достижения возраста 140 лет и более.