Gompertz–Makeham law of mortality
The Gompertz–Makeham law of mortality is a simple model that describes how the risk of death increases with age. It says the instantaneous death rate at age x is the sum of two parts: a part that grows with age and a small constant background risk from external causes.
The key formula is: the hazard (risk of dying at age x) μ(x) = α e^(β x) + λ. Here:
- α > 0 sets the starting level of the age-related risk.
- β > 0 controls how fast that risk grows with age (roughly exponentially).
- λ ≥ 0 is the background, age-independent risk from causes such as accidents or infections.
As people age, the first term α e^(β x) rises rapidly, so the overall death risk climbs steeply. The second term, λ, adds a steady background level of risk across ages.
A common takeaway is the idea that adult mortality risk roughly doubles every eight years. This comes from the Gompertz part and is often summarized by the mortality doubling time T = ln(2)/β. In many studies of adult humans, β is about 0.085 per year, which matches that eight-year doubling rule.
The model also connects to survival and life expectancy. From the hazard function, one can derive the survival function S(x) (the probability of surviving to age x) and related life expectancy measures. In special cases, the model reduces to simpler forms: if λ = 0, it becomes the pure Gompertz law; if α → 0 with λ fixed, it becomes an exponential model with constant hazard.
Gompertz–Makeham is widely used because of its simple form and clear interpretation. Actuaries use it for life tables and pricing life insurance and pensions; demographers and aging researchers use it to study adult mortality patterns; and it also appears in reliability theory for modeling aging in machines or other systems.
Notes and limitations: the model is not meant for infancy or early childhood, where mortality patterns differ. At very old ages, some studies report a slowing or plateau in mortality rates, but there is debate about whether this reflects biology or data quality and how best to model it. Despite these debates, the Gompertz–Makeham law remains a foundational tool for describing and analyzing adult mortality across populations and time.
This page was last edited on 2 February 2026, at 16:23 (CET).