Invariances in relations between the rates of mortality and aging captured by a generalization of the Gompertz-Makeham law: Lessons from Caucasus

Abstract

Abstract Among scarce biological relations qualifiable as laws, the Gompertz-Makeham law (GML) stands out being directly related to life-and-death issues. GML may be written as: -[dn(t)/n(t)]/dt≡µ(t) = e^(-v + γt) + C = e^(-v)*e^(gt) + C = µ0*e^(gt) + C, where µ(t) is mortality rate (MR), v captures vitality (resistance to mortality), g captures aging-associated v decrease, µ0 = 1/e^v is the initial MR, and C captures the MR part attributable to external (background) hazards irresistible at any age. GML status is questionable since, upon the common assumptions that vitality decreases linearly and C is constant, MR-vs-age trajectories violate GML, especially at later ages. A generalized GML (GGML) µ(t) = C(t)+µ0*e^[f(t)] suggests that MR increases exponentially IF vitality decreases linearly, i.e. IF f(t) = gt, and C = 0. GGML produces µ(t) changes from any vitality changes by exponentiation and, from any background hazardousness changes, in a linear way. Thus, f(t) may be deduced from µ(t), provided C(t) is treated properly. Based on this, it may be shown that a hump of the biological aging rate revealed through the lens of GGML at ages 65 to 90 years in low C(t) countries featuring high life expectancies may be discerned also in high C(t) countries by taking into account that C(t) there is increased mostly in the middle of age span, as in the North Caucasus and some other Russian Federation regions. Thus, GGML captures relational invariants inherent in the animate nature and discernable even beneath such multifactorial phenomena as human mortality and its such diverse manifestations as mortality kinetics. These invariants constrain advances in human life expectancy.