Age-Dependent Force of Mortality

• Age-Dependent Force of Mortality is defined as the instantaneous hazard of death at a given age, captured via life table models and the Gompertz–Makeham law.
• Deterministic ODE and stochastic models simulate how the accumulation of aging factors drives an exponential mortality rise and a plateau at extreme ages.
• The framework links molecular damage accumulation with population survival curves, highlighting potential intervention strategies to improve lifespan.

An age-dependent force of mortality refers to the age-specific rate at which individuals die, formally defined as the instantaneous hazard or risk of death at each age. In mathematical demography, the force of mortality λ(t) is central to life table modeling and is empirically characterized by systematic patterns—most notably the Gompertz–Makeham law, which asserts an exponential increase in mortality with age. Understanding the mechanistic basis for this age dependency is key to linking empirical mortality trajectories to biological processes such as the accumulation of deleterious factors or reliability failure of essential systems.

1. Fundamental Concepts and Definition

The classical force of mortality, λ(t), is defined as the negative derivative of the logarithm of the survival function, $l(t)$, with respect to age t: $λ(t) = -\frac{d}{dt} \log l(t)$ For humans and many other species, as observed since Gompertz, λ(t) exhibits a strongly exponential age dependency throughout mid- and late-adult life: $λ(t) = λ_0 + α e^{βt}$ where $λ_0$ is the age-independent (Makeham) term, and $α, β > 0$ characterize the age-dependent increase. This empirical “law” is robust across taxa and has motivated extensive search for its mechanistic underpinnings (Grüning et al., 2011).

2. Mechanistic Models: Accumulation of Ageing Factors

A central mechanistic explanation for the age-dependence of mortality is the accumulation of “ageing factors” (AFs)—molecular or cellular entities such as noxious metabolic products or genetic anomalies that arise through life and drive senescence. The accumulation theory formalizes this as a combination of constant creation and auto-catalytic replication:

• Deterministic ODE Model:

  $\frac{dc(t)}{dt} = r\,c(t) + p$

  where $c(t)$ is the abundance of AFs at age t, $r$ is the rate of auto-catalytic replication (self-reinforcing growth), and $p$ is the constant production rate. The closed-form solution:

  $c(t) = \frac{p}{r}(e^{rt} - 1) + c_0 e^{rt}$

  Taking mortality rate proportional to $c(t)$:

  $λ(t) = z c(t)$

  Thus, with $r > 0$, for large t, mortality grows as $e^{rt}$, precisely the exponential form observed in the Gompertz law.

• Threshold Stochastic Branching Model:

  Real biological systems with low copy numbers and stochastic effects are better described with discrete branching processes. Each AF unit at time t can replicate or not in a given step, leading to a random process for the number of AFs, $C_t$. Mortality is then associated with the probability of exceeding a critical threshold, $c^\dagger$:

  $λ(t) = z \cdot P(C_t \geq c^\dagger)$

  The stochastic formulation explains both the exponential middle-age mortality rise and the “old-age levelling off”: as the mean and variance in $C_t$ grow, the probability $P(C_t \geq c^\dagger)$ saturates, causing λ(t) to plateau rather than diverge in extreme old ages (Grüning et al., 2011).

Approach	Leading Behavior for λ(t)	Old-Age Plateau	Mechanistic Detail
Deterministic ODE	Exponential $e^{rt}$	No	Mean field only
Stochastic Threshold	Exponential, saturates	Yes	Accounts for fluctuations

3. Biological Interpretation and Systems

Ageing factors in this context may be self-replicating genomic anomalies (e.g., extra-chromosomal DNA in yeast), toxic metabolites, or cumulative macromolecular damage. Empirical data (such as yeast lifespan observations) support exponential increases in such deleterious entities with time due to the structure of their creation/replication. Critically, mortality is assumed not to scale linearly with AF number, but exhibits a threshold effect—death occurs when the AF burden exceeds the system’s capacity, reminiscent of organismal failure.

This paradigm unifies molecular/physiological accumulation with age-specific mortality, providing mechanistic support for the validity of the Gompertz-Makeham law across species (Grüning et al., 2011).

4. Deterministic vs. Stochastic Regimes: Old-Age Levelling Off

A notable empirical deviation from pure exponential growth is the observed flattening of the mortality rate at extreme ages in many datasets. The deterministic model, which neglects fluctuations, misses this phenomenon:

• In deterministic ODEs, exponential growth persists indefinitely.
• In stochastic models, once the mean and standard deviation of the AF count become large, the probability of threshold crossing (and thus λ(t)) reaches a limiting value. This quantitatively matches the old-age mortality plateau, a feature seen in both yeast and human cohorts.

Thus, stochasticity is essential for modeling realistic age-dependent forces of mortality at advanced ages.

5. Generalization: Creation and Replication, Biological Realism

The inclusion of both creation (p) and replication (r) processes in AF accumulation enhances the realism of the model:

• Creation only ($r=0$): $c(t)$ increases linearly, insufficient to explain the observed exponential mortality rise.
• Replication only: Exponential growth, but inconsistent with observed delays in mortality onset (i.e., negligible mortality at very young ages).
• Combined: Early “clean-slate” period with low AFs and mortality, exponential rise throughout mid-age, and stochastic levelling off at old age.

This architecture reproduces both the delayed mortality onset in early life and the Gompertzian regime in adulthood (Grüning et al., 2011).

6. Broader Implications and Applications

The accumulation theory of age-dependent force of mortality provides a mechanistic bridge from molecular/cellular processes to observable population survival curves:

• Explanation of Gompertz-Makeham Law: The law emerges as the natural consequence of self-reinforcing (auto-catalytic) biological damage and its stochastic fluctuations.
• Intervention Strategies: The critical role of threshold effects (c^\dagger) suggests that interventions increasing the threshold (improving the tolerance to damage) would increase lifespan, a pattern seen in public health improvements.
• Predictive Modeling: The framework offers testable predictions about the shape and plateauing of old-age mortality, allowing for model calibration on both deterministic and stochastic parameters using empirical lifespan data.
• Biological Plausibility: The formulation is compatible with both animal and microbial data, suggesting broad applicability.
• Explanatory Power: The explicit link between AF dynamics and force of mortality provides a deeper mechanistic explanation than phenomenological fits.

In summary, the age-dependent force of mortality is theoretically justified by the stochastic accumulation (and threshold-induced impact) of self-replicating aging factors, with deterministic ODE models capturing exponential rise and stochastic models accounting for levelling off at late ages. This provides a quantitatively robust and biologically plausible explanation for the empirical laws governing mortality patterns across diverse taxa (Grüning et al., 2011).