Mechanics of population aging and survival

Abstract

In this paper we extend the previous work of Witten and her team on defining a classical physics-driven model of survival in aging populations (Eakin, Bull Math Biol 56(6):1121–1141, 1994; Eakin and Witten, Mech Aging Dev 78(2):85–101, 1995; Witten and Eakin, Exp Gerontol 32(2):259–285, 1997) by revisiting the concept of a force of aging and introducing the concepts of a momentum of aging, a kinetic energy and a potential energy of an aging population. We further extend the analysis beyond the deterministic Newtonian mechanics of a macroscopic population as a whole by considering the probabilistic nature of survival of individual population cohort members, thus producing new statistical physics-based concepts of entropy and of a gerontological “temperature”. These new concepts are then illustrated with application to the classic parametric Gompertz survival model, which is a commonly used empirical descriptor for survival dynamics of mammalian species, human populations in particular. As a function of chronological age the Gompertz Model force, momentum, and power are seen to have an asymmetric unimodal peak profile, while the potential energy has a descending sigmoidal profile similar to that of the survival fraction. The “temperature” is an exponential function of age and the entropy for a future age at a current census age can be represented as a topological surface with an asymmetric unimodal hump.

Appendices

Appendices

Appendix A: Derivation of the Gompertz Model potential energy function

From the definition of potential energy V in Eq. (20) and the Gompertz Model force F as a function of gerontological distance as shown in Table 2 we start with the following expression:

$$\begin{aligned} V(x) + C_1 = - \int h_0 \gamma N_0 \left[ 1 + \frac{h_0}{\gamma } x \right] e^{-x} dx \end{aligned}$$

(72)

where $C_1$ is an arbitrary constant of integration for the indefinite integral. Thus,

$$\begin{aligned} V(x) + C_1 = - h_0 \gamma N_0 \int e^{-x} dx - {\gamma }^2 N_0 \int xe^{-x} dx. \end{aligned}$$

(73)

The elementary integrals remaining can easily be evaluated using calculus and integration by parts, or obtained from tables of integrals:

$$\begin{aligned} \int e^{-x}dx= & {} -e^{-x} + C_2 \end{aligned}$$

(74)

$$\begin{aligned} \int xe^{-x} dx= & {} -(1+x)e^{-x} + C_3 \end{aligned}$$

(75)

where $C_2$ and $C_3$ are additional arbitrary constants of integration for these indefinite integrals. Consolidating all of this,

$$\begin{aligned} V(x) = - [ - h_0 \gamma N_0 e^{-x} - {\gamma }^2 N_0 (1 + x)e^{-x} - C_1 + h_0 \gamma N_0 C_2 + {\gamma }^2 N_0 C_3 ]. \end{aligned}$$

(76)

After combining all the arbitrary constant terms by defining

$$\begin{aligned} C \equiv C_1 + h_0 \gamma N_0 C_2 + {\gamma }^2 N_0 C_3 \end{aligned}$$

(77)

the potential energy is given as

$$\begin{aligned} V(x) = \gamma N_0 e^{-x} ( h_0 + \gamma + \gamma x ) + C. \end{aligned}$$

(78)

Next, since C is arbitrary, we will specifiy a value to satisfy a desirable boundary condition, namely that when the gerontological distance x is infinite (which corresponds to an infinite age as well, when the populatiion is extinct), then the potential energy will be zero, i.e., $V(\infty ) = 0$. Since $e^{-x}$ becomes smaller faster than x becomes large, we end up with $V(\infty ) = 0 + C$. Thus, we choose $C= 0$ and the expression for the potential energy reduces to

$$\begin{aligned} V(x) = \gamma N_0 e^{-x} ( h_0 + \gamma + \gamma x ). \end{aligned}$$

(79)

Appendix B: Derivation of the Gompertz Model future age of maximum entropy

At any age a, when the number of surviving individuals in a Gompertz population is N(a), there is a future age $a + \Delta a$ dependent on a at which the entropy $Z(a, a+\Delta a )$ will be maximized. Because the entropy is related to the uncertainty about which particular microstate will be occupied, its magnitude will be proportional to the number of combinatorial permutations of living and dead individuals that could result in a survival fraction $S(a + \Delta a )$, which is given by the binomial coefficient

$$\begin{aligned} \left( {\begin{array}{c}N(a)\\ N(a + \Delta a)\end{array}}\right) = \frac{N(a) !}{N(a+\Delta a )! [N(a) - N(a+\Delta a )]!}. \end{aligned}$$

(80)

But in general $\left( {\begin{array}{c}n\\ m\end{array}}\right)$ is maximized when $m = \frac{n}{2}$ so that the entropy at future age $a+\Delta a$ is maximized when $N(a+\Delta a ) = \frac{1}{2} N(a)$, in other words

$$\begin{aligned} \frac{1}{2} N_0 e^{\frac{h_0}{\gamma } \left[ 1 - e^{\gamma a} \right] } = N_0 e^{\frac{h_0}{\gamma } \left[ 1 - e^{\gamma (a+\Delta a )} \right] } . \end{aligned}$$

(81)

After dividing out common factors we find that

$$\begin{aligned} 2 e^{-\frac{h_0}{\gamma } e^{\gamma a} e^{\gamma \Delta a }} = e^{- \frac{h_0}{\gamma } e^{\gamma a}}. \end{aligned}$$

(82)

Taking the logarithm of both sides we get

$$\begin{aligned} \ln (2) - \frac{h_0}{\gamma } e^{\gamma a } e^{\gamma \Delta a } = - \frac{h_0}{\gamma }e^{\gamma a}. \end{aligned}$$

(83)

Thus,

$$\begin{aligned} e^{\gamma \Delta a} = 1 + \frac{\gamma }{h_0} e^{- \gamma a} \ln (2) \end{aligned}$$

(84)

so that after taking the logarithm of both sides and dividing through by $\gamma$ we get as a final expression for $\Delta a$ the following

$$\begin{aligned} \Delta a = \frac{1}{\gamma } \ln \left[ 1+ \frac{\gamma }{h_0}e^{ - \gamma a} \ln (2) \right] . \end{aligned}$$

(85)

We now see that at age a the future age of maximum entropy occurs when the future age is

$$\begin{aligned} a + \Delta a = a + \frac{1}{\gamma } \ln \left[ 1+ \frac{\gamma }{h_0}e^{ - \gamma a} \ln (2) \right] . \end{aligned}$$

(86)