A joint explanation of infant and old age mortality

Abstract

Infant deaths and old age deaths are very different. The former are mostly due to severe congenital malformations of one or a small number of specific organs. On the contrary, old age deaths are largely the outcome of a long process of deterioration which starts in the 20s and affects almost all organs. In terms of age-specific death rates, there is also a clear distinction: the infant death rate falls off with age, whereas the adult and old age death rate increases exponentially with age in conformity with Gompertz’s law. An additional difference is that whereas aging and old age death have been extensively studied, infant death received much less attention. To our knowledge, the two effects have never been inter-connected. Clearly, it would be satisfactory to explain the two phenomena as being two variants within the same explanatory framework. In other words, a mechanism providing a combined explanation for the two forms of mortality would be welcome. This is the purpose of the present paper. We show here that the same biological effects can account for the two cases provided there is a difference in their severity: death triggered by isolated lethal anomalies in one case and widespread wear-out anomalies in the second. We show that quite generally this mechanism leads indeed, respectively, to a declining and an upgoing death rate. Moreover, this theoretical framework leads to the conjecture that the severity of the death effects, whether in infancy or old age, is higher for organisms which comprised a larger number of organs. Finally, let us observe that the main focus of the paper is the drastic difference of the age-specific death rates (i.e., decreasing versus increasing) because this difference is found in many species, whereas the question of the best fit (e.g., Gompertz versus Weibull) is rather specific to human mortality.

Appendix 1: Statistical versus probabilistic descriptions of the death process

In this appendix, we discuss some particular aspects of the theoretical framework. The first subsection clarifies the connection between age and time in the aging processes. The second subsection establishes the important connection between the density function of the ages of death and its frequency counterpart commonly referred to as the death rate.

1.1 1.1 Age versus time

Should we use age (noted x) or time (noted t)? At first sight, the question may seem irrelevant for if time is counted from the moment of birth the two variables are identical. It is not so simple however, as shown by the fact that in our min-max argument we had to introduce as many age of death variables X₁,X₂,… as there are vital organs. Moreover, for any organ, its real age is the time elapsed since it was created. For instance, in zebrafish, the heart appears and starts to beat some 20 h after fertilization. Most vital organs are created during embryogenesis which means that their age is not identical with time (measured after birth which for fish means hatching of the eggs). That is why X was a more appropriate variable than time.

However, in this appendix, our perspective is different for we wish to consider the evolution of a whole population or more precisely of a cohort of individuals born at the same moment. Taking this moment as origin of the time axis makes the age of each individual numerically identical with the time given by an external clock.

Actually, to describe the evolution of a cohort, time seems a better variable than its age. Why? Although the population exists at any time, for external observers, it becomes real only when we can know its size and that occurs only when a census (or a survey) takes place. Censuses are conducted at specific time intervals (e.g., in the USA every decade) and concern simultaneously all cohorts. That is why in this part calendar time seems to be the natural variable. Age will play a role only if we wish to consider different age groups.

1.2 1.2 Death rate versus probability density

As always, the tricky point is the relationship between the probabilistic notions and their statistical counterpart. The goal of this appendix is to recall the main notions and how they are related.

Let t denote the age of individuals in a cohort. Let y(t) denote the size of the cohort at time t. We wish to describe the decrease of the population in the course of time.

The probability that an individual would die in the time interval (t₁,t₂) (which is also an age interval) is:

$$ \text{(Number of those who die)}/\text{(number of those alive initially)} = [y(t_{1})-y(t_{2})]/y(t_{1}) $$

to get the probability of dying per unit of time we must divide by the length of the time interval Δt = t₂ − t₁.

$$ \mu(t)=(1/{\Delta} t)\ \left[\left( y(t_{1})-y(t_{2})\right)/y(t)\right] $$

Note that μ(t) represents what is usually called the death rate, sometimes also called the hazard rate or the force of death. Note also that: y(t₁) − y(t₂) = −Δy.

Now, let us consider the case of a constant probability of dying. One is led to: (1/y)(dy/dt) = −a which gives $y(t)=y_{0} \exp (-at)$. In other words, the survival function is a decreasing exponential.

Now let us consider a random variable T which represents the age of death of an individual. Its density function, defined by f(t)dt = P{t < T < t + dt}, is the derivative of the distribution function: F(t) = P{T ≤ t}. f(t)dt is the probability that the death of the individual occurs in the age interval (t,t + dt); f(t) is the probability per unit of time.

Note that: $f(t) \sim \mu (t)$. In words, μ(t) is the statistical counterpart of the probability density function of T.

If we consider again the case of a constant probability of dying (for ages in a bounded interval and zero elsewhere), it means: f(t) = μ(t) = a. Then, the distribution function is F(t) = at, at least until at is equal to 1. The decreasing distribution function is:

$$ G(t)=P\{ T>t \}=1-F(t)=1-at $$

Note that G(t) is different from the survival function.