Finite and instantaneous rates

Ecologists tend to talk about finite and instantaneous mortality rates rather than cumulative mortality and mortality rate. The finite mortality rate is the exact equivalent of cumulative mortality. The instantaneous mortality rate is what epidemiologists are estimating when they calculate mortality rate. Let us take an example.

Finite rates

Say we have a starting population (a cohort) of 100 animals of which 10% of the total die each month.

The monthly cumulative mortality or monthly finite mortality rate is simply the proportion, of those alive at the beginning of the period, that die during the period. For example, if in the first month 10% (0.1) of 100 die, then 0.1×100=10 and 10/100 = 0.1.

We have shown this graphically below:

{Fig. 1}

Over the full year the yearly cumulative mortality is 718 out of 1000, which is 0.718.

Note that the monthly cumulative mortality (10%, or 0.1) cannot just be multiplied by 12 to give the yearly cumulative mortality, that would give a mortality of over 100%. The correct way to work out cumulative mortality is to obtain it from the proportion surviving. In other words, if one-tenth of the population die each month, then ( 1 - 0.1 = ) nine-tenths survive. Over 2 months the proportion surviving would be 0.9 of 0.9, or 0.9×0.9, or 0.9². In which case, over 12 months, the proportion surviving would be 0.9¹², or 0.2824 - which would make the yearly cumulative mortality 1 - 0.2824 or 0.7176.

The problem with this attractively simple model is that the time interval (1 month) is entirely arbitrary - individuals seldom line up at convenient intervals to die! The simplest solution to this problem is to assume survival is a random process - and, if the population is very large, or we consider its average effects, we can therefore treat survival as a continuous process. In other words, we should consider the proportion surviving at any one instant in time.

Instantaneous rates

The monthly instantaneous mortality rate is the probability of death over a very short time period if we were to divide the time period (e.g. 1 month) up into many very short time periods. Mathematically, in order to do this we need to use calculus - or the formulae produced by it. If, however, we ignore how tangents are fitted to a curve, or the area under it is divided up, the important bit to remember is that calculus tends to produce equations that use natural logs (log_e or ln)- rather than log to the base ten (log₁₀).

1. One way you can estimate the instantaneous mortality rate is by plotting the natural logarithm of the number of animals against time. This is shown in the figure below:

   {Fig. 2}

   This produces a straight line relationship. The slope of this line (-.105) gives the required rate.

2. As you may suspect from looking at the graph, there is a much easier way to estimate the instantaneous mortality rate from the finite mortality rate, namely:

   Instantaneous mortality rate = ln (1.0 - finite mortality rate)

   For our example we get:
   Instantaneous mortality rate = ln (1.0 - 0.1) = ln 0.9 = -0.105 per month

Note that this is a true rate and not a proportion and can vary from -∞ to 0.

Contrary to the cumulative mortality rate, we can just multiply this value by 12 to give the yearly instantaneous mortality rate. Hence the yearly instantaneous mortality rate = 12 x -0.105 = -1.26 per year.

This can be converted to a finite rate using:

Finite mortality rate = 1.0 - e^{instantaneous mortality rate} = 0.716 which, allowing for rounding error, is what we obtained (0.718) from our first graph - by repeatedly multiplying the number alive by 0.9

Not that all these calculations are based on following a cohort of individuals. We are not considering any births that may occur during the period. The combined effects of birth and death rates are considered in the related topic on population change.