If a church has more people dying and leaving than joining, then it will decline. If over time the rates remain the same, then the overall rate of decline is a constant percentage (per capita) rate, leading to a negative exponential curve. This result assumes that the age distribution of joiners and leavers is similar to the remaining population, which keeps the average age of the population constant.

However, if there is a large discrepancy between the joining and loss rates, and especially if the losses are due to death, then aging of the population becomes a significant factor. The smaller the church becomes, the more pronounced the effect of population aging, which increases the death rate, reduces the birth rate, and thus increases the overall decline rate. Church decline then ceases to be negative exponential and becomes closer to a straight line. Therefore, care is needed when modelling churches near the end of their life as aging will tend to straighten the decline bringing extinction earlier than predicted by models based on constant percentages.

The following analysis compares different models of deaths and aging in a population to illustrate these effects.

### Birth Death Model of Church

Consider a simple model of births and deaths applied to the church. Additions are through the reinforcing feedback loop *R*: more people, more births, and thus more people added. This assumes all those born to church members join the church. Subtractions are through the balancing loop* B*: more people, more deaths, thus more people removed (if the population is increasing).

The model also assumes no-one from outside the church joins, and all in the church die rather than leave. This is the simple birth-death model, shown in figure 1. The stock *Church* represents the number of people in the church at any given time. Mathematically it reduces to the differential equation *dx/dt = (b-d)x* where *x* is the number in the church, represented by the stock in figure 1, and *b,d* the birth and death rates, represented by the converters in the figure.

Let the birth and death rates be those for the UK for 2013, b = 0.01226, d = 0.00933 (Index Mundi). The solution for the next 100 years is given in figure 2. It is a growing exponential, *x* = exp( (*b-d*) *t*), where *t* is time.

The exponential curve appears slight because the birth and death rates are close. If the birth rate was less than the death rate, the result would be exponential decline.

### Smoothed Death Model of Church

Assume now the church has no births, that is, all those born to church members no longer join the church. The model can be simplified to the pure draining process in figure 3, with the solution determined by the balancing loop B, and the per capita death rate. Such a process is called *first-order smoothing* (Sterman 2000) as it treats the attributes of all people in the church as equal, including their age.

The solution of this model is negative exponential decline, determined by the death rate, figure 4:

The result is clearly wrong as it predicts the church still has a healthy number of people a hundred years later in 2113! The mistake lies in the first-order smoothing because, as the church declines, the model has assumed the age distribution, e.g. the average age of the congregation, remains the same. Clearly, it will not. Instead, without any additions of younger people, the church will get older and cease to exist when the last person dies, probably before the 100 years are up.

### Cohort Death Model of Church

For a true model of the effect of deaths on the church, the stock church needs to keep track of the ages of the people in the church. This is done by splitting the stock into cohorts, figure 5:

This type of stock is called a conveyor and works like a conveyor belt. The people are assumed uniformly distributed in age, that is they have been in the stock for different lengths of times and thus are placed uniformly at appropriate places on the conveyor belt. Death is determined by the amount of time a person stays in the stock. In this case, 85 years is taken as a reasonable upper limit to lifespan. The resulting decline is now a straight line as not only does the church decline, but it also ages, figure 6.

The first-order smoothing process has now been replaced by a pure delay process, sometimes called a pipeline delay (Sterman 2000). Now the church ceases after 85 years, the lifetime of the population. Because the population were spread uniformly throughout the stock, there was at least one person who had just been born, at the back of the conveyor. They are the last one to die as it is assumed all people live the full lifespan. Crucially, the decline is not negative exponential but straight, which suggest that a church dominated by the death process will more closely follow a straight-line decline.

In such a church, the life expectancy of the remaining people is going down, that is the percentage/per capita death rate is rising, as the church declines. For example, if the church consists of all people in their eighties and the human lifespan is 85, then the church ceases in 5 years, however large it is. The per capita death rate in the first year is 20%, in the second year 25%, and so on until the final year when it is 100%.

If randomness is introduced, because some people do live longer than the lifespan, then the church may last a little longer, but now the results can only be given in probabilities.

### Death Model Represented by Feedback Loops

Alternatively, it is possible to look at the situation using the first-order smoothing of figure 3 if the death rate is allowed to rise as the population declines. There are now two loops, the balancing loop *B* for deaths, and the reinforcing loop *R2* for aging, figure 7:

The balancing loop* B*, representing deaths, exerts an opposing force on the stock *Church* leading to a negative exponential curve, with the church approaching equilibrium at zero. This is like a brake, slowing the decline. The reinforcing loop *R2*, representing aging, exerts a positive force on *Church* leading to a positive exponential curve. This accelerates the decline. If there are no other additions or subtractions from *Church* then the two forces of acceleration and braking are in balance, and there is no net force. Thus, the decline is a straight line, figure 8, line 1. Essentially, the effects of the two loops cancel out.

Note the per capita death rate, line 2, is increasing as the population declines, and thus ages. Towards the end of the church’s lifetime, the death rate increases dramatically to 100% as the last people die. Notice that the lifespan of the people in church declines as no people are joining, which would have lowered the average age, figure 9. If people did join, and an equal number of people left, then the age difference between joiners and leavers will affect the age distribution in the church, causing it to decline slower, though still a straight line. If the joining and leaving are not equal, then the decline will have some curvature, but not as much as figure 4 above.

Thus, care needs to be taken with models of churches in the last stages of decline as aging will tend to straighten the decline and bring extinction faster than predicted by models based on percentage, per capita rates, that is exponential/logarithmic models.

### Discrete Aging Chain

An aging chain is a model of a population split into cohorts to record the number of people in each age group as the population changes. Thus, a model of the population with deaths only, no births, additions or subtractions, could be modelled with, for example, three age groups, figure 10:

Note the width of the first and last age groups are both 20 years, with the middle group double – width 40 years. It is assumed that only the older age category have deaths. Thus, for the first 20 years, only the youngest age category (plot 1) declines as they age and join the middle group. In the other two categories, the inflows match the outflows, whether age progression or death.

From 20 years onwards, the middle age category (plot 2) starts to decline as there are no more young people to progress. The oldest age group does not start declining until the middle category is depleted. It has been assumed all age groups start uniform. If this were a church, it would be getting progressively older. The population declines as a straight line as in figure 6.

This aging chain is the type of model used in demographics and can be easily simulated with a spreadsheet. It models time in discrete steps according to the degree of age resolution required. Often they are annual, or quarterly. Thus it is called a discrete aging chain.

### Continuous Aging Chain

Most social models are constructed using continuous time. This is because the processes involved in the interaction of different population groups are mixed over the total individuals in the population, thus leading to very short time scales over which changes occur. Although this is not true for species subject to seasonal effects, it is nevertheless is a reasonable approximation. However, for human populations, the time step is usually a small fraction of a day, thus demanding a continuous time model. The types of nonlinear processes in these models behave very differently with continuous time as opposed to discrete. The latter may induce oscillations and chaos not present in the continuous time case, or in the real world.

Thus it would be wrong to use models that are discrete where processes that involve contact within populations are required. As a consequence, aging chains are often modelled continuously with first-order smoothing, figure 12:

Comparing the results with the discrete case, figure 11, it is seen that errors start to creep into the solution as the model smooths what should be strict age progression, figure 13.

Notice the middle age group declines immediately as more are leaving the category than entering it. Effectively, they are being sent into the next cohort too early because first-order smoothing treats all individuals in the stock the same at all times. Note also there are people still in the young category even at 50 years, long after they should have aged, as the first order smoothing keeps the aging progression rate the same even though the average age in the young category is advancing. This effect is referred to as **cohort blending**. The population declines exponentially, and wrongly, as in figure 4.

Of course, a population with only deaths is an extreme case. This is very rare in human populations and only seen in subgroups such as organisations and churches near their the end of their lifespan. In nearly all cases, when births and other additions and subtractions are included, the continuous and discrete models of aging give very similar results as cohort blending averages out. The age distribution is now changing very slowly and does not unduly affect the progression rates.

The two cases which require discrete aging are: (1) where age-related deaths dominate, as above; (2) where there is sudden migration in a specific age group. Mixing discrete aging with continuous processes is both analytically and computationally difficult, and usually compromises must be made. However, the issue has been solved in system dynamics by Eberlein & Thompson (2013) where they took the integration step as a very narrow cohort age group. People are working on other solutions to the problem of cohort blending.