To Main Proceedings Document
Growth and Decline of Religious
and Subcultural Groups
J ohn Hayward
University of Glamorgan
Division of Mathematics
University of Glamorgan
Pontypridd, CF37 1DL
Wales, UK
01443 482258
jhayward@ glam.ac.uk
Abstract
Within any population there are subcultural groups, based on religion, political be-
lief, or lifestyle, which hold the allegiance of people. Often such groups include
membership of an organisation such as a church, cult, political party, or some looser
group that encourages the beliefs and practices of the subculture. These groups may
survive for many generations, before eventually dying out. Many remain small mi-
norities with little potential for growth. However a small number not only flourish,
but eventually influence the bulk of society. A prime example of a successful group is
the Christian church which, having grown for two millennia, is well poised to con-
tinue growing in the future, except in Western Europe where it still declines.
This paper examines the dynamics of the growth of such religious and subcultural
groups and attempts to explain, using system dynamics, the causes of their growth
and decline. Based on a mathematical model of church growth, published by the au-
thor, a systems dynamics model is developed incorporating births, deaths, and rever-
sion. It is argued that the primary cause of growth is the existence of enthusiasts
within the group who spread the belief, and hence recruit, through contact with un-
believers. It is further argued that growth is limited because they are only enthusias-
tic for a limited length of time, falling back to ordinary members of the group, with
virtually no recruitment potential.
The model is applied to the spread of the Christian church and other subcultural
groups, discussing their potential for long-term survival.
1 Introduction
The growth of human populations has been extensively modelled from both a systems and a
mathematical viewpoint. The processes of births, deaths, immigration and emigration are standard
text-book models. However within the human population there exist subcultures which possess
some features that distinguish them from the remainder of the population. Foremost among these
are religious groups represented by churches, sects and cults which can have very rigid boundaries
separating them from the rest of society. Other subcultures could include political groups, or more
diffuse lifestyle groups such as vegetarians, youth subcultures such as punks, a hobbies society or
even mobile phone users.
The dynamics of these subgroups differ from a standard human population in that they can recruit
from a population of people who have yet to possess the features of the subgroup. In the case of a
church there is an explicit desire to actively recruit, called conversion, from the rest of the popula-
tion. For lifestyle groups, such as vegetarians that desire to see others like themselves is purely at
the individual level, or may not exist at all. However the lifestyle may prove attractive to others so
that they also convert and become part of the group. Such conversion is very common amongst
youth sub-cultures where there is a desire to be like their radical peers and different from an older
generation.
Thus religious and subcultural groups have the conversion process as a significant part of its dy-
namics. This needs to be balanced out by reversion, those who give up the church or lifestyle. In
addition it is possible to be born into such subgroups. Churches gain a lot of their recruitment
from the children of those already in the church. It is possible that the children of vegetarians are
more likely to become vegetarian themselves. However in the case of youth subcultures the re-
verse is often the case and children are more likely to abandon the culture of their parents. Thus
births (and deaths) also play a part in the dynamics of religious and subcultural groups.
In the last thirty years a considerable amount of effort has been expended in attempting to under-
stand why and how churches grow or decline. Much of the work is qualitative in nature usually
giving recommendations to a church as to how it may improve its growth. (Gibbs 1981, Pointer
1987, Schwartz 1998 are typical examples). However quantitative measures of growth are also
important and much attendance and membership data are produced (e.g. Brierley 199a). Any
modelling of the growth process is limited to statistical analysis or numerical arguments (Stark
1996). However in Hayward (1995 & 1999) a church growth model was developed using mathe-
matical population modelling. The model, based on the general epidemic model, was constructed
on two principles:
1. conversion growth of the church comes from contact between unbelievers and active believ-
ers, called enthusiasts;
2. enthusiasts only hold their enthusiasm, or potential for converting others, for a limited length
of time after which they become inactive, but are still part of the church.
The model exhibited growth behaviour very similar to that seen in a religious revival, where the
growth runs out because there are insufficient enthusiasts to maintain the level of conversion
among a dwindling number of unbelievers. The result is that the growth ceases without all the un-
believers being converted. Indeed there is a threshold that the number of unbelievers must be
above for significant growth to take place. That threshold depends only on the number of people
that an enthusiast is responsible for converting during their enthusiastic phase.
The purposes of this current paper are to re-express the model in a systems format in order to un-
derstand the dynamics of the conversion process more fully, and to include effects of births,
deaths and reversion. By taking a systems approach it is hoped that the behaviour can be under-
stood without the need for extensive mathematical analysis. In addition, as outlined above, the
model can be applied to subcultural groups other than churches and thus may be of interest to a
wider audience. Although it will be constructed with churches in mind, other applications will be
alluded to.
The paper is organised into two separate models. The first (Ch. 2) looks at the conversion process
where the enthusiasts who are responsible for the conversion process retain that enthusiasm
throughout their lifetime. The second (Ch. 3) then adds in the feature that enthusiasts only have
conversion potential for a limited time span. Both will add births, deaths and reversion. It is
hoped that this approach will make the effects of the different process clearer.
A fuller version of this paper is available in two technical reports (Hayward 2000) and the models
available at the author's website (Church Growth Modelling Website).
2 Unlimited Enthusiasm Church Growth Model
2.1 Conversion Model
2.1.1 Believers and Unbelievers
A church is a collection of people who have common religious beliefs, a common identity, and
who meet together regularly, usually for worship. As such they are an identifiable subgroup of
society, including those who are like-minded, but largely excluding those who have no part of that
religious life. Thus, at the simplest level, it is possible to split society up into two groups: those
who belong to the church - called believers, and those who don't - called unbelievers.
2.1.2 Conversion Through C ontact
In addition the believers have a common mission, which is to see unbelievers become believers.
Effectively this means recruiting people into the church, called conversion. For simplicity such
conversion will be deemed to be a one-way process, thus there is a drain from the unbelievers to
the believers over time, with no other losses to the system. The model is thus a two compartment
model with transfer from unbelievers to believers (figure 1).
As was argued in Hayward 1999, the primary means of recruitment is through contact between an
unbeliever and a believer. The contact itself may be the actual means of conversion, or it may be
an invite to a meeting where the conversion occurs. Although many of these contacts occur be-
tween people who know each other, nevertheless it is possible that a believer's word of mouth tes-
timony could inspire those they do not believe to seek out the church and become a believer. This
is a common feature during times of religious revival where those affected by the revival are no-
ticeably enthusiastic.
The number of conversions each believer is responsible for will depend on the probability of such
a contact. Assuming that believers and unbelievers are homogeneously mixed in a population, this
Unbelievers Believers
- Conversions +
Actual No converted per believer per unit time
Potential No converted per believer per unit time
Probability of finding unbeliever
Fig. 1. Unlimited Enthusiasm Model of church growth
probability will be the fraction of the population who are unbelievers. Homogenous mixing means
the church has to be geographically spread, rather than confined to certain places. This would be
true for much of the history of Christianity, apart from when it was introduced to a country. Even
in its early years Christianity became spread throughout the Mediterranean world quickly due to
good transportation between major cities.
Thus the conversion rate (the flow rate conversions in figure 1) is proportional to the probability
of finding an unbeliever and the number of believers. The more believers there are, the more con-
tacts with unbelievers take place.
One final assumption concerning the conversion process is needed. The believers do not cease
recruiting unbelievers. Thus they do not run out of evangelistic zeal, or run out of unbelieving
contacts before the pool of unbelievers is exhausted.
Thus, leaving out the effects of births, deaths and reversion the unlimited enthusiasm church
growth model based on conversion is give in figure 1.
The potential number converted per believer (or converion potential) is the number of unbelievers
a believer could convert if all their contacts were with unbelievers. Of course the actual number
will get lower as the church grows because more of their contacts are with believers. This tends to
be true of churches as much of a believer's time is spent interacting with those who are like
minded, so less of their time is spent recruiting unbelievers. The same is true of youth cultures and
lifestyle groups. Vegetarians may influence other people to become vegetarian at a shared meal.
However the more vegetarians there are at the meal the less people there are to influence.
2.1.3 Analysis
If the population of unbelievers is large in comparison to the believers, then growth of the church
is initially exponential. At some point however the growth slows down and eventually ceases
when the whole of the population has been converted. This occurs because the efforts of the be-
lievers are increasingly “wasted” on other believers as the number of potential converts declines.
In the early period a single believer may know, or come into contact, with many believers, but as
the church grows and the pool of unbelievers declines that believer will know fewer unbelievers.
This can be seen be examining the causal loop diagram for the model as in figure 2. This is an ex-
ample of shifting loop dominance and results in the S-shaped, or logistic, growth of the believing
Actual Number
Converted
Believers R
Prob. of finding
unbeliever
Fig. 2. Causal loop diagram of the unlimited enthusiasm model of church growth
population. A sample simulation (figure 3) shows this logistic curve.
In mathematical terms this is sometimes referred to as the simple epidemic model (Hayward
1999). However the model as it is presented here will be called the unlimited enthusiasm church
growth model. It can be shown mathematically that the model results is the logistic equation
(Hayward 1999), where the logistic limit is the total number of people in the population. The
Fraction of Total Population
1.004 oop
1
2: Believers
O50 af e--eeeeeeeceeeeeees
0.00--—= + + 1
0.00 : 15.00 30.00
Years
Fig. 3. Growth of the church through conversion with unlimited enthusiasm only
model is often used for the dispersal of innovations through a population, the so called Fisher- Pry
models (Kumar et al 1992). A similar model was used by Coleman (1964), and Bartholomew
(1982) for diffusion of social phenomena.
The time taken for the population to be converted depends on the initial numbers of believers and
unbelievers, and also on the potential number of converts per believer per unit time. In this simula-
tion the initial percentage of believers was 1%, and the potential number of people converted per
believer per year was 0.4. That is each believer has had two contacts in five years that has led to a
conversion. This results in the conversion of 99% of the population in just over 20 years. Such
dramatic growth seems surprising from such a small conversion potential. The reason is that each
convert also makes converts at the same rate, and does so indefinitely. To simulate more realistic
church growth these assumptions will need to be challenged.
This S-shaped growth is typical of the type of growth seen in religious revivals. A point comes
when the growth of the church explodes because there are so many enthusiasts whose contacts
are resulting in the conversion of unbelievers, many of whom also become enthusiasts. Such be-
haviour can be seen in the Welsh revival of 1904, where 100,000 people were converted on just
over a year. Likewise the current revival in South America is seeing explosive growth. Neverthe-
less no revival, however powerful, has ever resulted in the conversion of a whole population.
Thus some changes to the model are required.
The bulk of the conversions occur in the middle period of the growth. Thus early on in the growth
there are few contacts, a thus few conversions, because the percentage of unbelievers is so small.
Because growth is the main factor through which revivals are first noticed it can mean that a re-
vival can be underway some time before it comes to the attention of the population at large. This
was true of the early Christian church which remained small for the first 200 years. E.g. only 1%
of the city of Rome was Christian about 250 AD. Yet by 300 AD the church was so widespread
further persecution by the still Pagan state became impossible (Stark 1996). Similarly in Egypt
estimates of the size of the Christian church based on inscriptions of Christian names shows simi-
lar slow then rapid growth (Bagnall1982)
Likewise the growth of a fashion or subculture follows similar lines. The fashion for long hair
among males, which started in the 1960's was for a long time only possessed by a minority. Y et
within a decade the fashion was widespread throughout western society, so much so that young
people rebelled and adopted a different fashion! The adoption of colour TV’s has followed a simi-
lar pattern and there is every expectation that the adoption of mobile phones will follow suit.
2.2 Unlimited Enthusiasm with Births, Deaths.
2.2.1 Births and Deaths
Introducing births into the model demands a question. Are these additions the same categories of
people as their parents? That is, are the children of believers automatically believers, and those of
unbelievers automatically unbelievers? Clearly a child is not born with a religious belief. However
if a child is brought up in the faith and subsequently adopts that faith while still in childhood, or
adolescence, then they will be regarded as being born as believers. This is often called biological
growth to distinguish it from conversion growth. Such children are not converted into the church
having been an unbeliever.
Recognising the fact that not all the children of believers will become believers then a parameter
representing the fraction of children of believers who are born believers is introduced.
Deaths are more straightforward, providing we assume that believers and unbelievers die at the
same rate. Of course if the church is being persecuted it may have a higher death rate. If the be-
lievers are vegetarians they may live longer (or shorter!). If the subculture is of drug users then
the death rate may be higher than the rest of the population. For this paper it is assumed all death
rates are the same.
Figures 4 and 5 show the birth and death processes on the two population categories, with the
conversion process suppressed.
birth rate
births of believers
fraction born believers
ee
conversion
Believers
deaths of believers 5
Nees
death rate
Fig. 5. Dependence on birth and death rates for believers
Note that the believers now have an additional balancing loop due to deaths which is either equal
or larger in effect than the reinforcing loop due to births, thus giving a net balancing effect.
2.2.2 Analysis
Consider the case where the birth and death rates are equal, so that there is no change in the total
population. The solution can be investigated for different fractions born believers, as in figure 6.
The solution is still S-shaped.
Thus whatever the fraction of children kept by the church, the population of believers increases to
Unbelievers
iz) births of unbelievers
O
conversion
deaths of unbelievers
fraction born believers
birth rate death rate
Fig. 4. Dependence on birth and death rates for unbelievers
Believers as a Fraction of Total Population
1.00
o5o4---
Percentage Bom Believers:
1=0%
4 PLP gececrae seme RTERT 2=25%
3 = 50%
4=7%
5= 100%
0.00 75.00 112.50 150.00
‘Years
Fig. 6. Growth of church for different fractions born as believers
Number of converts per believer per unit time = 0.09
a stable fixed value. However the whole population is no longer converted unless all the children
of believers become believers themselves. Anything less and the church can only grow to a value
less than 100% of the whole population, a value which is independent of the initial percentage of
believers. Thus the inability of the church to hold its own children in the faith will prevent it from
converting the whole population over time. The result of the net balancing loop due to deaths and
births is to lower the possible growth limit of the church.
The lower the conversion potential the lower the growth limit the church reaches in society, and
the longer it takes to do so. Halving the potential number converted per believer per unit time re-
sults in a drastically slower growth. It also sees the believers’ final share of the population drop
from 84% to 68% for the case where no children are retained by the church. If the church cannot
hold its own children the conversion potential becomes crucial. Indeed it is possible that the con-
version potential could be so low that the church eventually becomes extinct.
Consider the case where the church is already 50% of the population, and is only able to hold
50% of its children. The church then requires a conversion potential of 1.4 just to stand still. Any-
thing lower and the church will decline. This has been the pattern of church decline in Western
Europe since 1900, where the failure to hold children has been a significant factor.
2.3 Reversion
2.3.1 The Reversion Process
The next assumption that needs to be challenged is that believers remain believers all their life-
time. Clearly some do renounce the faith and return to the world. This may be for a number of
reasons:
Physical persecution. At its most extreme believer's lives could be in danger, as frequently
happened in the early church. Christians would be asked to swear allegiance to Caesar and
curse Christ, or face death. Those who succumbed were no longer welcomed in the church.
The fear of this test caused many others renounce the faith also.
Psychological persecution. Some believers give up the faith due to pressure of unconverted
family or peers. The pressure may take the form of teasing or being shunned from normal so-
cial activities. The believer gives up the church to have an easier life.
Some of the believers were not truly converted in the first place and give up the faith once it
loses its initial attraction. The unbelieving world they left behind has proved more attractive
to them.
The believer may have fallen into sin and caused them to be to ashamed to be part of the
church anymore.
The believer may have been hurt by others in the church and it is now too painful for them to
be paer of the church with those who hurt them.
Thus believers only remain believers for a fixed length of time. This is an averaging effect. Many
believers will remain believers all their lives, but some do not and return to the unbelieving world.
This has the effect of reducing the average length of time a person remains a believer.
Can these people ever return to the church? In other words will a softening take place that allows
them to be converted again. It will be assumed that hardened people only remain hardened for a
fixed length of time, after which they are opened to re-conversion with the same likelihood as any
unbeliever. Thus that can be regarded as unbelievers.
Thus the outline model is given in figure 7. Reversion has introduced an additional balancing loop
on believers, reducing further the target level for the church. However there is also an additional
reinforcing loop as hardened people are re-cycled around to unbelievers again. Thus the target
will be reduced less.
2.3.2 Analysis - Short Term Behaviour
In the short term births and deaths can be ignored. Assume no softening on this time-scale. Thus
the model is effectively the epidemic model where the susceptibles are the unbelievers, the in-
fected are the believers and the removed are those hardened to conversion. Structurally this is the
same as the model used in Hayward (1999) with the two main differences in that the middle cate-
gory, the infected, constitutes the whole church and the final category, the removed, are people
outside the church.
In the epidemic model the infected category always dies away leaving only the susceptible and the
l
Fraction of Total Population
1.00—+; =: a
OS0g rece eeeeeeeveeees
Months
fg Wotowth of church due to ¢onversions and reversion at rate of 3%
avers!
: : ; ign
deaths of hardened with no Hirths and deaths
Fig. 7. Unlimited enthusiasm model with births, deaths and reversion with softening
(conversion details missing)
removed. Thus the church dies away after an initial growth, leaving some, but not all of society
hardened to conversion (figure 8). Clearly this is an unrealistic result as a time-scale has been cho-
sen so that births and deaths can be ignored.
However it is good model of short term fashion or behavioural patterns where there is no time to
pass them onto children, or where children would have no desire to copy adults, or where they
have been deliberately excluded from exposure to the behaviour. Many sects or cults follow this
pattern and die out due to a failure to recruit enough to balance out those they lose. The church
itself has such short-term behavioural patterns. One such example was an early charismatic
movement in the UK under the leadership of a Presbyterian minister Edward Irving in the 1820's.
For a brief period a number of people exhibited charismatic behaviour. In this application the “un-
believers” are church people and the “believers” are those with the behaviour. The movement
eventually burned out, not affecting the wider church or the later Pentecostalism. A church of
such people did remain, but only a shadow of its former self and little interest in recruitment due
to its belief in the immanent end of the world. These are the equivalent of the hardened. Many ad-
ventist sects follow a similar growth pattern and later become extinct.
2.3.3 Analysis - Long Term Behaviour
If births and deaths are included a more realistic picture for the growth of the church is achieved.
If the person is permanently hardened and never open to being converted back into the church
then the tendency is for the church to make part of society hardened to the faith. Figure 9 is typi-
cal. Like the cases without reversion the numbers in the church level out to a fixed value, often
declining to it after an initial overshoot depending on its initial percentage in society. Thus rever-
sion, like births and deaths, causes the church not to convert the whole of society even when new
unbelievers open to conversion are being born. However the church does not die away as the pool
of unbelievers is being replenished by births.
Adjusting the conversion rate or the fraction who are born believers makes little difference to the
final percentage in the church, or to the time-scale to reach stability, although it has a bigger effect
on the proportion of society hardened to conversion. The main effect on church numbers comes
from adjusting the reversion rate. This rate of loss is therefore crucial for the long-term success
of the church.
The reversion rate in figure 9 was chosen at 3%. This means that the average length of stay in the
church is around 30 years. This explains why it takes at least double that figure for the numbers to
stabilise. Of course this average is made up of a small number of people who stay much shorter
than this and many who never leave the church at all.
Keeping the same parameters as figure 9 and introducing a softening rate of 1% sees a consider-
able improvement in the final value of the church of 35% from 25% without softening (figure 10).
Increasing the softening rate further sees substantial improvements in the churches numbers e.g. a
softening rate of 6% sees the church up to 56%. Thus a church which is losing people back to the
world can make considerable improvements in its long term prospects if it seriously considers how
to win back those people, and makes the effort to keep the dialogue open.
1.004:
2 1: Unbeli
2: Believers
3: Har
0.504
0.00 50.00 100.00 150.00 200.00
Years
_ 900 ——— + - i— 7 - 1
Fig. 10. Growth of church ditedo conversions, neersion at rate of 3%o0aed a softeningpoate
of 1%. Fraction born as believers is zero and nugaber converted per person per unit time =
0.2
Fig. 9. Growth of church due to conversions and reversion at rate of 3%. Fraction born as
believers is zero and potential number converted per person per unit time = 0.2
Notice also that the peak of the growth without softening (figure 9) is more or less the same as
with softening (figure 10). Thus there is a temptation for the church in a successful period to ig-
nore reversion because it has little short term effect on numbers. However it is the long term
numbers that reversion effects. Notice in figure 9 the decline is much greater.
2.3.5 Church’s Prospects with No Conversion
Clearly with no conversions taking place then growth of the church is not possible unless the
church has a higher birth rate than surrounding society. Of course if there are no reversions and all
the children remain in the church then the church will not decline. However what if there is rever-
sion, could the church continue to maintain itself with a higher birth rate?
A typical situation is given in figure 11. The church can indeed maintain itself, but only at the ex-
pense of unbelievers growing. Although it is possible for a church to grow in proportion to soci-
ety by having a higher birth rate, once even a moderate amount of reversion takes place a substan-
tially higher birth rate is needed to maintain the church’s proportion in society. Trying to counter-
balance reversion by the birth rate is highly unstable. Small changes in birth rate can make the dif-
Fraction of Total Population
1.004
0.504
Fig. 11. Church growth model with no conversions, a reversion rate of 1%, and a birth rate
of believers 70% larger than the rest of society.
ference between survival and decline. Believers are subject purely to birth and death processes
which either grow exponentially or decline to zero unless they just balance out. The conversion
loop, which gave the stability by setting the target at or below the level of the total population, is
now missing.
The church losing a fraction of its children produces similar effects. Thus the effects of reversion,
or losing children, can only be safely counteracted by conversion from the unbelieving pool. The
prospects for any group that does not recruit is bleak unless it is so self-contained that virtually
no-one is lost and they consistently have larger than average families.
3 Limited Enthusiasm Church Growth Model
3.1 Limited Enthusiasm Without Births, Deaths and Reversion
3.1.1 Rationale for the limited Enthusiasm C hurch Growth Model
From the results of chapter 2 it is clear that growth in the unlimited enthusiasm church growth
model is only limited by the failure of the church to hold all its own young people and reversion
from the church. Even in these cases the church numbers levelled out to a fixed value. Although
this can be regarded as a good model of the growth of the church, particularly in periods of re-
vival, it doesn’t model the decline of the church. Currently the Christian church in Western
Europe is marked by decline and not growth which may well lead to its extinction from large parts
of that area within a generation or so. The unlimited enthusiam model has depended on an unreal-
istic assumption that believers continue to recruit in the same way indefinitely
Instead it is found that believers stop recruiting for a number of reasons:
1. The believers have been recruiting through their network of friends and relatives which is
now exhausted. There are three scenarios:
the people in this network have become believers themselves;
they have become immune to any further pressure to join the church;
they have ceased to have meaningful contact with the believer. Many new converts find
after a year or so that they have a new set of friends in the church and their old unbeliev-
ing set have drifted away. Often the new convert does this subconsciously because being
part of the church means taking on a new set of values leaving them uncomfortable with
the values of their old friends.
2. Churches do not just recruit or evangelise. After a while new converts find other work to do
within the church and spend less time on recruitment activities.
3. In periods of intense growth the pastoral demands of dealing with new converts prevent min-
isters from spending as much time on evangelism as they might like and thus their recruitment
potential drops.
4. Often believers run out of enthusiasm for recruitment and settle into a more subdued version
of belief where the zeal to see new converts has declined to the point of inactivity. Again
there are a number of scenarios behind this:
the believer has forgotten why they converted from unbelief to belief. They now have no
desire to see others converted;
often the believers gain status within the church and lose the real reasons why they joined
in the first place. Any enthusiasm they now have is centred on their own advancement;
the believers may find the church so enjoyable that their enthusiasm is for their own ex-
perience of it rather than to see others converted;
it may be that the church has not lived up to expectation and the believer has lost enthu-
siasm for anything to do with the beliefs. Instead they have settled into a nominal church
life.
Many of these reasons are summed up in Wesley's Law of the decline of pure religion. “Tak-
ing up the religion has produced benefits which makes missionary zeal too costly to engage
in” (Kelly 1986)
Thus it is assumed that a believer's enthusiasm, or recruitment potential, is limited in time. This
immediately leads to two categories of believers, active believers and inactive ones. The active
believers are active in recruitment and could be called enthusiasts where it is understood that their
enthusiasm includes zeal to see others converted and behaviour that actively does something
about it. This leads to the limited enthusiasm model of church growth. The dynamics of the
growth of the church is driven by these enthusiasts, who may he dedicated evangelists or lay-
people with a zeal for spreading the faith.
It should be stressed that it is purely the recruitment potential of the enthusiast that is limited. As
there are so many ways in which this may happen the inactive believers may still include people
who display enthusiasm for other aspects of the faith. Indeed this is characteristic of times of re-
vival where believers have a heightened experience of what it means to be a believer and thus a
much larger desire to see others so converted. This desire does not last long before they move
back to the level of enthusiasm for conversion that they had before, but it is sufficient to cause a
period of rapid growth in the church.
Thus inactive believers are only comparatively inactive compared to the believers. They may in-
deed have some recruitment potential, but much less than the enthusiasts. For the purposes of this
model it will be assumed that inactive believers play no part in recruitment.
3.1.2 Construction of Limited Enthusiasm Model
It will be further assumed that all new converts will be active believers until one of the above
processes makes them inactive. Also it will be assumed that there is no mechanism for making in-
active believers enthusiasts again, so once inactive they will never become active in recruitment
again.
This leads to a three compartment model where unbelievers get converted by active believers and
become, active in recruitment themselves. After a time active believers become inactive (see figure
12). This is the model described mathematically in Hayward 1999, and used by Coleman 1964 and
Bartholomew 1983 for the spread of rumours.
Society Church
Probability of finding unbeliever
Active Believers
loss of enthusiasm
Unbelievers Inactive
conversions
Duration
Actual No converted per believer
Potential No converted per believer
Fig. 12, Limited enthusiasm church growth model with no births, deaths or reversion
As in the unlimited enthusiasm model, the number of conversions depends on both the number of
unbelievers and active believers. Because enthusiasm is limited it is now more convenient to let
the conversion parameter be the potential number converted per believer during their enthusiastic
phase given all their contacts are unbelievers. Clearly the actual number will be less when the pro-
portions in church compared to society gets bigger, as in the unlimited enthusiasm model.
However now the number of conversions per unit time (conversions) will also depend on the
length of time spent in the active category. Thus if every enthusiast converts two others during
their enthusiastic phase the number of conversions in one year will depend on how long they are
enthusiastic. If it takes two years rather than one the dynamics will be slowed down. In Hayward
1999 it was shown that this approach could model the case where the number of potential con-
verts went up if there were more people (crowd model) and where the number of potential con-
verts were fixed regardless of population size (fixed contacts model). This was providing the total
population remained constant.
Thus the number of conversions per unit time are:
Conversions = Actual number converted per believer X Active Believers / Duration
A new parameter called duration is now introduced which effects the conversion rate. Duration
also determines the rate of transfer from active believers to the inactive believers, which is as-
sumed to be a draining process.
This model is the classic general epidemic model which has been extensively studied and devel-
oped (Bailey 1975, Anderson and May 1987). The mathematical foundations of this model and
further details of its relationship to church growth are given in Hayward 1999.
3.1.3 Analysis
Fraction of Total Population
1.004
1: Unbelievers
2: Active Believers
0.504
Fig. 13. Limited enthusiasm with no births, deaths or reversion. Enthusiastic phase is 1 year
and the number converted per believer is 2
The active believers are now subject to another balancing loop, the draining process to the inac-
tive believers, compared with the situation in figure 2. However this balancing loop includes a
negative flow, so that not only is its level limited but will now actually be reduced to a target
level. As this is a straight draining process the target is zero and the active believers eventually
decline and disappear after an initial growth. This is the standard behaviour of the epidemic
model where the active believers are like those infected with the disease. The inactive believers
are like those who have been “cured” from the disease, or have died, and are no longer open to
infection, or able to infect others. Figure 13 shows a typical pattern of growth.
Clearly the numbers in the total church, eventually reflected by just the number of inactives, is lim-
ited and the church has failed to convert the whole population. This was the result discovered by
Kermack and McKendrick (1927) where they showed that even in an epidemic the infection didn’t
spread through the whole population. It burned itself out due to a lack of infected people who had
less and less contact with those susceptible to the disease and more with those already immune. In
the church growth case the growth has run out due to a lack of enthusiasts.
It is this type of sudden growth behaviour which characterises times of religious revival both at
the local level, and at a national or international level. Examples of such local growth in the Chris-
tian church are well documented: Northampton Massachusetts in the 1730’s (Edwards 1965);
Nagaland North East India 1976 (Orr 2000) and more recently Pensecola Florida from 1995 on-
wards. Examples of such national growth include the First Great Awakening 18th century USA
and UK (Edwards 1990), Second Great Awakening Early 19th century USA. Much faster revivals
occured in the USA (1858) and then Wales, Scotland and Northern Ireland in 1859 and also
Wales in 1904-5. A much longer revival in the twentieth century has been the growth of Pentecos-
talism especially in South and Latin America.
Apart from those revivals still continuing all, show the characteristic rapid rise in total church
numbers with the growth eventually burning itself out. Of course data doesn’t record the number
of active believers. However the model was applied in Hayward 1999 to church membership data
for the Welsh revival of 1904-5, where it was shown that the potential number converted per ac-
tive believer was about 2.02 with the duration of the enthusiastic phase about 1 week. Clearly
these are average values, not true of every enthusiast in the revival.
In fact extending the enthusiastic period does not make any difference to the final numbers in the
church in this model, it simply takes longer for the growth to occur. The key parameter is the po-
tential number converted per (active) believer. Decreasing this will result in substantially less con-
versions. Indeed if this had been lower than 2 no revival would have taken place. It is remarkable
how such a small change in the effectiveness of those who recruit makes such a large difference to
the growth of the church.
The potential number of conversion per active believer is the parameter that governs the threshold
of the epidemic type growth. According to epidemic theory (Bailey 1975, Anderson and May
1987, Hayward 1999 equation 23) if the number of unbelievers times the potential number of
conversions per active believer is above one then the epidemic or revival will occur. If it is below
then there will be no such growth.
3.1.4 Not All Converts Become Enthusiasts
One criticism of the idealistic limited enthusiasm model is that all converts become such enthusi-
asts themselves. Even in times of religious revival the change in a person’s life may not extend to
zeal for recruitment. Indeed some new converts may come from families where most are already
believers and thus have few unbelievers they can make contact with. Thus not all new converts are
enthusiasts. This leads to an extra parameter to determine the proportion of converts who become
active believers. The dynamical model is now modified as in figure 14 .
It is now possible to fit the data from the 1904-5 Welsh revival to this new model where not all
the converts become active. in 1904 the combined total membership for Welsh churches stood at
48.94% of the total population (Hayward 1999 & 2000), and rose to 53.43% by the end of 1905.
The bulk of the converts came in a period of about 12 months.
Now assume only 10% of converts became active believers, the results of the simulation yields a
duration of enthusiastic phase in a believer of about 1 month and the number converted per active
believer at 18.3. Compared with the model where all converts became enthusiasts the enthusiastic
Society Church
conversion to inactive
Inactive
loss of enthusiasm
Unbelievers
Active Believers
conversion to active
Fig. 14, Limited enthusiasm model where not all the converts become enthusiasts
phase is four times longer and each enthusiast has nine times as many converts. Thus to explain
that the revival was spread by a smaller number of people requires a much larger effectiveness on
the part of the enthusiasts. If only 1% of converts become enthusiasts then each enthusiast needs
to make around 100 converts in 4 months!
It is more likely that many converts became enthusiasts and spread news of the revival to a smaller
number of people. That then brought the people to a place where they became converted. How-
ever it was the action of the enthusiasts who brought them to that place, rather than the evangelist
who spoke in that place which was the key to the dynamics of the growth of the church.
3.2 Limited Enthusiasm With Births, Deaths and Reversion
3.2.1 Construction of Model
It is now possible to include the effects of births, deaths and reversion in the limited enthusiasm
model. There are now 3 processes that will limit the growth of the church:
* — the limited enthusiastic period of believers;
* reversion from the church to the world;
* — inability of the church to keep all its children in the faith.
Now there are two categories of believers assumptions need to be made with regard to reversion.
It is assumed that those active believers who revert and become hardened don’t do so directly.
Instead they become inactive believers first.
Again with births a decision has to be made whether the children of active believers become active
themselves, or are inactive. A parameter is introduced to control what proportion of active believ-
Society Church
Hardened
Unbelievers 3Z Active Believers
63. = Births Active
On
births Unbelievers,
{ ) , IK) deaths Active
Fig. 15. Outline of limited enthusiasm conversion model with births, deaths and reversion.
Conversion details are suppressed but are similar to figure12.
ers are born believers and the proportion of these who are bom active. it may well be that active
believers may hold more of their children in the faith than inactive ones because generally they
have more enthusiasm for the beliefs themselves.
The basic stocks and flows are given in figure 15. The conversion process follows figure 12 for
recruitemt to both active and inactive believers. Births, deaths, reversion and softening processes
follow figures 4, 5 and 7.
3.2.2 General Comments
Simulations show that a steady state situation, where the church settles to a fixed percentage of
society, is possible, with a typically small percentage of the church being enthusiasts or believers
active in recruitment. Unlike the church growth model of Hayward (1999) these enthusiaists do
not die out but sustain themselves, and the rest of the church, at fixed levels. A small increase in
their recruitment potential can see a substantial rise in the number in the church. Typically the
numbers oscillate for 100-200 years before settling to a new level. Thus even with no change in
parameters, stability in numbers can take a long time to achieve
The steady state values correspond, more or less, to the threshold of the epidemic outlined earlier.
However in this case if the potential number of conversion is above the threshold, then the church
survives with numbers at stable values largely determine by that conversion potential. However if
it is below the threshold the church will eventually decline to extinction. The potential number of
converts per active believer is crucial for the church's survival
The full model is extensively analysed in Hayward (2000). Here applications will be limited to two
cases from the last 100 hundred years: the Christian church in Western Europe and also in South
America.
3.2.3 Church Decline in Western Europe
At the beginning of the 1900’s the church in many European countries accounting for about half
the population. However enthusiasm was noticeably running out, especially with a tide of disbelief
and liberalism within the church’s own ranks. A simulation of the subsequent decline appears in
figure 16, where the parameters:
potential number converted per active believer = 2.4
duration of enthusiastic phase = 4 years
were estimated by matching the graph of the total number in the church to data for the last one
hundred years (UK data was used Brierley 1999a).
Other parameters estimated by recent data or local anecdotal evidence:
reversion rate = 0.016, softening rate = 0.5, fraction made active = 0.5,
fraction inactive born unbeliever = 0.75, fraction active born unbeliever = 0,
fraction active bom believer and active = 0.5
Note that the number of enthusiasts in this model has been rising for the last 50 years of the 20th
century. In reality there has been a rise in the most enthusiastic part of the church, the evangelical
and charismatic parts. these are the section of the church that recruit the most. If this model is
close to the truth then the church will begin to start growing again in about 30 years due to the
increasing number of enthusiasts. This growth is without any change in the value of the parame-
ters and is a generational effect due to child losses and reversion achieving their true level rather
than a true revival. There is a long-term damped oscillation to a fixed value. Eventually it could
recover from its current low of 8% to about 14% of the total population at the end of the 21st
century.
What can be done within the church to improve the growth by changes made now? Changing the
softening rate has little effect. However if reversion is switched off the final value of 14% be-
comes 16%. It becomes 18% if all the children are retained and 20% if both effects are combined.
BO 00 ay ssstsssssstsssssstsssasssassisasassasasassasy TERRE pe
1: Church as Percentage of Population
2: Active Believers as Percentage of Church
30.00
Years
d.00 sooo — 40c.o0 150.00 200.00
Fig. 17. Typical church decline in Western Europe 1900-2000 followed by
increasing the conversion potential 2000-2100
Fig. 16 Typical church decline in Western Europe from 1900.
Clearly tackling loses makes some significant improvements.
The dramatic effects come from improving the recruitment to the church. If the potential number
converted per active believer goes from 2.4 to 3 the percentage in the year 2000 is now 30%, see
figure 17. Indeed the same improvement can be made by making 60% of the new converts enthu-
siasts rather than just 50%. Now the oscillations are taking longer to settle down.
Also of significance is the lag between the total number in the church and the number of enthusi-
asts. The number of enthusiasts starts to wane long before the church does. If the number of en-
thusiasts could be measured then early warnings of problems within the church could be given.
3.2.4 Church Growth in South America
60.004
1: Church as Percentage of Population
2: Active Believers as Percentage of Church
30.004
0.00 100.00
Years
Fig. 18. Typical church growth in South America 1900-2100
The Protestant Christian church in South America has seen explosive growth especially during the
1970's and 1980’s (Brierley 1999b). This has been largely the result of the activity of Pentecostal
churches which display a high degree of enthusiasm. A typical simulation is given in figure 18,
where:
potential number converted per active believer = 3.3
duration of enthusiastic phase = 10 years
have been adjusted to fit typical South American church membership data. Other parameters are
kept the same as in the European case.
Clearly the percentage of enthusiasts is now past its peak leading to a levelling off and slight de-
cline for the total church in the next 50 years. If this situation is true the South American churches
should be seeing a rise in the level of nominalism in the church. If so tackling this nominalism
could prevent the church's future decline. If the church wants to see growth beyond these pro-
jected levels it must improve its conversion rate. It cannot expect to see the whole of the popula-
tion converted on its current levels of conversion.
4 Conclusion
4.1 Main Conclusions
Realistic patterns of growth are seen when the recruitment to the church is driven by enthusi-
asts responsible for the conversions.
If all the church are such enthusiasts, and they are able to keep all their members, and bring
all their children into the faith, then the whole population gets converted.
If a church loses some of its children, or members, then even if they are open to re-
conversion, the church does not convert the whole of society.
If not all the church are enthusiasts, either because they have lost their enthusiasm, or never
had it in the first place, then the church fails to convert the whole of society.
If the church survives it always tends to fixed stable values as long as parameters do not
change.
If the conversion potential of the enthusiasts is not high enough the church will eventually
become extinct.
In the short term the number of enthusiasts can be become extinct as they increasingly fail to
reach a shrinking pool of unbelievers. This occurs before all the unbelievers get converted.
The church may rise to a high percentage before the effects of reversion cause it to stabilise
at a lower value. Thus success in the church may initially mask the long term damage that re-
version will cause.
4.2 Further Work
The models need to be applied to more situations involving churches and others subcultures.
Applications to the growth of drug use may be particularly relevant
In particular the limited enthusiasm model needs to be investigated for non-religious subcul-
tures. Recruitment to such groups as vegetarians is passive rather than active, i.e. by people
observing their lifestyle In this case “conversion” will not depend on a believer's enthusiasm.
Thus the limited enthusiasm model may not be appropriate.
In the case of lifestyle and youth subcultures the effect of advertising, which does not depend
on contact, needs to be built in.
A mechanism is needed to allow inactive believers to be “re-infected” by active believers and
thus have their enthusiasm restored.
5 Acknowledgements
All graphs, diagrams and simulations were produced with “Stella” software produced by High
Performance Systems Inc. The author would like to thank the University of Glamorgan for sup-
porting this research.
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