Tuesday, 2 June 2015

Limits to Church Growth - Part 2

The Reproduction of Enthusiasts

In a previous post I investigated two barriers to church growth: Lack of supply of religion by the church; and lack of demand for religion in society [1]. For those barriers to be removed a church needs to take action to increase recruitment and not take demand by the population for granted. In particular, the church needs to create demand for religion, “to seek and save the lost”, to engage with the population and convince those who don’t want the religion to embrace it.

In this blog a model is presented where the church creates demand by contacting unbelievers and persuading some to accept Christ, and thus become Christians. This is the limited enthusiasm model, which has been published and tested with data a number of times [2]. Here, it is re-interpreted in terms of supply and demand.  However creating demand brings about another barrier to church growth, one connected with the behaviour of the Church members involved in demand creation, the enthusiasts of the church.

Limit 3: Lack of Enthusiasts

Enthusiasts are the name given to those in the church who actively engage in spreading the gospel, in particular in recruitment to the church. These enthusiasts are the ones “supplying religion” to society by “creating demand” through their persuasive actions within that society. As such, unlike the previous models [1], supply matches demand. However not all the church are such enthusiasts, thus supply is proportional to the number of enthusiasts, rather than the whole church.

The reason not all the church are enthusiasts is that few Christians remain enthusiasts throughout their lifetime. The most effective enthusiasts are new converts, as these have the most contacts amongst unbelievers. After a time they settle in to church life and exchange their old unconverted friends for new converted ones. As such many eventually cease to be enthusiasts. Even if they keep unconverted friends, those friends get used to the new religious ways of the convert, whose witness becomes less effective.

Added to that, some enthusiasts also lose enthusiasm for the faith; perhaps the novelty has worn off, or the new religion has not met their expectations. Yet again it means that enthusiasts do not remain so for ever.

The model is expressed in system dynamics form in figure 1 [3]. Feedback loop R represents the action of the enthusiasts supplying the church’s beliefs by creating demand in society. Loop B1 is the reaction of society, the extent to which society really demands religion. B2 represents the fraction of enthusiasts who become inactive over time. The model is similar to that of the spread of a disease with the enthusiasts the “infected” Christians.

Figure 1: Limited Enthusiasm Model
For supply and demand to match, assuming contacts between enthusiasts and society are uniformly mixed, then the enthusiasts become less effective as the number outside church falls, making potential converts harder to find.  Thus the loss of enthusiasts creates a barrier to growth as that loss eventually exceeds the ability of enthusiasts to reproduce themselves from the diminishing pool of unbelievers. The result is that not all outside church are converted, and church reaches a limit much less than the size of the population.

Let a church number 60 people initially, with 5 of them enthusiasts. Let the number outside church be 2000.  The results are in figure 2. The enthusiasts peak about time 20 (curve 3), after which church growth slows (curve 1). Church stops growing at 540, because it has run out of enthusiasts, with over 1,500 in society remaining unconverted, curve 2. The inability for enthusiasts to reproduce themselves is a barrier to church growth.

Figure 2: Church Growth Limited by Behaviour of Enthusiasts
The results in figure 2 illustrate the epidemic metaphor, and are a pattern seen in a range of social phenomena, such as the spread of protests, ideologies, rumours, languages or fashions [4]. If a contagion, physical or social, spreads evenly, i.e. without targeting the susceptibles, and with a fixed contagion strength, then growth will always be limited. The enthusiasts will end up at zero.

If people leave the church, and in the future become open to joining again, then it is possible for enthusiasts to remain non-zero, but the barrier to growth remains. The church still can’t reproduce enthusiasts fast enough.

Removing the Growth Barrier

To remove, or raise, the barrier to growth a number of options are open to the church:
  1. The enthusiasts could selectively target those who are outside the church, spending more time with them rather than with church members. Practically this may prove difficult as time does need to be spent on believers in a growing church for their nurture and retention. This gives enthusiasts limited time for participation in non-church groups. Also, actually making contact with those outside church can be a problem if there are groups of people hostile to religion, and whose lives never overlap with anything church related. Whatever the strategy, as church grows, the people remaining outside become increasingly harder to reach.
  2. The enthusiasts should also reproduce themselves from inactive church members, not just new converts. This can be done by training, raising expectations of church membership, and especially by renewal of Christians in the Holy Spirit. This approach raises the barrier, but there is still a limit to church growth, albeit higher [5].
  3. Enable Enthusiasts to remain enthusiastic for longer. There could be many strategies, but if the church could just appreciate and encourage enthusiasts, rather than putting them down then ….. Enough said!
  4. Widen the pool of susceptibles by extending the church’s influence into another area, perhaps a through a church plant. This strategy quickly generates more demand, a new epidemic, and keeps up the flow of new enthusiasts. This is a standard strategy for newer churches, those yet to be established and become widespread. Unfortunately the unconverted people in the original community remain unconverted. Not ideal as the church is meant to seek and save the lost, not just grow!
  5. The enthusiasts should seek to increase their effectiveness, so the “contagion” is no longer fixed but can become larger. This is not so much about training enthusiasts but increasing their spiritual life. More life gives more growth, and thus gives even more life. This is the stuff of revival, outpourings of the Holy Spirit, and will be examined in the next blog on limits to church growth.



[1] See Blog, Limits to Church Growth, part 1

[2] See Hayward (1999, 2000, 2002, 2010) at 

[3]  For the conventional epidemiological construction of the Limited Enthusiasm Model see the above papers and

[4]  See examples among the publications at

[5] This is explored in the Renewal Model http://www.churchmodel.org.uk/Renmodel.html

Monday, 27 April 2015

Institutionalism and Church Decline

In a recent post, Gillan Scott, deputy editor of the Archbishop Cranmer blog, suggested that the Church of England might be more interested in managing decline than engaging in mission [1]. He quotes Peter Broadbent, Bishop of Willesden:
“Unfortunately, there are bishops around the place who think: ‘Well actually, we’ve just got to cater for this ongoing decline in our Church.’ And I worry about that.”
Gillan goes on to illustrate the point with his own growing church, which has been unable to appoint a much-needed assistant because of the shortage of clergy elsewhere in the diocese. Put simply, the needs of the institution have priority over the needs of an individual congregation. The implication is that institutionalism is a barrier to church growth and is a contributor to church decline. Gillan states:
But often it would appear that those churches which are growing are doing so despite rather than because of the structures and hierarchy.
Institutionalism happens when the organisation has grown to the point that it must maintain its “structures and hierarchy”, and the role of its individual parts is to service that maintenance. It stifles local innovation and thus limits growth. If decline sets in, then the organisation cannot produce sufficient new growth to recover.

Organisational Lifecycle

Institutionalism is an example of a stage in a lifecycle that can affect any organisation, religious, political, corporate, or cultural; a church, a company, or even a nation. Adizes [2, p.103] spells out the corporate lifecycle, to summarise: Courtship, Infant, Adolescent, Prime, Aristocracy, Bureaucracy, and Death. It is at the aristocracy stage where institutionalism sets in through internal politics [2, p.106], despite, or perhaps because of its success. Formality replaces informality, money is spent on control rather than sales and innovation, and the emphasis moves on to how things are done, rather than why [2, p.64].

The organisational lifecycle has been applied to individual congregations [3]. McIntosh gives the stages as: Emerging, Growing, Consolidating, Declining, and Dying [3]. By the declining stage the purpose of the church has been forgotten because of the work needed to keep afloat what they have left. Applying this lifecycle to a denomination, like the Church of England, it can be seen that most in the UK are now institutions somewhere between the declining and dying stage.

A Model of Decline By Institutionalism

Because of the generic pattern of an organisational lifecycle, the situation is ideal for a system dynamics explanation. System dynamics is a modelling methodology that links behaviour to cause and effect. To keep things as simple as possible, just consider two variables: Church, the number of people in the denomination; and Institutionalism, the collection of variables that indicate the church’s emphasis on the wider corporate needs, rather than the local work where growth takes place [4]. Institutionalism is an example of a soft variable, one that is hard to measure, but whose meaning is generally understood [5].

Consider 3 hypotheses:
1.              The more people in church, the more are added to the church. This is feedback loop R1,  figure 1.
2.              The more people in church, the more leave, feedback loop B1, figure 1.
3.              The  more people in church, the more it becomes institutionalised, thus the less are added to the church, loop B2, figure 1.
Figure 1: Causal Loop Diagram of Church and Institutional Growth

The feedback loops are causally circular, with the effect “feeding back” to change the original cause. R1 is a reinforcing loop, a virtuous cycle, giving exponential church growth. B2 is a balancing loop, limiting the growth of the church due to rising institutionalism. Lay people and clergy move from innovative agents of evangelism to people whose role is merely to “turn up, pay up and shut up”, servicing the institutional needs. B1 is also balancing, reducing the size of the church.

R1 can be thought of as a positive force, with B1 and B2 as negative forces opposing it. The future size of the church depends on which force “wins”.

In order to examine the outcome of the hypotheses, a system dynamics model is required. This will enable computer simulation to illustrate the models behaviour. Readers who prefer to avoid technical details can skip the next section!

System Dynamics Model

The model is given in figure 2. There is one stock for the church, where both R1 and B1 come from connections to its flows. Institutionalism is also a stock, the loop B2 being formed, through the flows: growth of institutionalism, and add to church.
Figure 2: System Dynamics Model of Church and Institutional Growth
Three further hypotheses are needed. B3 is a resistive force that represents the difficulty of increasing institutionalism when it gets near the organisational capacity. R2 is a positive force coming from people within the church who want to increase denominational institutionalism, perhaps for their own self-preservation or power. B4 represents attempts to reduce institutionalism.

Model Results

Assume there are no attempts to reduce institutionalism. Instead it is allowed to grow to capacity, the situation that perhaps represents many denominations.  A new denomination grows rapidly, through R1, for around 50-70 years figure 3, curve 1. Institutionalism also grows although its growth is delayed compared with that of the church, figure 3, curve 2.
Figure 3: Results of Church and Institutional Growth Model
 Around the 70 year mark church growth slows and reaches a peak because the effects of institutionalism, B2, have slowed the growth, allowing additions to the church to just about balance the number leaving (which includes death), figure 4. After that point the losses exceed the additions and the church denomination continues to decline until it is eventually extinct, figure 3. In reality the extinction is faster, a straight line rather than the slowing curve of figure 1, as aging becomes a factor. This extinction has come about because institutionalism has been allowed to saturate at capacity, and no attempt has been made to reduce it.

Figure 4: Comparison of Additions to, and Losses from, Church

Note that extinction has resulted regardless of the size of the target population (unlimited in this model). Extinction in this model is due to a lack of supply, not alack of demand.

As this model is for illustrative purposes only, the values of timescales and the church size should not be taken literally. They are merely relative. They may have different values depending on the denomination, or their social setting. It is in the shape of the curves that the model illustrates reality [4].

Combating Institutionalism

A policy is introduced whereby the church attempts to reduce institutionalism (B4) in proportion to the amount of net decline it experiences. That is, the policy is not enacted until decline takes place. The policy is allowed an average of a 30 year delay to take full effect; a high number because it needs to effect most of the denomination, not just a few parts [6].

One such result shows church decline slowing from about 110 years, but it is insufficient to bring about growth, figure 5, curve 1. The oscillation in institutionalism is due to the delay between policy implementation and effect. Once it is perceived to have some effect on reducing decline, the policy backs off, before it has time to have full effect.

Figure 5: Attempt to Reduce Institutionalism in Proportion to Net Church Losses

 Of course the effect of the policy on halting decline depends on its effectiveness in dealing with institutional resistance. Figure 6 shows the base case of no such policy, curve 1, compared with 3 policies of differing effectiveness, curves 2-4. It shows that it is possible for a declining church to get back to growth. However the policy should be continue to be applied with the same intensity, and not applied less just because numbers recover a bit, as in these simulations. Complacency in results will breed oscillations, instability and eventually decline, as many companies know to their cost.

Figure 6: Comparison of Attempts to Reduce Institutionalism (2-4), with No Attempt (1).

Denominational Decline

What does actual denominational decline look like? The Church of England does not have consistent membership figures over a long period of time, and it is only in the last few decades that attendance has been measured. However the Methodist Church has good membership statistics over most of its lifetime [7]. The graph is shown in figure 7.

Figure 7: Membership of Methodist Church of Great Britain and Northern Ireland
Some of the growth from 1760 to 1900 in figure 7 was population growth, but the bulk of it was due to conversion. The length of the growth phase indicates that the Methodist church successfully dealt with issues of institutionalism during its early stages, especially in the transition from the first generation of leaders. Note a split in the 1850s and the effect of revival in 1904-5, which contrary to popular belief had considerable impact in England as well as Wales.

However from 1900 onwards the numbers plateaued, and then fell from the Second World War onwards, with a blip in the 1950s, probably due to Billy Graham crusades. Comparing figure 7 with figure 3 it is clear that if institutionalism is one of the causes of Methodist decline, then it has not been tackled, and extinction is not far away.

This is not just a Methodist issue. Statistics indicate that most pre-20th century denominations will be extinct by 2050, except the Church of England, whose decline is slower, and the Baptists, who are independently organised. By the middle of the century the Christian landscape will be dominated by what are now Pentecostal and independent churches, who may well have changed and have their own issues with institutionalism by then.

Such decline is not just a church phenomenon. By 2050 the inability of most Western societies to deal with their huge debts may have led to their downsizing, (euphemism for becoming poorer!) And all nations will be hit by dwindling natural resources and climate change, with a likely decline in world population. The lack of a few Christian denominations may be the least of the problems faced by people in the middle of the century!

Tackling Institutionalism and Recovering Growth

Can institutionalism in denominations be tackled and growth recovered? Given countries’ inability to deal with their debts, energy needs, and climate change – always too little too late – it does not bode well. Church is an even more sluggish institution!

Gillan Scott suggests that the battle for the church’s very existence, its numerical survival, is more important than its current struggle on how it deals with LGBT issues [1]. As a “gay-affirming” ideology continues to spread in the West, the church has become dominated by debates on introducing same-sex marriage, and falling into line with government policy, rather than how it can avoid decline, extinction and see growth. Which issue is more important? Perhaps the two issues are connected.

The policy to introduce same-sex marriage in the church could be construed as a force for institutionalism, as it assists the desire of a denomination to remain relevant to society. I do not know about the Church of England, but in the Church in Wales the policy is being driven by the denominational leadership, and those it employs [8]; the ones with the most to lose if the denomination becomes irrelevant, and the least to lose if the revised marriage policy is introduced. Introducing same-sex marriage seems a classic case of a policy designed to service the needs of an institution, rather than help the individual and congregational agents of growth.  As such, if introduced, it will be a force to maintain the institutionalism that is resisting the church’s attempt to avoid extinction, part of feedback loop R2, figure 2 [9].

What could be a way forward for denominations with institutionalism issues?

If we accept that most historic denominations are heading for extinction in their current form [10], then, rather than make minor changes to that form, perhaps it is better to discontinue the form altogether. That is, policies are needed to deregulate how congregations operate. Let a denomination divide up into smaller groupings with different beliefs of liberal and conservative persuasions. Allow congregations to join the group they identify with best, or go independent.

Allow congregations to pay for their own ministers and not have to send money into a central pot. Let them keep all their income, so that if successful they can invest in their work, or that of their chosen associates. Let congregations choose ministers from outside denominational ranks, and adapt their operational management and clergy structures. What is left of central denominations can provide support services, pensions, advice etc, on a consultancy basis.

Such deregulated denominations would allow spiritual renewal to flourish with less hinderance, with healthy competition driving up standards. Enthusiasts would be generated, conversions would follow. This I think would give the best chance for avoiding extinction and encourage church growth. It would probably look a bit like the early church.

References & Notes 

Scott G. (2015). Church of England Mistakes Mission for the Management of Decline. Archbishop Cranmer Blog, 23/04/15.

The blog refers to:
Davies M. (2015). Church growth: Bishop Broadbent rounds on the critics of Reform and Renewal. Church Times, 21/4/15.

Adizes, I. (1992). Corporate Lifecycles: How and why corporations grow and die and what to do about it. NYIF.

Arn, W. (1985). Five Stages in the Life-Cycle of Churches, Pasadena, CA: American Church Growth.
Davies, G. Understanding Parish Growth Stages, Diocese of Sydney.
McIntosh G.L. (2009). Taking Your Church To The Next Level, Baker Books.
Saarinen, M.F. (2001). The Life Cycle of a Congregation, MD:Alban Institute.

This is an example of a metaphorical model, one whose purpose is to provide transferable insight, rather than exactly replicate a specific situation. See Morecroft J. (2007). Strategic Modelling and Business Dynamics. Wiley. P.414.

Hayward J., Jeffs R.A., Howells L. & Evans K.S. (2014). Model Building with Soft Variables: A Case Study on Riots. (2014). 32nd International Conference of the System Dynamics Society, Delft, Netherlands, July 2014.

For an example of a delay between introducing a policy and its effect, consider charismatic renewal. It started in the early 1960s, but it was not until the 1990s did the cultural change it introduced become widespread in the church. The Alpha Course, the movement’s most influential tool, came about in the 1990s. Most of the church has still not embraced the cultural and spiritual change and probably never will.

Data for 1767-1970 is taken from: Currie R. Gilbert A.D. & Horsley L. (1977). Churches and Churchgoers: Patterns of Church Growth in the British Isles since 1700, table A3. Before the formation of the Methodist Church of GB and NI the different church streams have been added together.
Data for 1980-1990  is taken from Brierley P. (1999). Religious Trends 2000/2001 No.2,  table 9.10.2 Christian Research.

Data for 2000, 2010, and estimates of 2015, 2020 are taken from Brierley P. (2014). UK Church Statistics 2015-2020, Brierley Consultancy.

The current consultation on same-sex marriages in the Church in Wales comes from its governing body and Bench of Bishops. There has been no movement of lay people or clergy calling for change; no protests at the current status quo; no congregational petitions to the leadership; no emergence of prayer-groups praying for change of the definition of marriage. The call for change is top-down, not bottom-up, suggesting it is driven by institutional needs, not congregational or individual. 

It could be argued that introducing same-sex marriage in church would attract more people because of the marriages, and the church’s increased relevance to society. These hypotheses are not in the model. They could be added by allowing the church to draw from a limited pool of people, rather than the unlimited pool in figure 2. The pool could then be divided into people who would favour the policy, the ambivalent, and those opposed. In addition, the effect of the policy on church leaving rates would need to be added. Model calibration would be difficult. My conversations with researchers in the USA denominations that have introduced same-sex blessings and marriages is that the effect on people leaving is larger than that of people joining, and that neither were major factors in the denominations’ decision to implement the policies.

It is denominationalism that is heading for extinction in the UK, not Christianity, and not all congregations currently part of historic denominations. Some will survive and grow.

Friday, 3 April 2015

Nagaland Revival 1976-1982

Nagaland, an Indian state bordering Burma, saw dramatic growth in the number of Christians during the 20th century [1]. Composed of 16 separate tribes with different languages, the Naga people use English as their predominant language, which has undoubtedly helped the spread of Christianity.  The growth of Christianity among 13 of the tribes has been documented by Paul Hattaway [1], showing substantial church growth through the 1950s and 1960s, with a massive increase in growth from 1976. Reading the narrative [1] there had been a series of evangelistic campaigns with sporadic and regional revival for 20 years, but then from 1976 there was a nationwide outpouring of the Holy Spirit, a national revival. By 1980 the church had increased by 55% in 4 years; from 26% of the Christian population  in 1976, to 41% of the population in 1980 [2].

The Naga churches were largely Baptist, having been founded by such missionaries earlier in the 20th century. The Baptist church membership data for the period was obtained by Edwin Orr [3]:

Year        1975     1976     1977     1978     1979     1980     1982
Membership  132,495; 134,682; 162,000; 179,047; 188,779; 209,382; 220,617

At the time of the revival the Naga state was largely closed to outsiders due to tensions with the Indian government. Thus the revival was a purely internal affair and ideal to compare with the Limited Enthusiasm model [4]. The starting date for the revival was1976 [1], where the church had experienced a moderate increase from the previous year.  Thus the comparison between data and model runs from 1976 to 1982, with the 1979 data excluded, as it is lower than expected [5].

The model simulation, given in figure 1, shows a good fit to all points (except 1979), with a reproduction potential of around 1.46. The enthusiastic period for the Nagaland revival is 6 months, which is 8 times larger than that of the 1904/5 Welsh revival, which explains why the revival in Nagaland lasted so much longer than in Wales.  This longer enthusiastic period may well be due to a lower population density in Nagaland, compared with Wales, with the associated longer journey and communication times. 

Figure 1: Church Membership Nagaland and Limited Enthusiasm Model

The huge impact of the revival on the growth of the church can be seen by the reduction in the number of unbelievers, figure 2.  The graph also shows an unusual nature of this revival, that in 1976 the church started with a large number of enthusiasts, who reached their peak in 1977 (figure 2), even though most of revival, and the growth, was after that date. Thus the revival appeared to be different to the normal pattern, where revivals start with only a few enthusiasts and take time to build. In this sense the enthusiasm was skewed towards the beginning of the revival, with much of the converts due to that initial enthusiastic momentum.

Figure 2: Unbelievers, Enthusiasts & Inactive Believers, Nagaland, from Limited Enthusiasm Model

 The skewed nature of the revival is also seen in figure 3, which shows the reproduction potential above the threshold for only one year after the start of the revival (indicated by the arrow) [6]. Once a sufficient number of unbelievers were converted the threshold rose above the reproduction potential and the revival growth slowed. 

Figure 3: Reproduction Potential of Enthusiasts Compared with Revival Threshold

An attempt was made to fit the Limited Enthusiasm model to the data from 1975 in order to include a slow build up of numbers, but no satisfactory fit could be found. That leaves the question as to the origin of the large number of enthusiasts in 1976, given there had been little church growth before that year. There are at least three possibilities:

 1.   Between 1975 and 1976 a small number of enthusiasts made many new enthusiasts from the inactive believers - the renewal process;

 2.  Between those dates the spiritual life of the enthusiasts was building, raising the reproduction potential until it tipped the church into revival and produced a large number of enthusiasts;

 3.  There was a sudden mass baptism with the Holy Spirit of many believers between 1975 and 1976 - a Nagaland Pentecost.

All three theories could be tested by extensions to the limited enthusiasm model. However the reality may go back further than 1975. Hattaway notes [1, p.84] that the revivals of the 1960s had left the churches with a lack of trained leadership for the host of new converts. Thus it is possible that growth was being stifled for organisational reasons, rather than a lack of the Holy Spirit. They could not disciple converts fast enough, or retain them in church.

However in 1972 the evangelist Billy Graham visited Nagaland and was responsible for many leaders dedicating themselves to Christian service and to being appropriately trained. Over 100,000 had gathered to hear him preach in the capital, Kohima. The following 4 years saw an increase in prayer meetings for revival with “tremendous demonstrations of God’s power” [1, p.85]. It is likely that by 1976 the new leaders were both spiritually and practically prepared to cope with revival and were a crucial part of the large number of enthusiasts at its beginning. 

In that case it can be concluded that Billy Graham was a key catalyst in the huge Nagaland outpouring of 1976-1982, even though he was not there during the period! But his preparatory work was critical. Revivals come from the work of the Holy Spirit and good preparation.

References & Notes

[1] Hattaway P. (2006), From Head-Hunters to Church Planters, Piquant.

[2] The population here refers to the nominal Christian population, about 75% of the total population. The remainder of the total population were tribal religions, Muslims, Hindus etc. The 1970s revival was a phenomenon within the Christian community. The previous decades saw Christian growth at the expense of the indigenous tribal religions, but had left many with the name Christian but without the commitment and church attendance.

[3] Orr J. Edwin (2000) The Outpouring of the Spirit in Revival and Awakening and its Issue in Church Growth. Church Growth Association. Available from Church Growth Modelling.

[4] Hayward J. (1999). Mathematical Modeling of Church Growth, Journal of Mathematical Sociology, 23(4), 255–292.

[5] There are at least three possibilities for the discrepancy in 1979.

Firstly, the reporting of church membership data can be delayed in a particular year if there were some mitigating circumstances. Given the ongoing conflict between the Indian government and the Naga state, and the difficulty of travel in the country, it easy to see how this can happen. If the 1979 figure had been genuinely low then the pattern would have carried on until 1980. However the 1980 figure follows from 1976-1978.

Secondly, there could be two phases to the revival; one ending in 1979, and another starting in 1980. This is not mentioned in Hattaway’s book [1], but that does not mean it did not happen.

Thirdly, the delays in the geographical spread of the revival could have hindered growth in one year. The Limited Enthusiasm church growth model, like the SIR epidemic model, assumes homogenous mixing. The Naga terrain, and varied ethnicity, is anything but homogeneous!

[6] The reproduction potential measures the strength of the revival. It is the number of enthusiasts, not just converts, that one enthusiast could potentially make during the whole of their enthusiastic period. The Limited Enthusiasm model shows that revival growth takes place if the reproduction potential exceeds a threshold. This threshold increases as the pool of potential converts, the unbelievers, gets smaller. Thus revivals end before everyone is converted.

Thursday, 29 January 2015

Limits to Church Growth - Part 1

All churches want to grow. It is part of the mission of the church to spread the faith, to make converts, to see new people come to Christ. For that mission to be fulfilled then either congregations become larger, or new congregations get planted. Thus growth is a legitimate expectation of any church.

However it is the experience of many growing churches that the larger they get, the harder it becomes to maintain that growth. There is a limit to church growth, especially in a congregation. These limits are barriers that stop a church from growing indefinitely and hindering its attempts to fulfil the commission that Christ gave.

There are many reasons why church growth is limited, and there may be more than one present in any given church. Whatever the reasons, each one will require the church to do something to remove the barrier so that growth can continue. It is not a sin to plateau. But failing to remove an obvious limit that causes the plateau is an issue. God does not expect us to sit back and let things happen, he gives instructions, e.g. “Go into all the world and preach the gospel to every creature” Mark 16:15. Christians are not passive in church growth!

This blog, one of a series, explores two simple limits to church growth using the system dynamics methodology. The two limits are connected with supply and demand. The church supplies religion, society demands it.

Limit 1: Lack of Supply

If a church relies on new people coming to church by their own choice, and puts no effort into seeking new people, then it will eventually stop growing. It is assumed that there is some demand for religion in society, but the church does nothing to meet people and supply what they are looking for. Instead it relies on people searching the church out.

Assuming there are a constant number of new people coming to church every year, i.e. demand for religion remains constant, why then does the church stop growing? The answer is that growth is not the only process taking place – people are leaving as well. Some people move for jobs, some people fall away. Others of course die. These cause the growth to be limited.

Figure 1: Constant Demand Model of Church Growth

A simple system dynamics model will illustrate this effect, figure 1. The stock, Church, measures how many people belong to the church. The inflow, come to church, is the fixed number who come to church each year, presumably because they are seeking the Christian religion. Assume it is 5 people per year.

The outflow, leave church, represents the people who leave each year. It is not constant, but depends on the number in the church. The connector from church to leave church indicates this. The reasons people leave are personal, i.e. individual. Thus if there is an average chance of someone leaving, then the more people, the more will leave. So assume a fixed percentage leave every year, say 5%. This effect is indicated by the feedback loop B in figure 1, where B means balancing. The bigger the church the more people leave, thus the less in church. Changes in the church feed back on itself, and in this case oppose growth.

Figure 2 shows what happens to a church of 60 people. It grows over the following 50 years, but the growth slows because as it gets bigger the number of people who leave is larger, figure 3. It will stop growing at 100 people because 5% of a 100 is 5; the outflow matches the inflow!

Figure 2: Limit to Church Growth Caused by Constant Demand and Rising Number Leaving
Figure 3: Demand, "come to church", Matches Leaving at the Limit

It is like driving a car and taking your foot off the accelerator. The car comes to a halt, slowing down through air resistance and friction. The situation with the church is the same; the “force” due to people leaving slows church growth down until it stops.

To remove the barrier the church could reduce the losses, but with death and geographical mobility among the reasons for leaving then there is only so much that can be done here. The key to dealing with the limit cause by lack of supply is to do something to increase recruitment. The principle must be the bigger the church, the bigger the effort in recruitment. More people in the community are met, thus greater sharing of the gospel, more persuasion. Don’t take demand for religion for granted.

From a system dynamics point of view the balancing feedback loop B needs opposing. Somehow a form of feedback that reinforces growth must be introduced.

Limit 2 Lack of Demand

Assume that the church supplies religion to the population according to its size. In plain English, the larger the church, the more people go into the community advertising the faith, actively finding those who demand religion, “seeking and saving the lost”, Luke 19:10. This is indicated by the feedback loop R in figure 4. It is a “force” exerted by the church on those outside, causing some to join. It is called R for reinforcing. The bigger the church, the more join – growth accelerates. The supply rate is the percentage of the church who recruit one person each year, given there is sufficient demand. Let the supply rate be 8%.
Figure 4: Supply and Demand Model of Church Growth

However there is not a constant demand for religion. Society is finite, and the number outside the church is a limited pool on which to draw, the stock Outside Church, figure 4. Instead, any demand for religion is a fixed percentage, demand rate, because as the pool of those outside goes down, less people will remain who demand church. Let the demand rate be 0.5%. Thus there is another force opposing growth, the loop B1.

Assume that those outside the church are 2000 and the leaving rate stays the same, now called loop B2 and still opposing growth. Thus a church that starts with 50 people (2.4% of society), grows initially, figure 5, as demand exceeds supply, figure 6. There are plenty of people demanding religion outside the church, more than the church can recruit each year, thus the church’s supply, R, controls the growth.
Figure 5: Limit to Church Growth Caused by Failure to Increase Demand

However at some point supply exceeds demand, figure 6. As hard as the church tries, there are no longer sufficient people demanding religion for them to find in the random pool of people they contact each year. Now demand, B1, controls the growth, and eventually equals the leaving rate where growth stops, around 160 in this case, about 7.8% of society.

Figure 6: Demand Eventually Falls Below Supply, Limiting Growth

Incidentally, if those who leave the church are not open to rejoining, then the church will transition to decline. Only if some ex-members are recycled can the church keep its numbers steady.

To remove the barrier it must first be noticed there is a serious flaw in one  of the model assumptions above. “Seeking and saving the lost” does not refer to only gathering in those who demand religion, those open to becoming Christian, those interested in church. The “lost” in that gospel passage means everyone, and seeking and saving them means persuading everyone to accept Christ, even if they are disinterested or hostile. It is convincing them that they should demand the religion even if they don’t at present.

Thus to remove this barrier the church should be creating demand; not just advertising the faith, but witnessing their faith – doing things that cause people to change their mind. Church members must have a quality of life that makes others want the same. It can be evangelism, ministries of compassion and mercy, whatever – as long as those outside the church realise they are missing something they don’t have, never thought of having, or even hated, and that church is the place they find it. The demand rate cannot be left constant; it is the church’s job to increase it.

The next blog on limits to church growth will explore how the church may influence the demand, and the further barriers to growth these uncover.  Limits to Church Growth - Part 2

Thursday, 20 November 2014

The Eyam Plague of 1666: A System Dynamics Model

At present the news is full of the rather frightening Ebola outbreaks in West Africa. Other infectious diseases also get into the news often, for example AIDS/HIV and flu. What is less well know is that such infectious diseases spread according to fairly precise mathematical rules. This follows from the person-to-person contact involved in the spread of the disease.

It is this process that the limited enthusiasm hypothesis of the church growth models is based on. In that case the “disease” is faith, and it is spread by word of mouth contact. It is not only churches that grow this way, the same epidemiological mechanism has been used to model the spread of languages, scientific ideas, riot behaviour, bulimia, cigarette smoking and even Facebook [1].

To illustrate how this principle works I want to use a standard case study in mathematical epidemiology: the spread of the plague in the Derbyshire village of Eyam in 1666.

Eyam Plague

The years 1665-1666 saw the great plague hit England, notably London, which lost about 15% of its population [2]. This was the last epidemic of bubonic plague in the UK, a disease that had been an ongoing problem since the days of the Black Death in the 14th century. The primary mechanism of spread of the disease is through the bite of an infected black rat flea. However once established the disease can spread person-to-person, which gave rise to the popular rhyme “Ring a ring of roses, a pocket full of posies, atishoo, atishoo, we all fall down” [3].

Although largely confined to London, an outbreak occurred in the Derbyshire village of Eyam due to a person acquiring the disease from a piece of infected cloth sent from London in 1665. Once the Eyam outbreak took hold in 1666 the local clergyman took the precaution of isolating the village, as best he could, to prevent the spread of the disease. This action of his made it an ideal case study to mathematically model the spread of a disease, as migration could be ruled out as a major mechanism in its spread.

Such a mathematical model of the plague was carried out by GF Raggett [4] using methods based on differential equations. Raggett explained why the spread of the disease in Eyam must have been largely person-to-person rather than rat fleas, as over the period of a year infected rats would have left the area and infected the wider area. No such cases occurred. Using mathematics Raggett then showed how the model predicted the number of deaths due to the epidemic, and demonstrated some important results [5].

What follows is a system dynamics version of Raggett’s work to help explain how a disease spreads without using mathematics. The model is often called the SIR model, after the symbols in the equations, the epidemic model, or the Kermack McKendrick model, after the first people who published it [6].

System Dynamics Model

The model assumes the population of people are split into 3 categories of people: the Susceptibles, who could potentially catch the disease; the Infected, who are carrying the disease and could infect others; and the Removed who have had the disease and cannot catch it again, either because they are cured and immune, or have died. The letters SIR stand for these three categories.

In system dynamics this model can be expressed as stocks and flows:

Figure 1
The removed category has been renamed Deceased as most cases of bubonic plague ended in death.

There are two processes (called flows) involved: catch the disease, which moves susceptibles into infected; and deaths, which moves infected into deceased.

The catch disease process is subject to two social forces: R1 and B1. R1 causes the increase in the number of infected to accelerate as more infected gives more new cases each day, thus more infected. This is called reinforcing feedback and is the first phase of growth in the infected, (figure 2).

Figure 2

In addition the force B1 slows that growth as the pool of susceptibles is depleted, making it harder for infected people to make new cases. This slowing force is balancing feedback and opposes the force R1. B1 eventually dominates over R1, the second phase of growth (figure 2) [7].

Eventually the number who catch the disease drops below the deaths and B1 now causes the infected to decline faster and faster, the first phase of decline (figure 2).

The deaths process is subject the social force B2 as the more infected there are the more die, thus depleting their numbers. This force only dominates at the end causing the decrease in infected to slow down, the second phase of decline, (figure 2).

Raggett [4] showed from the recorded deaths that when the main period of the plague epidemic started, June 19th 1666, there were 7 people infected. The population was known to be 261 at that time. By the end of the epidemic, in the middle of October that year, only 83 people had survived.

From these figures, and knowing the infectious period of the plague is about 11 days, it is possible to simulate the system dynamics model, and compare it with the data for cumulative deaths (green curve, figure 3) [8]. Comparing Deceased with recorded deaths shows a good fit. It is remarkable that something that involves people, and random behaviour, gives such predictable results. This predictability is what allows modern day epidemics to be so successfully tackled, and the consequences of not tacking action computed [9].

Figure 3

Note the following:

1. The epidemic burns out before everyone gets the disease. There are susceptibles remaining at the end of the epidemic (black curve figure 3).

2. At the peak of the epidemic, where the number of daily cases is at a maximum (about 45 days, figure 3), even though the daily death rate is slowing down the epidemic is not at an end and a significant number of deaths are still to come, (green curve figure 3).

3. At any one time the number of infected people is quite small compared with the population (blue curve, figure 2 and figure 3). It is their cumulative number over time that is large. It does not take many infected people at a given time to keep an epidemic going [10].

Reproductive Ratio

The strength of an epidemic is measured by the reproductive ratio, called R0. At its simplest it is the number of people one infected person could potentially infect during their infectious period, if the whole population were susceptible [11]. The number has to be bigger than one for an epidemic to happen. The larger this number then the bigger the epidemic becomes. Different diseases have different reproductive ratios [12].

Using the numbers above, the reproductive ratio comes out at about R0 = 1.6 [13]. This is much less than highly infectious diseases such as Measles (range 12-18) and Smallpox (5-7) [14]. Nevertheless 1.6 was still large enough for well over half the population of Eyam to get the disease. A value of R0 of 1.6 is similar to Ebola (1.5-2.5). However because Bubonic Plague is spread through fleas, and through the air, it is harder to take action to reduce R0 compared with Ebola, which is only spread through contact with bodily fluids.


What turned out to be an ideal case study to test a mathematical model for the spread of a disease turned out to be a tragedy for the people of Eyam. The majority of the population died, including the wife of the brave clergyman who isolated the village and performed all the burials [15]. However his action saved many more lives of people in the region, and the lessons learned, which mathematicians can now explore, gives confidence to models that have given strategies to combat epidemics and save millions of people. That studies of this sort can help understand social diffusion processes such as religion is a bonus.

References & Notes

[1] For a selection of social modelling papers that use the epidemic/disease analogy see references at:

[3]  For a history of the Great Plague of London see Wikipedia
  http://en.wikipedia.org/wiki/Great_Plague_of_London and the references contained within.

[4]   Raggett, Graham F. "Modelling the Eyam plague." Bull. Inst. Math. and its Applic 18, no. 221-226 (1982): 530. http://math.unm.edu/~sulsky/mathcamp/Eyam.pdf
Note Raggett used the burial records to estimate deaths. There is a slight time delay between the two, but not enough to seriously affect his results.

[5] For a history of the Eyam plague see:
Wallis, Patrick. "A dreadful heritage: Interpreting epidemic disease at Eyam, 1666–2000." In History workshop journal, vol. 61, no. 1, pp. 31-56. Oxford University Press, 2006.

For the demography of the Eyam plague see:
Race, Philip. "Some further consideration of the plague in Eyam, 1665/6." Local population studies 54 (1995): 56-57.

[6] Kermack, William O., and Anderson G. McKendrick. "Contributions to the mathematical theory of epidemics. Part I." In Proc. R. Soc. A, vol. 115, no. 5, pp. 700-721. 1927.

[7] The structure of the feedback loops, or forces, on those who catch the disease can be broken down into a number of parts where it is assumed the populations are proportionally mixed. The connection between population density and the likelihood of contact requires further assumptions.

[8] The model was constructed and simulated in the Software Stella, available from ISEE Systems http://www.iseesystems.com/.

[9] Similar models for Ebola in West Africa, 2014, have already been constructed and are informing policies to reduce its impact. For example:

Meltzer, Martin I., Charisma Y. Atkins, Scott Santibanez, Barbara Knust, Brett W. Petersen, Elizabeth D. Ervin, Stuart T. Nichol, Inger K. Damon, and Michael L. Washington. "Estimating the future number of cases in the Ebola epidemic—Liberia and Sierra Leone, 2014–2015." MMWR Surveill Summ 63, no. suppl 3 (2014): 1-14.

Kiskowski, Maria. "Description of the Early Growth Dynamics of 2014 West Africa Ebola Epidemic." arXiv preprint arXiv:1410.5409 (2014).

Team, WHO Ebola Response. "Ebola virus disease in West Africa—the first 9 months of the epidemic and forward projections." N Engl J Med 371, no. 16 (2014): 1481-95.

[10] All these epidemiological principles are replicated in church growth, and other forms of social diffusion. Not all people in a population are converted. Substantial church growth still comes after the peak in the growth is over. At any one time there are very few infected, called enthusiasts, spreading the faith.

[11]  The reproductive ratio (or reproductive number)  is called the reproduction potential in church growth and measures how many people one enthusiast can potentially convert and make an enthusiast. Not all converts become enthusiasts.

[12] For most diseases the reproductive ratio is given as a range as its value can depend on population density and particular population behaviours. It is believed that Ebola in West Africa in 2014 started with a much higher than normal R0 due the particular burial practices used, allowing dead bodies to transmit the disease, thus extending the infectious period.

[13] A simple formula can be used to compute the reproductive ratio in terms of the population number, and initial and final number of susceptibles alone. This calculation was done in the software Mathcad:

For the computation of this formula for the reproductive ratio see:

Brauer, Fred. "Compartmental models in epidemiology." In Mathematical epidemiology, pp. 19-79. Springer Berlin Heidelberg, 2008.  http://quiz.math.yorku.ca/chap2.pdf

Brauer, Fred. "Compartmental models for epidemics." (2008). http://health.hprn.yorku.ca/epidemicnotes.pdf

There are numerous methods to compute the ratio, sometimes giving different answers, see [14] below. This is not an exact science.

Mathcad is available from http://www.ptc.com/product/mathcad

[14] Wikipedia and references within. http://en.wikipedia.org/wiki/Basic_reproduction_number

[15] There is a museum in Eyam where the visitor can learn about the outbreak. http://www.eyam-museum.org.uk/ Note that there had been cases and deaths in 1665 and early 1666 before the period of study used by Raggett starting June 19 1666. Thus the total deaths, and the original village size, are larger than used in Raggett’s study.