Wednesday, 30 November 2016

Infectious Church Growth

An “Agent-Based” View

The central hypothesis I use to model church growth is that religious belief spreads like an infectious disease. This principle is built in to the limited enthusiasm model of church growth. The church contains enthusiasts, who pass their faith on to unbelievers, who convert to the faith. Some of those new converts also become enthusiasts for their newfound faith. Eventually enthusiasm wanes, the convert runs out of people to positively influence; they cease to be enthusiasts, thus becoming inactive believers.

The essence of this model is expressed in the stocks and flows of system dynamics, figure 1, and it works well for periods of revival, and, with a few additions, sustained periods of growth or decline [1,2].

Figure 1: Outline of Limited Enthusiasm Model in System Dynamics

This type of model is called macroscopic, because people of the same type are treated as a single unit. The stock, “Enthusiasts”, stands for the total number of enthusiasts at any one time. What this type of model does not do is give a picture of events at the individual level. For this we need a microscopic model; one example of which is called an agent-based model.

Agent-Based Model

In an agent-based model each person is modelled individually – they are called agents. Every agent is capable of being in more than one state, and the hypotheses of the model determine how an agent changes their state. There are also hypotheses that describe how agents relate to each other – their network [3].

For the limited enthusiasm model of church growth, the agents are the people who can have one of three states, depending on whether they are: unbelievers, enthusiasts, or inactive believers. The simplest form of network is to use a rectangular grid so that each person, a mini square, relates to 8 neighbours. In figure 2, the green squares are unbelievers (U), the red squares are enthusiasts (E), and the blue squares are inactive believers (B).

Figure 2: Outline of Limited Enthusiasm Model in an Agent-Based Model 

The unbeliever at the centre of figure 2 can be potentially influenced by any of the 8 surrounding agents. Conversion may occur in a given time period if there is at least one enthusiast connected to the unbeliever. The more enthusiasts in that network, the more likely the conversion. Thus in figure 2, there is a 2 in 8 (= 25%) chance of the central unbeliever having a contact that may lead to conversion. The green turns red. That conversion is still not inevitable, but the longer an unbeliever has an enthusiast in their network, the more likely a conversion becomes.

The process of an enthusiast becoming inactive does not depend on neighbours. Instead, after a fixed period of time, there is a chance the enthusiast will cease to have influence. Thus the red turns blue.

The two preceding hypotheses are called transition rules. They are the algorithms that drive the model. The transitions green to red to blue are the individual level equivalent of the stock/flow diagram of figure 1.

A Simulation.

For a simulation I will use the agent-based simulation software NetLogo [4]. Let us start with a church of enthusiasts in one block, figure 3, plot 1. The world is the entire green square of 121 by 121 cells, 14641 agents.  Some of the initial enthusiasts have no contact with unbelievers, but the ones on the edge do. Thus as time progresses the church grows, plot 2, leaving a church of mainly inactive believers in its wake. These are inactive in conversion, though they may be active in other aspects of the church.

Figure 3: Four Snapshots From Agent-Based Version of the Limited Enthusiasm Model. Green Unbelievers; Red – Enthusiasts; Blue – Inactive Believers

As time runs on further, the church grows, figure 3, plots 3 and 4, though there is always a chance that it could stall as it runs out of enthusiasts. Many of its boundaries with society have no enthusiasts – thus no conversions. Also there are unconverted people who can no longer be reached, the green agents surrounded by blue ones.

The church eventually stops growing leaving many people unconverted, figure 4. A fundamental result of the spread of an infection is that it burns out before everyone gets the disease.  Likewise the limited enthusiasm model predicts that revivals burn out before everyone is converted. Figures 3 and 4 show this principle at the microscopic level.

Figure 4: Final State of a Simulation of the Limited Enthusiasm Model. No Enthusiasts Remain and Church Stops Growing.

The total number of people in the church (red and blue) can be plotted over time, figure 5; along with the total number of enthusiasts (red), figure 6. The growth patterns of the agent-based model are very similar to the system dynamics model, figure 7, but the former has more randomness due to its microscopic nature [5, 6]. To fully replicate the smooth system dynamics predictions, the agent-based model would need to be run many times and the results averaged [7].

Figure 5: Growth of Church Over Time in Agent-Based Limited Enthusiasm Model

Figure 6: Number of Enthusiasts Over Time in Agent-Based Limited Enthusiasm Model

Figure 7: Results of System Dynamics Limited Enthusiasm Model


Agent-based models give a very visual view of how the church grows and running a simulation can really bring a model to life. Try the online version of the agent-based model to get feel for its behaviour:

(Instructions under Model Info)

The real drawback with agent-based models is that it is very hard to describe the model. Unlike system dynamics, the agent-based methodology has no intuitive and visual representation of the model structure and its hypotheses. Also you do need to run an agent-based model many times to see a clear conclusion. A system dynamics model can achieve this in one run, and quickly connect behaviour with model structure. Try the system dynamics version of the limited enthusiasm model yourself and see the contrast:


Hopefully this quick introduction to agent-based modelling has given further insight into the nature of church growth, and the mind of the mathematical modeller!

John Hayward
Church Growth Modelling


[1] The limited enthusiasm model of church growth is explained in:

[2] I have other models of church growth, whose main aim is to explain why the effectiveness of believers in conversion may vary over time. Issues include limited resources, the generation of spiritual life and the accumulation of institutional baggage.

[3] For introductions to agent-based modelling see:

[4] NetLogo is freely available from

[5] The argument runs that the behaviour of an individual cannot be exactly predicted. Prediction only becomes possible when the behaviour of large numbers of individuals are combined and the unpredictability is smoothed out. Getting the nature and extent of randomness in individual behaviour is far from straightforward.

[6] The system dynamics simulation was performed in Stella Architect available from isee Systems

[7] Averaging does not give a complete replica of the system dynamics results. This is partly due to the unrealistic network used. Some people have many contacts; some have few. Scale free and small world networks give better results. Further replication would require an improved model of how long an agent remains infectious, i.e. an enthusiast.


Wednesday, 16 November 2016

The Rise and Decline of British Methodism

Application of the Institutional Model of Church Growth

Some while ago I introduced a model of church membership that explain the rapid growth of a church, only to be followed by decline as institutionalism set in [1]. The primary effect of institutionalism is on the conversion rate. Thus it is possible for a once vibrant, growing church, be it congregation, or denomination, to end up heading for extinction, as it is unable to sustain enough conversions to counteract its losses.

The primary purpose of the model was to provide an alternative explanation of church decline to the limited enthusiasm model. The latter uses the metaphor of the spread of a disease and can work well for periods of revival, and as a model of decline on its own. However it can’t model decline whose cause is a loss of vitality.

By contrast the institutional model uses the metaphor of overshoot and collapse due to the production of a resource with unintended consequences. As the church grows it becomes more organised, more professional, and becomes a pillar of society – all traits of organisational intuitionalism. After some delay, this institutionalism restricts recruitment, eventually causing decline. But because of the delays the church is not able to spot the unintended effects of a professional church and cannot take action fast enough to correct the problem. It knows it needs revival not regulation, but can’t quite bring itself to let the chaos of a spiritual outpouring spoil its neatly ordered structures. The ultimate result is extinction.

Most of the pre-1900 denominations in the UK display the growth and decline pattern, and are thus candidates to fit the institutional model [2]. Unfortunately few have consistent data over the time span required, the one church that does are the Methodist Church [3]. As they were birthed in revival, and are now declining, they are therefore an excellent candidate to have their membership pattern explained by institutionalism.

The Institutional Model

The model is constructed using system dynamics, a methodology that explains behaviour in terms of feedback [4]. There are two variables: Church meaning church membership in this case, and Institutionalism, figure 1. These are indicated by rectangles, called stocks in the diagram. The more people in church the more people are added to the church, feedback loop R1. People are the primary resource of the church, contacting those outside the church, and bringing them to faith.

Figure 1: Institutional Model of Church Growth
Of course the more people in church then the more will leave each year, balancing loop B1, figure 1. R1 needs to be bigger than B1 for the church to grow.

As the church grows there is a build up of institutionalism, which in turn restricts the number added to the church, loop B2, figure 1. Whereas loops R1 and B1 only involve one stock, B2 involves two. Thus its effects are subject to more delay as the marks of institutionalism take time to build. Indeed there are delays in the church building the institution and in its effect on conversion/recruitment (not shown in diagram). These delays make the effects of institutionalism much harder to handle.

There is only so much institutionalism any organisation can take, it becomes harder to generate as it gets bigger, loop B3. Indeed there may be attempts to reduce institutional, loop B4. However there will be people in the church who want to maintain the institution regardless of its size, loop R2.

Data Fitting

The model of figure 1 is fitted to membership data for the Methodist church in Great Britain, that is excluding Northern Ireland [5], figure 2. The earliest reliable data is 1767, which is well within the church’s early revival period, thus a good starting point [6]. The data fit is given in figure 2, where the membership figures of the different branches of Methodism have been combined in the periods where they were organisationally separate [7].

2: Data Fit of GB Methodist Church 1767-2014

The model gives a remarkably good fit, except around the times where a major division occurred (1850), and the subsequent rejoining of different streams (early 20th century), figure 2. This may be connected difficulty in collecting data in such periods, or genuine loss to the church.  

The model takes into account population growth, and the higher death rate and lower birth rates in an aging church [8]. However these effects are hard to estimate accurately as the church get smaller. Note the discrepancy in the last two data points; the model is more optimistic than the actual data, which currently predicts extinction before the middle of the century.

Generally membership the church peaks around 1900. However if the growth of the population is factored out then real growth, i.e. proportional growth, ended mid-1860s,  figure 3 [9]. After this time the church is making no further inroads into the population, losing ground instead. The year of this turning point compares well with some other denominations [3].
Figure 3: Methodist Membership as a Percentage of British Population 

Thus Methodist church dynamics can be divided into three periods: revival growth, saturation, and decline. What is the reason for this change to decline?

Reasons for Methodist Decline

The model explains decline in terms of rising institutionalism. For the Methodist church the model indicates that institutionalism rose fast late 1700s to early 1800s, around the time it moved from being an Anglican renewal movement to a separate denomination. Since the mid 19th century, the transition from revival growth to saturation,  institutionalism has remained intact, figure 4.

Figure 4: Simulation of Methodist Church Membership & Institutionalism for Best fit to Membership Data
The effects of rising institutionalism can be seen in the various rates of change, figure 5:

recruitment to church,
biological addition to the church, i.e. children of church members who enter membership;

leaving church,
deaths of church members.

Figure 5: Rates of  Addition and Loss, Methodist Church GB

From its inception up to the mid 19th century Methodist dynamics were dominated by conversion, that is recruitment from outside the church. This accounts for the church growing faster than the population during that period. These conversions cannot be explained by transfer growth from other denominations alone, as all denominations grew during this revival growth period.  Note conversion is much bigger than biological additions. The sum of the two is much larger than the losses due to deaths and leaving.

From the middle of the 19th century, the saturation period, recruitment stops growing and is slowly caught up by the leaving rate of a still growing church. Meanwhile the biological additions peak in 1900 as the church starts to age due to there being a smaller margin between conversion and leaving. However the death rate continues to rise. The transition from the saturation period to decline around 1900 is largely due the changing balance of these additions and losses

In the decline period conversion/recruitment and leaving rates are more less the same, thus conversion no longer drives the dynamics. Instead, with falling national birth rates, and aging congregations, the decline dynamics are dominated by the church death rate, which becomes the largest from 1975 onwards. Churches never retained more than about 50% of their children [3], thus without a vibrant conversion rate, unhindered by the effects of institutionalism, the church cannot possibly make up for its losses.

The Future

The time for the church to deal with its institutionalism was in the 1860s, where the end of proportional growth, figure 3, would have been sufficient evidence that something was wrong. By the early 1900s when decline starts, and a generation had past without revival, it was too late to make effective changes to its dynamics. Such is the effect of delays. So what can be done now?

At this stage of denominational decline it is best to think in terms of congregations rather than the denomination. There will be strong Methodist congregations, even in 2016. These could be identified and allowed more autonomy to act independently of the denomination, especially in terms of finance and theology. Such strong congregations could then be free to seek partnerships with other church groupings, either local or national. They could also be encouraged to take control of other Methodist congregations whose prospects are poor perhaps by a church plant.

I realise such deregulation of denominational structure might lead to the end of the Methodist church as a denomination. But what it could lead to is new church movements that emerge from its now freer, stronger congregations, who would be the seeds of future denominations. This is not a just a Methodist issue, but a strategy that could be replicated across all declining denominations.

The Christian church in the UK is in a state of transition, the old is passing and the new will come. Yet managed well the best of the old will survive and flourish in the future. The Methodist denomination may go [10] but the spirit of Methodism will emerge, perhaps stronger, in churches with new names, but believing the age-old beliefs that made Methodism so strong in its early days.

References & Notes

[1] Blog: Institutionalism and Church Decline. 27/4/2015.

[2] The Baptist Union of Great Britain is an exception to the growth and decline pattern. This may be due to it being a collection of independent churches – often very independent!

[3] I have already discussed the reasons for growth and decline using data from the Welsh Calvinist Methodist Church, aka the Presbyterian Church of Wales. Like the Wesleyan Methodists they have a rich data set, recording births, deaths, joining and leaving rates. Indeed their data shows the stability of leaving rates, and child retention rates, over long periods. Instead it is the change in conversion rate that affects growth and decline.

Blog: Church Decline Caused by Lack of Conversions

Blog: Church Decline cause by Lack of Revivals

Blog: Why Revivals Stopped in the UK

[4] For more information on System Dynamics see:
Sterman J.D. (2000). Business Dynamics: Systems Thinking and Modeling for a Complex World, McGraw Hill.

Morecroft J. (2007). Strategic Modelling and Business Dynamics: A Feedback Systems Approach. John Wiley and Sons.

Systems Dynamics Society,

[5] The division of Ireland into North and South in the 20th century makes it difficult to include figures for Northern Ireland over a 250 year timescale.

[6] Currie, R., Gilbert, A. D., & Horsley, L. S. (1977). Churches and Churchgoers: Patterns of Church Growth in the British Isles since 1700. Oxford University Press, table A3, pp. 139-144.

Brierley P. (2014). UK Church Statistics 2010-2020, and previous editions. Brierley Consultancy. Also Religious Trends, Christian Research.

Statistics for Mission, various editions, Methodist Church.

[7] Data fit was performed using optimisation in the system dynamics software Vensim. Ventana Systems

[8] The growth of the GB population is given in the graph below. Note how population growth accelerates in the 1800s with the industrial revolution, then slows a bit in the twentieth century.

The GB birth rate peaks early 1800, see graph below. It drops significantly from late 19th century onwards. Even with the falling death rate no church in the 21st century can expect to grow as fast as those of 19th century, family size is so much smaller. Likewise the church cannot expect to be as young as those of the 19th century, and are made even older by the longer life expectancy. However the church can expect revival as in the 19th century, and it can expect to grow faster than the population.

In the 18th and 19th century, the church was generally younger than the population average, hence the birth ratio being above one, and the death ratio below one, see graph below. The most likely explanation is that the converts into the church were younger than the population average. Modern studies on conversion shows that it peaks in the twenties age group. When conversion rates were high, and the church growing, the church was relatively young, even though it only kept half its own young people. As conversions fell the church became older. Thus from 1900 the death rate rises and the birth rate falls, graph below. With the current levels of birth and death rates in church, even a large revival would not turn decline into growth overnight. The effects of revival now may not be fully seen for 30 years or more.

Further details at:

[9] Analysis graphs produced in Stella Architect, also used to draw the model in figure 1. ISEE Systems

[10] Although statistically most pre-1900 denominations are heading for extinction, this does not mean they will inevitably go extinct. Some may survive in a much smaller form, probably in large towns and cities, providing a valuable link with the past. Indeed there is no reason why a church that is almost extinct cannot grow again if it turns back to God and receives his power. The Lord looks at the heart not the denominational name, or the statistics!