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 into 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].
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.
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 .
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).
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.
For a simulation I will use the agent-based simulation software NetLogo . 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.
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.
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 .
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:
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!
Church Growth Modelling
 The limited enthusiasm model of church growth is explained in:
- Mathematical Modeling of Church Growth, Journal of Mathematical Sociology. 23(4), 255-292, 1999.
- A Dynamical Model of Church Growth and its Application to Contemporary Revivals. Review of Religious Research, 43(3), 218-241, March 2002.
- A General Model of Church Growth and Decline. Journal of Mathematical Sociology, 29(3), 177-207, 2005.
- Church Growth via Enthusiasts and Renewal. Presented at the 28th International Conference of the System Dynamics Society, Seoul, South Korea, July 2010.
- Tipping the Church into Growth. https://www.churchmodel.org.uk/CGTipping.pdf
- Church Growth Modelling Website.
 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, Church Growth Models.
 For introductions to agent-based modelling see:
- From the author of NetLogo http://www.intro-to-abm.com/
- Open Agent Based Modelling https://www.openabm.org/book/export/html/3443
- An Introduction to Agent Based Modeling for Undergraduates
 NetLogo is freely available from https://ccl.northwestern.edu/netlogo/
 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.
 The system dynamics simulation was performed in Stella Architect available from isee Systemshttp://www.iseesystems.com/
 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.