Every mathematical model is an attempt to replicate aspects of reality. Mathematics is the language used to express that model and capture the complexity of the real world. The type of replication required will determine the nature of the model produced – the model’s intended purpose. Likewise, the ability of the mathematical framework to handle real-world complexity will affect the model – the theoretical and practical limitations of mathematics. Complexity and purpose help determine the type of model produced.

Models can be classified according to their:

• Fidelity, the extent to which models are intended to replicate reality;
• Theoretical Support, a logical framework of consistent laws which enable models to explain a wide range of behaviour;
• Methodology, a logical collection of methods and principles, mathematical and computational in this case, that allows a model, and any associated theory, to be expressed and analysed.

In summary: “One or more methodologies are used to express a theory of a certain strength which in turn supports a model of a certain fidelity.”

A model can be described as “successful” by the degree to which it replicates reality and usefully fulfils its purpose. A model does not need to use the richest methodology, have the strongest theoretical support or the highest fidelity to be useful. For example, a simple back-of-the-envelope calculation that improves one process, in one industry, at a given time is highly useful and thus successful. Even if a model is not “rocket science”, it can still instil confidence.

### Model Fidelity

Model fidelity describes the extent to which a model can replicate reality. Fidelity depends on the purpose of the model. Some models, such as an aircraft flight simulator need to strongly replicate reality to fulfil their purpose of training people. Thus, they have very high fidelity, sometimes called analogue. However, other models may only need to illustrate a principle, such as the motion of a train. It could be made analogue for a specific train, but people often learn more from a simpler illustration.

Model fidelity also depends on the ability of the mathematics used to handle complexity. A more illustrative model may only be possible, such as the interaction of many species, as the governing laws may not be fully known, or data is not detailed enough to calibrate completely.

Further details in Morecroft J. (2007). Types of fidelity:

##### Applications

The Limited Enthusiasm model of church growth is towards the high fidelity end of the spectrum as all its variables and parameters have well-defined meaning and measurable values, more analogue than illustrative.

With demographics added the model becomes a little less analogue as some of the parameters are harder to estimate. Nevertheless, they are, in principle, measurable. The absence of cohorts makes the birth/death process a little more illustrative than analogue. The model provides principles and can be calibrated for congregations and denominations and fits within the standard genre of population modelling.

The Spiritual Life model of church growth is illustrative. Realistic values can be given for the population numbers and compared with data, but Spiritual Life, which is a soft variable, is in principle hard to quantify. However, the model can tell plausible stories with the level of Spiritual Life being compared with proxy variables. The model provides principles, but would not be easy to calibrate for a particular church grouping.

The Limits to Church Growth models are metaphorical. Although they could be made illustrative, and perhaps even analogue, that is not their purpose. They are constructed to show how growth limits from other sciences, such as the environment and business, are also significant for understanding barriers to church growth.

### Theoretical Support

Not all models can be elevated to the lofty position of a “theory”. However, all models have a degree of theoretical support. For some situations, strong theoretical support is essential to build confidence. This is particularly true for models which are hard to calibrate or have soft variables, ones that are hard to measure. However, for models of specific situations, where a particular problem needs to be solved, and/or where there is a wealth of data to calibrate the model, then strong theoretical support is not always necessary. Confidence is built from the model’s ability to solve the problem.

What follows is my own rough outline of the degree of theoretical support for a model, no doubt incomplete!

##### Applications

The Limited Enthusiasm model is part of the general theory of word-of-mouth models, which are in turn based on models of the spread of disease. These are based on the mass action principle that expresses how two differing subpopulations interact. Other examples of models based on this theory are in the social diffusion references and the epidemic modelling references. The model with demographics added has the same theory.

The Discipleship model is phenomenological. It is a model of a generic church congregation. It could be applied to most congregations (so not “specific”), but has no applicability outside a church context (thus not “general”). It uses various familiar model constructs.

Of the various limits to church growth models, the Bounded Resource model is general as it is a typical limits-to-growth model. However, the Supply and Demand models are common sense.

### Methodology

A methodology is a collection of rules and principles that allow a model to be constructed, expressed and analysed. The rules must make logical sense and thus be expressed in terms of mathematical and computational constructs. Mathematics contains a range of constructs from calculus to statistics, probability to symbolic logic. A modelling methodology must have rules that relate its mathematics to the real world.

The following are three of many dynamical modelling methodologies. They are dynamical as the type of modelling concerns behaviour over time.

To date, all the church growth models presented on this site are built using either differential equation or system dynamics. However, there are also agent-based versions of some of these models.