The central hypothesis of the limited enthusiasm model is that conversion growth in the church is driven by a sub-group of church members called enthusiasts. In addition, the demographics model includes births, deaths and reversion, allowing for church decline through lack of child retention, ageing and people leaving the church. The unbelievers are split into those open to conversion and those hardened, allowing for secularisation in society.

The following model details supplement the Demographic Model Overview.


The model consists of four groups of people: 

  • enthusiasts who are responsible for spreading the faith, and inactive believers, who together make up the church; 
  • open unbelievers who are open to conversion;
  • hardened unbelievers who make up the rest of society who are outside the church.

The hardened unbeliever category includes people who are antipathetic or hostile to Christianity, former open believers who have secularized, or believers who have turned away from the church either by abandoning the faith or being hurt.

For simplicity, all church members are assumed to be believers and vice versa. There is no data on who among a church membership are true believers in Jesus Christ and little prospect of it being reliably obtained.

It is further assumed that members are identical to the attendees of the church. This is because there are only reliable data for attendance. Membership data is less reliable as different denominations and churches use different definitions of membership. Thus the model deals with attendance and recruitment issues only. Sometimes membership data needs to be used because no attendance data is available.

In reality, there will be members and attendees who are not believers, members who do not attend and attendees who are not members. There may even be believers who are neither members nor attendees. However, these effects will tend to cancel each other out, and the above assumptions imply that their net effect is negligible.

System Dynamics Model

The above hypotheses are expressed in a system dynamics model, with the central ones indicated in figure 1:

Demographics diagram
Figure 1

A number of flows connected with births and reversion are missing. This model is an extension of the original Hayward (2005) model, allowing open unbelievers to become hardened.

Dynamic Hypotheses

The dynamic hypotheses are:

Conversion Through Enthusiasts (R)Enthusiasts are responsible for spreading the faith, i.e. conversion to the church. The more they convert, the more enthusiasts. This accelerates growth. Spreading the faith can be by numerous means.
Loss of Enthusiasm (B2)After some time, the enthusiasts lose their potential to convert. This slows and limits growth.
Diminishing Susceptible Pool (B1)As people are converted, the effectiveness of the enthusiasts on the remaining unbelievers become less as proportionally more of the enthusiast’s time is spent on believers. This slows growth.
Not All Converts Become EnthusiastsNot every convert becomes an enthusiast. Some become immediately inactive.
Enthusiasts inactive believers leave the church (B3) and become open unbelievers open to rejoining the church (R2).The reasons for leaving are not important but could include varying commitments, pressures on time, switching denominations or geographical movement. Thus recycling, R2, is possible.
Enthusiasts inactive believers revert, i.e. leave the church and become hardened unbelievers, not open to rejoining.The reasons for reversion may include abandoning the faith or being hurt by the church, in varying degrees.
Hardened unbelievers may become open after a time, and open unbelievers may become hardened.Both softening and hardening processes may be due to actions of the church in society or actions within society independent of the church, such as wealth or poverty. They are fixed rates in this model. Hardening can include secularisation.
All populations have births, but children may change to a different population group before adulthood.For example, some children brought up in church become believers, some drift and become open unbelievers, and some dislike religion and become hardened. Likewise, children of open unbelievers may become harder than their parents, secularisation; and children of hardened unbelievers may become softer as they do not have the same baggage in their background.


The differential equations of the original Hayward (2005) model are:

Demographics details equations

where U, A, B, H are open unbelievers, enthusiasts, inactive believers and hardened unbelievers, respectively. N is the total population. Cp is the conversion potential. g is the fraction of converts who become enthusiasts. Thus, Rp = g Cp is the reproduction potential. τa is the duration enthusiasts are active. ε is the softening rate of hard unbelievers. αa and αb are the leaving rates from enthusiasts and inactive believers. The various fractions labelled f relate to the fraction of children born into the different populations. b and d are the birth and death rates. See the parameters below.

The model is generalised to include secularisation effects:

Demographics details extended equations

where μ is the hardening or secularisation rate. φah and φbh are the fractions of church leavers who become hardened unbelievers instead of open ones. bc and dc are the birth and death rates within the church.

What is Meant by “Spreading the Faith”?

All aspects of conversion and spreading the faith are discussed in the limited enthusiasm model.

Why Do People Leave the Church?

People leave a church for many reasons. Some renounce the faith and return to the world. Some still claim to be Christian but no longer attend church. This model does not make any spiritual judgment on those that leave the church, in the same way as it makes no assumptions of those who belong. It merely divides leavers into those who become open to rejoining and those who do not.

Reasons for people leaving church include:

  • Physical persecution. At its most extreme, believers’ 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 to renounce the faith also.
  • Psychological persecution. Some believers give up their faith due to the pressure of unconverted family members or peers. The pressure may take the form of teasing or being shunned from normal social activities. The believer gives up the church to have a more comfortable life.
  • Some of the believers were not genuinely converted in the first place and gave up the faith once it started to lose its initial attraction. The unbelieving world they left behind has proved more attractive to them.
  • The believer may have fallen into sin, which caused them to be too ashamed to be part of the church anymore.
  • The believer may have been hurt by others in the church. It is now too painful for them to be part of the church with those who hurt them. Sometimes it is too painful to be in any church as it reminds them of the past hurt.

How Do the Children of Believers Become Part of the Church?

Children are not normally born with a faith. However, if a child is brought up in the Christian faith, Christian parents hope that they will believe for themselves and join the church. This is often called biological growth to distinguish it from conversion growth. Of course, the children, at some point, should become converted! However, their conversion has taken place within the Christian community rather than from the world outside. In this model, such children will be regarded as being “born as believers”.

There is a time lag between birth and the formal admission of such children into the church. This time lag will vary according to church policy – anything from aged 6 to 18! It will also vary depending on whether the model is compared with attendance, membership or communicant figures. In this model, this time lag is ignored.

Why Do the Children of Believers Fail to Become Part of the Church?

  • Some are not converted and thus see no reason for belonging to the church. Although familiar with the church’s spiritual culture, it is meaningless to them.
  • Some churches offer so small that even converted children see little point in joining. However, the absence of Christian friends does not make it any easier for them to join another church, which is a huge step for a young person.
  • Young people are very sensitive to hypocrisy. The adult generation’s perceived and actual hypocrisy will cause young Christians to prefer to be believers “who do not attend church”.

Why Do Unbelievers Harden or Soften?

An unbeliever may be hardened because they have had a bad experience of the church, either as a former member or through contact with church members. Any message they hear from the church may only reinforce their negative views rather than lead to conversion. They may harden through negative media coverage of the church or through the opinions of other unbelievers they meet. However, over time, such hardened attitudes may soften, either through age, the fading memory of what caused them to be hardened, or through a different church experience.

Hardened unbelievers may include people in rival religions, denominations, or those for whom Christianity is a culture they do not understand. Time is needed to bridge these gaps. The model does not include details of why hardening or softening occurs, only that they occur at fixed rates. The rates can be set to model the effects of secularisation: people drifting from a Christian culture over time. Using different rates can capture a society becoming more accommodating of Christianity.

Children of open unbelievers may become hardened, perhaps due to the increasing cultural distance of a new generation. Children of hardened unbelievers may become open, maybe due to not acquiring their parents’ cultural baggage.

Model Parameters

The behaviour of the model is controlled by parameters that reflect the church’s effectiveness and the response of society:

Reproduction Potential RpThis is the number of unbelievers converted and made enthusiasts, through one existing enthusiast, given the whole population are unbelievers. It measures how much an enthusiast can “reproduce” themselves from the pool of unbelievers.
Duration of Enthusiastic Phase τaThe average length of time an enthusiast is active in conversion before they become an inactive believer.
Fraction of Converts Enthusiast gThe fraction of new converts who become enthusiasts. The remaining people become inactive believers immediately on conversion.
Leaving Rate
αa and αb
The total rate at which people leave church regardless of whether it is reversion or leaving to the open category. The fractions of those who leave who become open or hardened unbelievers, φah and φbh.
Fraction of Believer’s Children Retained in ChurchA fraction each for inactive believers fbu and enthusiasts fau who leave the church before adulthood. fbh and fah are the fractions that are born open. faa is the fraction of children of enthusiasts who become enthusiasts.
Fraction of Unbeliever’s Children Who are Either Hard or OpenA fraction of children of hardened unbelievers may be open fhu and a fraction of children of open parents may be open fuu.
Hardening Rate μThe fraction of the open unbelievers who become hardened per year.
Softening Rate εThe fraction of the hardened unbelievers who become open per year.
Birth and Death Rates b, d unbelievers,
bc, dc for the church
Church and outside society can have separate birth and death rates to model ageing and religions with higher than average family sizes.
Initial Fraction of Church EnthusiastThe fraction of the church that are enthusiasts at the start of the model.
Initial Fraction of Unbelievers HardenedThe fraction of unbelievers that are hardened at the start of the model.

Model Results

Applications to UK Denominations

Church of EnglandChurch of ScotlandChurch in Wales
United Reformed ChurchMethodist ChurchWelsh Presbyterians
Welsh IndependentsElim PentecostalUK Baptists
NewfrontiersRoman CatholicFIEC
Scottish Episcopal Church
Growth, Decline and Extinction of UK Churches