Results of the Limited Enthusiasm Model with Spiritual Life

In the Spiritual Life model, enthusiasts generate the spiritual life of the church, as well as make converts. In addition, the effectiveness of the enthusiasts in conversion improves with more spiritual life. The model captures the feedback loop: more life gives more growth and thus gives more life. The Spiritual Life model is an extension of the Limited Enthusiasm model.

The model is applied to the membership data for Welsh churches 1902-1906 to show how a build of spiritual life in the churches prior to the 1904/5 revival tipped those churches into revival growth. Consequently, spiritual life decreased as the revival subsided as there were insufficient enthusiasts to maintain it.

Data Fitting & Calibration

The total membership numbers in the Welsh churches are taken from Williams (1985) with estimates of the Anglican church, see Explanatory Notes on “Mathematical Modeling of Church Growth”, section 5.6. The combined membership is compared with the Spiritual Life model to give an optimum fit, figure 1. Because membership figures are given at the end of each year, the numbers on the horizontal axis refer to the END of each year.

Figure 1

The churches had been growing slowly from 1902 through most of 1904, after which the effects of the revival on membership accelerated giving a massive increase by the end of 1905. After that date, there was a moderate fall. The data fit is achieved with a natural reproduction potential of 1.915 enthusiasts per enthusiast and a duration enthusiastic of around 1 week. This short timescale may reflect the fact that many revival meetings were on consecutive nights and the influence of a revived Christian had fallen once they went into a second week. It can also appear artificially short as the “week of enthusiasm” may be spread over many weeks of nonconsecutive periods.

The natural reproduction potential is insufficient for revival growth. Instead, the actual reproduction of enthusiasts is influenced by a rise in the spiritual life of the churches, allowing it to rise above the natural level. That is church growth is like a disease with increasing infectiousness. The effect of spiritual life on conversion was 1.176, the maximum number of extra converts per enthusiast possible. The effect of church enthusiasts on spiritual life was 7.54, the maximum fractional increase in spiritual life possible per year. Spiritual life would decay at a rate of 70% a year if there were no attempt to maintain it.

Revival and Spiritual Life

The growth in the churches is driven by enthusiasts who make converts, some of whom also become enthusiasts. The reviving work of the Holy Spirit is passed on from person to person as they are converted. The simulation shows that in 1902 the number of enthusiasts in the church had been falling, figure 2. However the level of spiritual life of the enthusiasts was increasing; this reflects the renewing work of the Holy Spirit in the various prayer meetings for seeking a deeper life that were taking place. Spiritual life is rising steadily 1902-4 and ultimately enhances the ability of enthusiasts to make converts, hence the rise in enthusiasts.

Figure 2

Halfway through 1904, the number of enthusiasts explodes, figure 2, resulting in a rapid increase in the church due to conversion in 1905, figure 1. Note the rise in spiritual life, precedes the rise in enthusiasts which in turn precedes the growth of the church. Revivals start with a rise in the work of the Holy Spirit and its effect on existing believers and the church.

The enthusiasts peak in halfway through 1905, well before the church reaches its peak in early 1906. However spiritual life keeps growing until late in 1905. The peak in spiritual life is a good marker of the end of the revival as most of the growth in the church has taken place, figure 3. After this point, the life in the church, and thus the effectiveness of individual enthusiasts, starts to fall quite significantly. By 1906 there are no enthusiasts left, figure 2. Though there were many still feeling the effects of the revival, their enthusiasm was no longer channelled into making converts, probably because they had lost influential contacts with unbelievers.

Figure 3

A significant level of spiritual life remains well after the growth phase of the revival. Post revival churches are no doubt much livelier with more works of the Holy Spirit taking place than at the beginning. A spiritual church but with few conversions. Thus the continuing presence of spiritual life is not a predictor that a revival is coming to an end. By the time spiritual life is falling it is too late. The early predictor for the end of revival is the number of enthusiasts not the liveliness of the church. Revival growth could be extended by ensuring revived Christians and new converts are committed to evangelism and witness.

Revival Threshold

At the end of 1902, the reproduction potential of the enthusiasts was still below the threshold for revival growth, indicating the inadequate reproduction of enthusiasts to give revival growth, figure 4. However early in 1903, the reproduction potential rises due to the effects of the increasing spiritual life in the churches. The enhanced enthusiasts are able to make sufficient enthusiasts to more than replace themselves thus giving the revival growth in the church.

Figure 4

he revival growth threshold (line2 ) rises as the number of unbelievers shrinks due to conversion; it is harder for enthusiasts to find fresh unbelievers. Half way through 1905 it rises above the reproduction potential, leading to a decrease in enthusiasts and a slowing down of church growth. (N.B. the years on the horizontal axis refer to accumulated numbers at the end of that year, due to membership statistics being released at the end of a year.)


The limited enthusiasm model fits the data from the Welsh Revival if just run from 1904/5, but does not fit if the simulation starts earlier in 1902. This is because the reproduction potential is higher during the revival than before. Adding the variable spiritual life provides a mechanism for explaining why the reproduction potential increases in terms of the action of a small number of church members seeking and receiving more of the Holy Spirit. This extended model gives an excellent fit to the data over that period, figure 1.

One drawback is that spiritual life starting at near zero in 1902 is unrealistic. This may be due to the absence in the model renewal mechanism, where enthusiasts make new enthusiasts out of existing church members. Such a further extension is possible but the extra number of parameters may make data fitting complicated. The results presented above are sufficient to drive home the importance of seeking the work of the Holy Spirit to help tip a church into revival.

This work was part of an M.Phil project pursued by L. Howells at the University of South Wales, thesis Dynamics of Church Growth and Spiritual Life, 2015.