Thursday, 20 November 2014

The Eyam Plague of 1666: A System Dynamics Model

At present the news is full of the rather frightening Ebola outbreaks in West Africa. Other infectious diseases also get into the news often, for example AIDS/HIV and flu. What is less well know is that such infectious diseases spread according to fairly precise mathematical rules. This follows from the person-to-person contact involved in the spread of the disease.

It is this process that the limited enthusiasm hypothesis of the church growth models is based on. In that case the “disease” is faith, and it is spread by word of mouth contact. It is not only churches that grow this way, the same epidemiological mechanism has been used to model the spread of languages, scientific ideas, riot behaviour, bulimia, cigarette smoking and even Facebook [1].

To illustrate how this principle works I want to use a standard case study in mathematical epidemiology: the spread of the plague in the Derbyshire village of Eyam in 1666.

Eyam Plague

The years 1665-1666 saw the great plague hit England, notably London, which lost about 15% of its population [2]. This was the last epidemic of bubonic plague in the UK, a disease that had been an ongoing problem since the days of the Black Death in the 14th century. The primary mechanism of spread of the disease is through the bite of an infected black rat flea. However once established the disease can spread person-to-person, which gave rise to the popular rhyme “Ring a ring of roses, a pocket full of posies, atishoo, atishoo, we all fall down” [3].

Although largely confined to London, an outbreak occurred in the Derbyshire village of Eyam due to a person acquiring the disease from a piece of infected cloth sent from London in 1665. Once the Eyam outbreak took hold in 1666 the local clergyman took the precaution of isolating the village, as best he could, to prevent the spread of the disease. This action of his made it an ideal case study to mathematically model the spread of a disease, as migration could be ruled out as a major mechanism in its spread.

Such a mathematical model of the plague was carried out by GF Raggett [4] using methods based on differential equations. Raggett explained why the spread of the disease in Eyam must have been largely person-to-person rather than rat fleas, as over the period of a year infected rats would have left the area and infected the wider area. No such cases occurred. Using mathematics Raggett then showed how the model predicted the number of deaths due to the epidemic, and demonstrated some important results [5].

What follows is a system dynamics version of Raggett’s work to help explain how a disease spreads without using mathematics. The model is often called the SIR model, after the symbols in the equations, the epidemic model, or the Kermack McKendrick model, after the first people who published it [6].

System Dynamics Model

The model assumes the population of people are split into 3 categories of people: the Susceptibles, who could potentially catch the disease; the Infected, who are carrying the disease and could infect others; and the Removed who have had the disease and cannot catch it again, either because they are cured and immune, or have died. The letters SIR stand for these three categories.

In system dynamics this model can be expressed as stocks and flows:

Figure 1
The removed category has been renamed Deceased as most cases of bubonic plague ended in death.

There are two processes (called flows) involved: catch the disease, which moves susceptibles into infected; and deaths, which moves infected into deceased.

The catch disease process is subject to two social forces: R1 and B1. R1 causes the increase in the number of infected to accelerate as more infected gives more new cases each day, thus more infected. This is called reinforcing feedback and is the first phase of growth in the infected, (figure 2).

Figure 2

In addition the force B1 slows that growth as the pool of susceptibles is depleted, making it harder for infected people to make new cases. This slowing force is balancing feedback and opposes the force R1. B1 eventually dominates over R1, the second phase of growth (figure 2) [7].

Eventually the number who catch the disease drops below the deaths and B1 now causes the infected to decline faster and faster, the first phase of decline (figure 2).

The deaths process is subject the social force B2 as the more infected there are the more die, thus depleting their numbers. This force only dominates at the end causing the decrease in infected to slow down, the second phase of decline, (figure 2).

Raggett [4] showed from the recorded deaths that when the main period of the plague epidemic started, June 19th 1666, there were 7 people infected. The population was known to be 261 at that time. By the end of the epidemic, in the middle of October that year, only 83 people had survived.

From these figures, and knowing the infectious period of the plague is about 11 days, it is possible to simulate the system dynamics model, and compare it with the data for cumulative deaths (green curve, figure 3) [8]. Comparing Deceased with recorded deaths shows a good fit. It is remarkable that something that involves people, and random behaviour, gives such predictable results. This predictability is what allows modern day epidemics to be so successfully tackled, and the consequences of not tacking action computed [9].

Figure 3

Note the following:

1. The epidemic burns out before everyone gets the disease. There are susceptibles remaining at the end of the epidemic (black curve figure 3).

2. At the peak of the epidemic, where the number of daily cases is at a maximum (about 45 days, figure 3), even though the daily death rate is slowing down the epidemic is not at an end and a significant number of deaths are still to come, (green curve figure 3).

3. At any one time the number of infected people is quite small compared with the population (blue curve, figure 2 and figure 3). It is their cumulative number over time that is large. It does not take many infected people at a given time to keep an epidemic going [10].

Reproductive Ratio

The strength of an epidemic is measured by the reproductive ratio, called R0. At its simplest it is the number of people one infected person could potentially infect during their infectious period, if the whole population were susceptible [11]. The number has to be bigger than one for an epidemic to happen. The larger this number then the bigger the epidemic becomes. Different diseases have different reproductive ratios [12].

Using the numbers above, the reproductive ratio comes out at about R0 = 1.6 [13]. This is much less than highly infectious diseases such as Measles (range 12-18) and Smallpox (5-7) [14]. Nevertheless 1.6 was still large enough for well over half the population of Eyam to get the disease. A value of R0 of 1.6 is similar to Ebola (1.5-2.5). However because Bubonic Plague is spread through fleas, and through the air, it is harder to take action to reduce R0 compared with Ebola, which is only spread through contact with bodily fluids.


What turned out to be an ideal case study to test a mathematical model for the spread of a disease turned out to be a tragedy for the people of Eyam. The majority of the population died, including the wife of the brave clergyman who isolated the village and performed all the burials [15]. However his action saved many more lives of people in the region, and the lessons learned, which mathematicians can now explore, gives confidence to models that have given strategies to combat epidemics and save millions of people. That studies of this sort can help understand social diffusion processes such as religion is a bonus.

References & Notes

[1] For a selection of social modelling papers that use the epidemic/disease analogy see references at:

[3]  For a history of the Great Plague of London see Wikipedia and the references contained within.

[4]   Raggett, Graham F. "Modelling the Eyam plague." Bull. Inst. Math. and its Applic 18, no. 221-226 (1982): 530.
Note Raggett used the burial records to estimate deaths. There is a slight time delay between the two, but not enough to seriously affect his results.

[5] For a history of the Eyam plague see:
Wallis, Patrick. "A dreadful heritage: Interpreting epidemic disease at Eyam, 1666–2000." In History workshop journal, vol. 61, no. 1, pp. 31-56. Oxford University Press, 2006.

For the demography of the Eyam plague see:
Race, Philip. "Some further consideration of the plague in Eyam, 1665/6." Local population studies 54 (1995): 56-57.

[6] Kermack, William O., and Anderson G. McKendrick. "Contributions to the mathematical theory of epidemics. Part I." In Proc. R. Soc. A, vol. 115, no. 5, pp. 700-721. 1927.

[7] The structure of the feedback loops, or forces, on those who catch the disease can be broken down into a number of parts where it is assumed the populations are proportionally mixed. The connection between population density and the likelihood of contact requires further assumptions.

[8] The model was constructed and simulated in the Software Stella, available from ISEE Systems

[9] Similar models for Ebola in West Africa, 2014, have already been constructed and are informing policies to reduce its impact. For example:

Meltzer, Martin I., Charisma Y. Atkins, Scott Santibanez, Barbara Knust, Brett W. Petersen, Elizabeth D. Ervin, Stuart T. Nichol, Inger K. Damon, and Michael L. Washington. "Estimating the future number of cases in the Ebola epidemic—Liberia and Sierra Leone, 2014–2015." MMWR Surveill Summ 63, no. suppl 3 (2014): 1-14.

Kiskowski, Maria. "Description of the Early Growth Dynamics of 2014 West Africa Ebola Epidemic." arXiv preprint arXiv:1410.5409 (2014).

Team, WHO Ebola Response. "Ebola virus disease in West Africa—the first 9 months of the epidemic and forward projections." N Engl J Med 371, no. 16 (2014): 1481-95.

[10] All these epidemiological principles are replicated in church growth, and other forms of social diffusion. Not all people in a population are converted. Substantial church growth still comes after the peak in the growth is over. At any one time there are very few infected, called enthusiasts, spreading the faith.

[11]  The reproductive ratio (or reproductive number)  is called the reproduction potential in church growth and measures how many people one enthusiast can potentially convert and make an enthusiast. Not all converts become enthusiasts.

[12] For most diseases the reproductive ratio is given as a range as its value can depend on population density and particular population behaviours. It is believed that Ebola in West Africa in 2014 started with a much higher than normal R0 due the particular burial practices used, allowing dead bodies to transmit the disease, thus extending the infectious period.

[13] A simple formula can be used to compute the reproductive ratio in terms of the population number, and initial and final number of susceptibles alone. This calculation was done in the software Mathcad:

For the computation of this formula for the reproductive ratio see:

Brauer, Fred. "Compartmental models in epidemiology." In Mathematical epidemiology, pp. 19-79. Springer Berlin Heidelberg, 2008.

Brauer, Fred. "Compartmental models for epidemics." (2008).

There are numerous methods to compute the ratio, sometimes giving different answers, see [14] below. This is not an exact science.

Mathcad is available from

[14] Wikipedia and references within.

[15] There is a museum in Eyam where the visitor can learn about the outbreak. Note that there had been cases and deaths in 1665 and early 1666 before the period of study used by Raggett starting June 19 1666. Thus the total deaths, and the original village size, are larger than used in Raggett’s study.

Thursday, 6 November 2014

Is the Charismatic Revival Over?

The Changing Nature of Worship Songs

In a recent article for Christian Today, worship leader Noel Richards passed comment on the style of worship songs written in the 1990s and those written now[1]. His central thesis is that there were songs sung then, especially in the events like March for Jesus, that are not appropriate now. His article was a response to a previous one by Martin Saunders who called for ten 90s worship anthems to be revived, songs such as Called to a Battle and Champion, both Richards’ songs[2].

My intention in this blog is to examine Richards’ claims as evidence for a tentative thesis of mine: that the revival, known as charismatic renewal, which started around 1960, is now over[3]. The thesis is only tentative, as it will take more evidence than presented here to prove it. Up to now it has been a thesis I have alluded to in articles and talks just based on my personal observations.

So to Noel Richards’ article.

Evidence 1

The focus of the movement has shifted from spiritual internals to human externals.

Richards sets his article’s context with this sentence:

The charismatic/evangelical churches were at the cutting edge of worship in those days.

Now I know exactly what he means, I was there at the time, leading worship, sometimes with his songs (thank you Noel!). But I think he has revised history a bit. The charismatic churches were at the cutting edge of a movement where the Holy Spirit transformed people. Worship was part of that movement, but only part. The changes in worship style were an external and human-centred expression of the God-centred and internal work in people’s hearts.

Once people look back and focus on the externals, such as worship, rather than heart changes, such as baptism with the Spirit, then it is a sign that the movement or revival is over. Think how people still praise the wonderful hymns of Wesley from the 18th century. But the Methodist revival was about conversion and being saved, not about the hymns[4]! Only once the conversions ended and the evangelistic, and evangelical, fire went out did people start lauding the hymnology.

Likewise the elevation of worship songs and styles now is a sign that the charismatic revival that spawned them is over. The external has taken over from the internal, the work of man rather than the work of God. Richards says:

We believed that worship and prayer events would help pierce the darkness over our towns and cities.

Not really! We believed that people filled with God’s Holy Spirit would be used by him to pierce the darkness. The events were merely the means to receive from God.

Evidence 2

The movement ceases to be lay and voluntary, but has become the domain of the professional and specialist.

Richards states:

The 1990s saw a generation of pioneering hymn-writers such as Chris Bowater, Graham Kendrick, Dave Fellingham, Dave Bilbrough, Martin Smith.

In reality the true pioneers of the worship of charismatic renewal were in the 1960s and 1970s. People like the Fisher Folk from the Church of the Redeemer Houston[5], or Keith Green. They were a small part of the “worship in the Spirit” pioneers, two of the few who became known as they managed to produce albums and song books. The bulk of the pioneers were not professionals, produced no media, and only achieved local prominence, but they were used by God to transform thousands of lives. History does not record who most of these pioneers were, but they wrote songs that ordinary church people could play, the key dynamic that took worship leading out of the hands of the musical expert and placed it in the hands of the Spirit-filled believer.

What the people mentioned by Richards actually pioneered was the rise of the worship music industry, along with many Vineyard and Calvary Chapel musicians in the USA. Undoubtedly their songs were better written, the musical standards higher, and they have been a huge inspiration to many, but from a Holy Spirit point of view they were building on the work of others now forgotten. Their songs were undoubtedly used by God to pursue the revival, but their legacy now is a worship music genre that is so professional in its standards that it is beyond the reach of all but the best musicians. A complete reversal of the charismatic revival’s origins! [6]

That the worship of charismatic renewal is now firmly in the hands of professionals and focuses on the production of media, such as albums, rather than producing Spirit-filled people, is a sign the revival is over. In sociological terms the “worship song” has been “routinized”, set into a fixed form. What was spontaneous has been codified, almost developed its own liturgy. Worship songs really do sound like “worship songs”, slightly dated versions of modern pop music [7] whose formula can’t be challenged. Revival challenges such things. Thus I conclude the revival is over.

Evidence 3

The movement has become timid and unable to proclaim its message boldly.

A feature of any revival is its willingness to engage with the world, proclaim its message publicly and face the consequences. Think of the origins of Christianity or the Methodist revival in the 1700s. The Holy Spirit drives believers into the open.

Richards says that:

The church and the world of 2014 is very different to that of the 90s.

That is certainly true with regard to the world. There is now far more vocal and organised opposition to Christianity in the West than back in the 1980s. Dare I suggest that some of that opposition is a fruit of the boldness of Christians back in the height of the revival? That rather than seeing the current onslaught on the church by the Western political establishment and non-faith groups as a negative thing, it is actually positive, because at some point the boldness of a Spirit-led movement rattled some cages.

Richards goes on:

Whereas in the 90s I would have been comfortable with the phrase "we are going to take the nation for Jesus", I would not use that language today. Why? How would we feel if a group of people were marching through the streets of the UK singing about taking the nation for Allah?

Another change in the western world has been the demise of widespread protest movements. Gone are the days of the 1960s with student and political demonstrations, and the strikes of the 1970s and early 80s. Such mass challenges to the nation no longer happen as few of the causes have enough enthusiastic supporters to carry it out. By the time Christians took to the streets in March for Jesus, we were about the last ones left! 

Have we forgotten that it is healthy in a free country for ideological groups to publicly challenge a system? That it is actually good for all of us that groups take to the streets saying they want to change the nation? Even groups we do not agree with? Why should Christians be uncomfortable with people singing they want to take the nation for Allah, just because they disagree with it? Could it be that Christians have become uncomfortable with boldness and the certainty of their beliefs, and thus fear such qualities in others? If so that is a sure sign the revival has ended. The change in the content of worship songs away from definitive belief statements that Richards describes also suggests the church has lost confidence in its fundamental message.

It is ironic that as Christians are no longer comfortable with taking the nation for Jesus, humanists are quite comfortable with taking the nation for secular humanism and are very vocal about it. They, and similar “rights” groups, are quite willing to engage in the public arena and influence policy makers and the media to change society into their version of it. They set the agenda for what is politically correct, and lobby to get laws changed, even “laws” that pre-date human history! There is a boldness and revival in humanism and lifestyle ideologies, but it seems there is no longer one in the Christian church.

So I agree with Richards the church of now is not the same as the 1990s, the peak of the charismatic revival. In short it has lost boldness, lost its fighting spirit, and lost the heart of what that revival was about, being saved and filled with the Spirit. Hence I conclude the charismatic revival is over.

Richards asks:

how do people of other faiths and no faith feel, when they hear our militant declarations of Christian dominion?

My guess is they feel the same way I felt in the late 70s when Christians kept telling me about Jesus; offended, irritated and angry. But I am glad they did, I found Jesus! It is never acceptable to society to proclaim Jesus. So when revivals come and society is disturbed by noisy, blunt, uncompromising Christians, rather than apologise for it, accept that this has always been God’s way. Every revival in history has had this effect and always will. It makes revived Christians unpopular with society and often with the church, but it is God’s way. Jesus needs enthusiasts!

Don’t Lament the End of Revival

If the charismatic revival has ended, don’t lament it. Revivals do end. Most end very quickly as they brought rapid changes in individuals, thus spread rapidly and end rapidly. The Welsh revival of 1904 is one such example. A revival that seeks to change churches, as well as individuals, needs time to enact those changes, thus takes longer to spread, so last longer.  But they still end.

The good news is they start again. History, the Bible, and my church growth models show that it only takes a small number of people to receive a fresh move of the Spirit and it will spread and lead to widespread conversions. I can understand why the church is embracing new ways of engaging with society and trying to make itself relevant. But it is a new work of the Holy Spirit and revival that will save the church from its current decline, and save the world, not new methods. In 1960 that revival was only a prayer away, it still is [8].


[1] Christian Today, 30/10/14

[2] Christian Today, 27/10/14

[3] I know not everyone sees the late 20th century charismatic renewal as a revival, but consider the evidence: It was about the Holy Spirit changing people; it spread and transformed mainstream denominations with Holy Spirit ministries; it was behind the Jesus people revival; it spawned many new church movements, notably the UK restoration churches, some of which are still growing; it changed the nature of Pentecostalism; many of the largest congregations today originate in this movement; it is behind the Alpha Course, probably the most successful evangelistic course ever; it has transformed the way churches across the world worship; a high proportion of ordinands for ministry have come from this background. Charismatic renewal as revival needs an article in its own right!

[4] The “Wesley” in “Wesley’s hymns” refers to Charles Wesley. Of course the movement was led by his brother, John, the more famous of the two and the one whose surname gives name to “Wesleyan” as a theology. Charles was also a fine preacher, but his fame is due to his song writing, an important part of the movement, but only a part.

[6] Don’t get me wrong. I really appreciate contemporary worship music. All the songwriters Noel Richards mentioned, and many others, had a huge impact on the charismatic renewal and on many people, myself included. But there is always a danger of the means becoming more important than the end. It happened to hymns, choirs, and even preaching. The same could now be happening with worship music.

[7] Pop music, rock music etc. has also had its movements. It has its periods pioneered by the rank and file, exploding nationally and internationally, only to end in professional routineness. Mass media played a large role in propagating such movements as Rock n Roll, Mersey Beat, Punk and Britpop. But the rise of the Internet, and the vast number of TV and radio channels, has changed the way music is heard and distributed. Together with an extensive back catalogue of songs and styles to access, it is now much harder for new distinctive musical movements to form. As such modern pop music has become quite a mish-mash of styles that is hard for modern worship songs to emulate and stay “modern”. Nevertheless even 2014 worship songs still sound like a “best-of” compilation from an Indie band of 10 years ago!

[7] To read of the beginnings of charismatic renewal as far as mainstream Christianity is concerned read Dennis Bennett, Nine O Clock in the Morning.  Still available new and secondhand.