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 14

^{th}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).
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 19

^{th}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].
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.### Conclusion

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:

http://www.churchmodel.org.uk/Diffrefs.html

http://www.churchmodel.org.uk/Diffrefs.html

[2] National Archives http://www.nationalarchives.gov.uk/education/resources/great-plague/

[3] For a
history of the Great Plague of London see Wikipedia

http://en.wikipedia.org/wiki/Great_Plague_of_London and the references contained within.

http://en.wikipedia.org/wiki/Great_Plague_of_London 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. http://math.unm.edu/~sulsky/mathcamp/Eyam.pdf
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 http://www.iseesystems.com/.

[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."

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

Brauer, Fred. "Compartmental models for epidemics." (2008). http://health.hprn.yorku.ca/epidemicnotes.pdf

Brauer, Fred. "Compartmental models in epidemiology." In

*Mathematical epidemiology*, pp. 19-79. Springer Berlin Heidelberg, 2008. http://quiz.math.yorku.ca/chap2.pdfBrauer, Fred. "Compartmental models for epidemics." (2008). http://health.hprn.yorku.ca/epidemicnotes.pdf

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 http://www.ptc.com/product/mathcad

[14] Wikipedia and references within. http://en.wikipedia.org/wiki/Basic_reproduction_number

[15] There is a museum in Eyam where the visitor can learn
about the outbreak. http://www.eyam-museum.org.uk/
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.

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