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 known 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 that 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.

Modelling

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

Stocks & Flows

The model assumes the population is 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:

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.

Feedback Loops

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 to the social force B2. 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).

Deaths

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 taking action computed [9].

Results:

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, 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 the 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. The lessons learned, which mathematicians can now explore, give confidence to models that have given strategies to combat epidemics and save millions of people. A bonus of these studies is that they can help us understand social diffusion processes such as religion.

References & Notes

[1] For a selection of social modelling papers that use the epidemic/disease analogy, see Social Diffusion References.

[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 Application, 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 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.” 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 to 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:

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