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.

### Eyam Plague

^{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].

### System Dynamics Model

*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.

Figure 1 |

*Deceased*as most cases of bubonic plague ended in death.

*catch the disease*, which moves susceptibles into infected; and

*deaths*, which moves infected into deceased.

*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).

*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].

*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).

*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).

^{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.

*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].

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

*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].

*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

### References & Notes

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

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

*Bull. Inst. Math. and its Applic*18, no. 221-226 (1982): 530. http://math.unm.edu/~sulsky/mathcamp/Eyam.pdf

*History workshop journal*, vol. 61, no. 1, pp. 31-56. Oxford University Press, 2006.

*Local population studies*54 (1995): 56-57.

*Proc. R. Soc. A*, vol. 115, no. 5, pp. 700-721. 1927.

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.

*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.

__and__make an enthusiast. Not all converts become enthusiasts.

*R0*due the particular burial practices used, allowing dead bodies to transmit the disease, thus extending the infectious period.

Brauer, Fred. “Compartmental models in epidemiology.” In

*Mathematical epidemiology*, pp. 19-79. Springer Berlin Heidelberg, 2008. http://quiz.math.yorku.ca/chap2.pdf

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

**Tags:** Black death, Bubonic Plague, Disease, Ebola, epidemic, Eyam Plague, SIR