It is believed that plague (initially in bubonic form and later in pneumonic form) originated in Eyam when a box containing plague-carrying rat fleas arrived from ...
A Mathematical Model of the Eyam Plague using R Programming Language Hayedeh Jahandideh and Ashok Krishnamurthy, Ph.D. Department of Mathematics, Physics and Engineering Faculty of Science and Technology
Introduction
Results
The Village of Eyam, Derbyshire, England
Data Source
• The village of Eyam lying approximately twelve miles South-West of Sheffield, England, suffered an outbreak of plague epidemic between September 1665 and December 1666. • Plague is an infectious disease caused by the bacteria Yersinia Pestis. • It is believed that plague (initially in bubonic form and later in pneumonic form) originated in Eyam when a box containing plague-carrying rat fleas arrived from London. The virulence of the disease back then was severe, which explains the extremely high number of deaths in a short period of time. • The villagers courageously quarantined themselves to stop the plague infecting other villages. • The epidemic model that we used for this study is the common susceptible-infectiousremoved (S-I-R) compartmental model of epidemiology. • The removed compartment includes those who have died, have been quarantined, or have recovered from the disease and become immune. • The deterministic S-I-R model incorporating direct (human-to-human) transmissions of the infection is used to recreate the evolution of plague epidemic in a population of roughly 350 Eyam residents.
• Historical burial data indicates that there were actually two outbreaks, of which the first was relatively mild. Here we try to fit the deterministic S-I-R model over the period from mid-June to mid-October 1666, measuring time in months with an initial population of 254 susceptibles and 7 infectives, and a final population of 83. • The final size relation with S0 = 254, I0 = 7, S∞ = 83 gives β/α = 6.54×10−3, α/β = 153. • The infective period was 11 days, or 0.3667 month, so that α = 2.73. Then β = 0.0178. Generally it is very difficult to estimate the infection rate which depends on the particular disease being studied. • In most cases these parameter estimates can be derived only retrospectively after the epidemic has run its course. Historical Data
Numerical Results of the Deterministic Model
Date
Susceptibles, S(t)
Infectives, I(t)
Time
Susceptibles, S(t)
Infectives, I(t)
June 18
254
7
0
254
7
July 3/4
235
14.5
1
190.98574
26.2828157
July 19
201
22
2
115.02277
24.47636177
August 3/4
153.5
29
3
86.22273
9.07548112
August 19
121
20
4
78.62842
2.52887833
September 3/4
110
8
5
76.70584
0.65471027
September 19
97
8
6
76.22034
0.16638066
October 4/5
Unknown
Unknown
7
76.09774
0.04208429
October 20
83
0
8
76.06678
0.01063203
• Only 83 people survived before the epidemic stopped spreading in late 1666. Density Plot and S-I Phase Plane using R Statistical Programming Language
Plague Pit Circa 1665, Tipping bodies from a cart into a communal grave in London during the Great Plague. Original Artwork: Engraving by J Franklin. (Photo by Hulton Archive/Getty Images)
Deterministic S-I-R Model Figure 1: Density of S(t), I(t) and R(t).
Susceptible
β SI
αI
Infected
Removed
I max = S0 + I 0 −
The deterministic form of our S-I-R model satisfies:
dS dI = − β SI = β SI − α I dt dt
α α α α log S0 − + log β β β β
= 30.4
dR = αI dt
with initial conditions:
= S (0) S0
= I (0) I 0 > 0
R = (0) 0
S0 = + I0 N
The positive constants β and α represent the infection rate and removal rate, respectively. This system of three ordinary differential equations uses three variables to define the state of the epidemic at time t : S(t) = density of the susceptible population, I(t) = density of the infected population, and R(t) = density of the removed population. There are no vital dynamics in this model, meaning that there are no new births or non-disease related deaths in any of the three compartments. α α I (t ) + S (t ) − log S (t ) =N − log S0 β β
• The quantity ℜ = β/α is called the basic t →∞ reproduction number which determines whether there is an epidemic. If ℜ < 1, the β S∞ infection dies out, while if ℜ > 1, there is an = [ N − S∞ ] =ℜ 1 − N epidemic. α • R∞ = (N – S∞) is called the final size of the epidemic.
As t → ∞, I ∞ = lim I (t ) = 0 S0 log S∞
S0 β log = [ N − S (t ) − I (t ) ] S (t ) α
Figure 2: S-I phase plane solution, model and data. The particular trajectory chosen is for S0 = 254, I0 = 7, simulating the plague effect at Eyam.
Comments and Future Work • The R statistical programming language was used to model the spread of the epidemic in Eyam. R is freely available from http://www.r-project.org/ • If it is to have credibility, our deterministic epidemic model should be able to reproduce the principal historical features of disease spread. • Our preliminary results agree remarkably well with the historic Eyam plague data. • Our future work is to extend this work to implement a stochastic differential difference equation.
References Kermack and McKendrick, (1927), Raggett (1980, 1981 & 1982), Krishnamurthy et al., (2010).
