Epidemics: a special case of an exploitative
interaction.
Bubonic plague (the black death, pestilence)
Yersinia pestis
Population of Europe in
millions
1000
1100
1200
1300
1347
1352
1400
1430
38
48
59
70
75
50
45
36
"How many valiant men, how many fair ladies, breakfast
with their kinfolk and the same night supped with their
ancestors in the next world! The condition of the people
was pitiable to behold. They sickened by the thousands
daily, and died unattended and without help. Many died in
the open street, others dying in their houses, made it
known by the stench of their rotting bodies. Consecrated
churchyards did not suffice for the burial of the vast
multitude of bodies, which were heaped by the hundreds in
vast trenches, like goods in a ships hold and covered with a
little earth."
-Giovanni Boccaccio (author of The Decameron)
Dracula – the plague carrier
Ring around a rosie
Pocket full of posies
Ashes ashes
We all fall down
 Plagues described in the old testament consistent with the black
plague
 First certain outbreak 430 BC in Athens estimated death toll 30%
(beginning of the end of Greek Empire)
 First pandemic: a series of repeated epidemics between 541 and
750 affected the Middle East to the Mediterranian (weakened the
Byzantine Empire).
 Second pandemic: 1346-1351. Estimated dead in China: 50%,
estimated dead in Europe; 30%. Followed by cyclical outbreaks
for >200 years
 Third pandemic: 1855 – 1900. From China to India, Africa,
Australia, Europe, Hawaii, India, Japan, the Middle East, the
Philippines and North and South America
Caffa, (today’s Feodosia in the
Ukraine), Genoese traders had a
fortress which was besieged in
1347 by Mongol armies.
The Mongols carried the disease
and catapulted dead bodies over
the fortress wall.
The Genoese fled home…
After the original epidemic, it revisited Europe once in almost every
generation for 600 years.
•
Most cases were more localized hitting individual cities
•
continuing high death toll of 30-40%
•
Last outbreak of that era in Marseilles (1720), when a commercial
ship named the `Grand Saint-Antoine' arrived from Syria and
Lebanon with cases of plague aboard. This epidemic killed 50,000
people.
Questions arise from the study of European epidemics:
 How do new epidemics suddenly appear?
 Why do they disappear as suddenly?
 Why can they come back?
 Can one eradicate diseases totally?
Epidemics introduced to the Americas from
Europe:
Smallpox
Bubonic plague
Typhus
Mumps
Influenza
Yellow fever
Measles
Scarlet fever
Estimated death toll uncertain. Up to 90% by some estimates.
The population biology of disease
i
I
S
I
c
infected
b
b
dead
v
b
R
resistant
S
susceptible
D
Birth rate (assumed =
combined death rate)
D
dead
N: total population size: N = S + I + R
b: rate of mortality not due to infection
c: mortality of infecteds due to disease
Assumption 1:
total population size remains constant:
birth rate (susceptibles) = bN + cI
Assumption 2:
Susceptible-Infecteds encounter rates depend only
on their respective population densities:
Susceptible loss rate = aIS + bS
a: transmission probability
b: rate of mortality not due to infection
Rate of change for population of Susceptibles:
dS
 bN  cI  aIS  bS
dt
Death rate, same
as birth rate
Susceptibles that
become infected
Susceptibles
that die of
other causes
Rate of change for population of Infecteds:
dI
 aIS  (b  c  v) I
dt
a: transmission probability
b: rate of mortality not due to infection
c: mortality of infecteds due to disease
v: rate of recovery
Rate of change for population of Recovered:
dR
 vI  bR
dt
b: rate of mortality not due to infection
v: rate of recovery
The SIR model of infectious disease
with direct transmission:
dS
 bN  cI  aIS  bS
dt
dI
 aIS  (b  c  v) I
dt
dR
 vI  bR
dt
Two-dimensional model, because S+I+R = const.
Infecteds
Infecteds
isocline
Susceptibles
isocline
I*
Susceptibles
As long as I*>0, the disease stays alive.
To keep the disease alive, the equilibrium
number of Infecteds has to be >0:
I*  0
if
aN
1
bcv
aN
is called the basic reproductive rate of the infectious disease
bcv
To keep the disease alive, the population
must exceed a minimum size.
bcv
N
a
The larger the population, the easier for the disease to survive.
Adding immunization:
I
S
I
D
infected
dead
immunization
R
resistant
S
D
dead
The SIR model of infectious disease
with direct transmission and immunization:
dS
 bN  cI  aIS  bS  iS
dt
dI
 aIS  (b  c  v) I
dt
dR
 vI  iS  bR
dt
Infecteds
Infecteds
isocline
immunization
rate
I*
Susceptibles isocline
Susceptibles
The disease dies out when I*<0.
Too few Susceptibles remain to keep the disease alive.
To keep the disease alive, the equilibrium
number of Infecteds has to be >0:
I*  0
if
b(aN  b  c  v)
i
bcv
abN
1
(b  i ) * (b  c  v)
is the critical immunization rate
that would eradicate the disease.
 How do new epidemics suddenly appear?
Origins: often jump host from animal to human. Host-jumping is
facilitated where people live in close quarters with livestock.
Epidemics spread and evolve fast in large, concentrated
population centers.
Trade, warfare, famines, which put people on the move, spread
the disease away from their origin.
Disease
Origin
Measles
Cattle
Tuberculosis
Cattle
Smallpox
Cattle
Anthrax
Cattle
Flu
Pigs & Ducks
Whooping cough
Pigs & Dogs
Malaria
Chicken & Ducks
 Why do they disappear as suddenly?
Everybody who can get infected was infected.
Diseases can die out when no carrier survives.
 Why can they come back?
After original infection, epidemics are prone to come every
generation as Susceptibles replace Resistants in the population.
 Can one eradicate diseases totally?
In theory, by immunization. In the US, many epidemic diseases
have practically been eliminated. However, many live on in
developing countries (and some in weapons laboratories).
New diseases evolve all the time.