Dynamical Models of Epidemics:
from Black Death to SARS
D. Gurarie
CWRU
History
Epidemics in History
– Plague in 14th Century Europe killed 25 million
– Aztecs lost half of 3.5 million to smallpox
– 20 million people in influenza epidemic of 1919
Diseases at Present
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1 million deaths per year due to malaria
1 million deaths per year due to measles
2 million deaths per year due to tuberculosis
3 million deaths per year due to HIV
Billions infected with these diseases
History of Epidemiology
. Hippocrates's On the Epidemics (circa 400 BC)
. John Graunt's Natural and Political Observations made upon the Bills of
Mortality (1662)
. Louis Pasteur and Robert Koch (middle 1800's)
History of Mathematical Epidemiology
. Daniel Bernoulli studied the effect of vaccination with cow pox on life
expectancy (1760)
. Ross's Simple Epidemic Model (1911)
. Kermack and McKendrick's General Epidemic Model (1927)
Schistosomiasis
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Chronic parasitic trematode infection
200-300 million people worldwide
Significant morbidity (esp. anemia)
Premature mortality
Life-cycle is complex, requiring species-specific
intermediate snail host
Optimal control strategies have not been established.
Geographic Distribution -1990
Smallpox: XVIII century
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Known facts:
– Short duration (10 days), high mortality
(75%)
– Life-long immunity for survivors
– Prevention: immunity by inoculation (??)
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Problem: could public health (life
expectancy) be improved by
inoculation?
“I simply wish that, in a matter which so closely
concerns the well-being of mankind, no decision shall
be made without all the knowledge which a little
analysis and calculation can provide.”
Daniel Bernoulli, on smallpox inoculation, 1766
Daniel Bernoulli
1700-1782
Bernoulli smallpox model (1766)
1) Population cohort of age a, n(a), mortality m(a)
2) Small pox effect
Caveat: if inoculation mortality f is included
one would need f<.5% for success!
Modeling issues and
strategies
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State variables for host/parasite
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“mean” or “distributed” (deterministic/stochastic)
Prevalence or level/intensity
Disease stages (latent,…)
Susceptibility and infectiousness
Transmission
– Homogeneous (uniformly mixed populations): “mass action”
– Heterogeneous: age/gender/ behavioral strata, spatially structured
contacts
– Environmental factors
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Multi-host systems, parasites with complex life cycles, ….
Goals of epidemic modeling
– Prediction
– Risk assessment
– Control (intervention, prevention)
Box (compartment) diagrams
SI
S – Susceptible
I – Infectious
E – Exposed
R - Removed
V – Vaccinated
…
S
S
SIR
Birth
I
I
SEIR
R
S
Death
E
I
V
SEIR
S
E
I
R
V
Total population: N = S+I+…
recruitment
R
SIR-type models
Ross, Kermak-McKendrick
•Population size is large and constant
•No birth, death, immigration or emigration
•No recovery
•No latency
•Homogeneous mixing
SI
b
S
I
SIR with immunity
b
S
m
I
R
Basic Reproduction
number: R0=bN/m
R0>1 – endemic
R0<1 - eradication
Residual S(∞)>0
SIR with loss of
immunity
m
b
I
S
R
d
endemic
epidemic
Control
(i) R0=“transmissiom”x”pop. density”/”recovery”<1. Hence critical density N>m/b
to sustain endemic level
(ii) Vaccination removes a fraction of N from transmission cycle: so eradication
(equilibrium I<0) requires (1-1/R0) fraction of N vaccinated
“Smallpox cohort” SIR
(1-n)d
lX
Z
X
m
Y
m+dn
m
Growth models: variable population N(t)
Const
recruitment
Linear growth
due to S
(Voltera-Lotka)
Linear growth
rate due to S,I
HIV/AIDS and
STD
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Variable population N=S+I
Natural growth a for S
Mortality m=10/year for I
Transmission: b S I/(S+I)
b= mean number of partners/per I
bS/(S+I) probability of infecting S (S-fraction of N)
Typical collapse
Conclusion:
Transm. treatment
Treatment w/o prevention of spread
can only increase g (collapse!)
AIDS for behavioral groups: 6D model
Parameters:
bh H bh T bt F bf T f
H
.1 .01 .4
.6 5 .01
Initial state
Shom
.1
Shet
1
Sfem
1
Ihom Ihet
.01
0
Ifem
0
Data (trends) of several African countries
Heterogeneous transmission for
distributed populations
• SIR type are only conceptual models
• Idealize transmissions and individual characteristics
(susceptibilities)
• Real epidemics requires heterogeneous models:
• age structure
• spatial/behavioral heterogeneity, etc.
Age structured models (smallpox)
Continuous population strata n(a,t), age “a”, time “t”
Discrete population bins: n=(na)
Normal growth
Infection
Example: 15-bin system with linear growth and
structured transmission
Age bins: red (young) to blue
(old)
High survival
Low survival
Fisher’s Equation (1937)
Original motivation: spread of a genetype in a population)
Infection: S(x,t), I(x,t) – (distributed) susceptibles and infectives
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Population density is constant N
No birth or death
No recovery or latent period
Only local infection
Infection rate is proportional to the number of infectives
Individuals disperse diffusively with constant D
Solutions: propagating density waves
Spreading wave in uniform
medium with const pop.
density
Spreading wave with
variable pop. density (red)
Problems:
•Equilibrium, Basic Reproduction Number?
•Speed of propagation (traveling waves)?
•Parameters for control, prevention?
Some current modeling issues
and approaches
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Spatial/temporal patterns of outbreaks
and spread
Stochastic modeling
Cellular Automata and Agent-Based
Models
Network Models (STD)