Compartmental Modeling: an influenza epidemic
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Compartmental Modeling: an influenza epidemic
AiS Challenge Summer Teacher Institute
2003Richard Allen
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Compartment Modeling
Compartment systems provide a systematic way of modeling physical and biological processes.In the modeling process, a problem is broken up into a collection of connected “black boxes” or “pools”, called compartments. A compartment is defined by a characteristic material (chemical species, biological entity) occupying a given volume.
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Compartment Modeling
A compartment system is usually open; it exchanges material with its environment
I
k01 k02
k21
k12
q1 q2
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Applications
Water pollution
Nuclear decay
Chemical kinetics
Population migration
Pharmacokinetics
Epidemiology
Economics – water resource management
Medicine Metabolism of
iodine and other metabolites
Potassium transport in heart muscle
Insulin-glucose kinetics
Lipoprotein kinetics
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Discrete Model: time line
q0 q1 q2 q3 … qn |---------|----------|------- --|---------------|---> t0 t1 t2 t3 … tn
t0, t1, t2, … are equally spaced times at which the variable Y is determined: dt = t1 – t0 = t2 – t1 = … .
q0, q1, q2, … are values of the variable Y at times t0, t1, t2, … .
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SIS Epidemic Model
Sj+1 = Sj + dt*[- a*Sj*Ij + b*Ij]
Ij+1 = Ij + dt*[+a* Sj*Ij - b* Ij]
tj+1 = tj + dt
t0, S0 and I0 given
S IInfectedsSusceptibles
a*S*I
b*S
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SIR Epidemic model
Sj+1 = Sj + dt*[+U - c *Sj*Ij - d *Sj]
Ij+1 = Ij + dt*[+c*Sj*Ij - d*Ij - e*Ij]
Rj+1 = Rj + dt*[+e*Ij - d*Rj]
tj+1 = tj + dt; t0, S0, I0, and R0 given
S RInfectedsSusceptibleI
Recovered
U
Infectedc*S*I
d d d
e*I
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Flu Epidemic in a Boarding School
In 1978, a study was conducted and reported in British Medical Journal (3/4/78) of an outbreak of the flu virus in a boy’s boarding school.
The school had a population of 763 boys; of these 512 were confined to bed during the epidemic, which lasted from 1/22/78 until 2/4/78. One infected boy initiated the epidemic.
At the outbreak, none of the boys had previously had flu, so no resistance was present.
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Flu Epidemic (cont.)
Our epidemic model uses the1927 Kermack-McKendrick SIR model: 3 compartments – Sus-ceptibles (S), Infecteds (I), and Recovereds (R)
Once infected and recovered, a patient has immunity, hence can’t re-enter the susceptible or infected group.
A constant population is assumed, no immigration into or emigration out of the school.
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Flu Epidemic (cont.)
Let the infection rate, inf = 0.00218 per day, and the removal rate, rec = 0.5 per day - average infectious period of 2 days.
S RInfecteds
ISusceptibles RecoveredsInfedteds
inf*S*I rem*I S I R
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Flu Epidemic (cont.)
Model equations
Sj+1 = Sj + dt*inf*Sj*IjIj+1 = Ij + dt*[inf*Sj*Ij – rec*Ij]Rj+1 = Rj + dt*rec*IjS0 = 762, I0 = 1, R0 = 0inf = 0.00218, rec = 0.5
S RInfectedsSusceptible
IRecoveredInfected
Inf*S*I rem*I
epidemicmodel
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Possible Extensions
Examine the impact of vaccinating students prior to the start of the epidemic. Assume 10% of the susceptible boys are vac-
cinated each day – some getting the shot while the epidemic is happening in order not to get sick (instant immunity).
Experiment with the 10% rate to determine how it changes the intensity and duration of the epidemic.
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References
http://www.sph.umich.edu/geomed/mods/compart/
http://www.shodor.org/master/
http://www.sph.umich.edu/geomed/mods/compart/docjacquez/node1.html