Transcript Mathematical Modeling / Computational Science

Compartmental Modeling:
an influenza epidemic
AiS Challenge
Summer Teacher Institute
2003
Richard Allen
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.
Compartment Modeling
A compartment system is usually open; it
exchanges material with its environment
I
k21
q1
q2
k12
k01
k02
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
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, … .
SIS Epidemic Model
S
Susceptibles
a*S*I
b*S
I
Infecteds
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
SIR Epidemic model 
U
S
Susceptible
d
c*S*I
I
Infected
e*I
Infecteds
R
Recovered
d
d
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
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.
Flu Epidemic (cont.)
Our epidemic model uses the1927 KermackMcKendrick SIR model: 3 compartments – Susceptibles (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.
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
S
Susceptibles
inf*S*I
II
Infedteds
Infecteds
rem*I
R
R
Recovereds
Flu Epidemic (cont.)
Model equations
S
Susceptible
Inf*S*I
I
rem*I
Infecteds
Infected
Sj+1 = Sj + dt*inf*Sj*Ij
Ij+1 = Ij + dt*[inf*Sj*Ij – rec*Ij]
Rj+1 = Rj + dt*rec*Ij
S0 = 762, I0 = 1, R0 = 0
inf = 0.00218, rec = 0.5
R
Recovered
epidemic
model
Possible Extensions
Examine the impact of vaccinating students
prior to the start of the epidemic.
Assume 10% of the susceptible boys are vaccinated 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.

References
http://www.sph.umich.edu/geomed/mods/co
mpart/
http://www.shodor.org/master/
http://www.sph.umich.edu/geomed/mods/co
mpart/docjacquez/node1.html