Compartmental Modeling: an influenza epidemic

AiS Challenge Summer Teacher Institute

2003Richard 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

k01 k02

k21

k12

q1 q2

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

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

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

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 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.

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

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

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.

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