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Compartmental Modeling: an influenza epidemicPowerPoint Presentation

Compartmental Modeling: an influenza epidemic

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Compartmental Modeling: an influenza epidemic

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Compartmental Modeling: an influenza epidemic

AiS Challenge

Summer Teacher Institute

2003

Richard Allen

- 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.

- A compartment system is usually open; it exchanges material with its environment

I

k21

q1

q2

k12

k01

k02

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

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, … .

S

I

Infecteds

Susceptibles

a*S*I

b*S

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

U

S

I

R

c*S*I

e*I

Recovered

Susceptible

Infected

Infecteds

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 R0given

d

d

d

- 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.

- 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.

Susceptibles

Infedteds

Recovereds

- 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

I

R

I

R

inf*S*I

rem*I

Infecteds

Model equations

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

S

I

R

Inf*S*I

rem*I

Infecteds

Susceptible

Infected

Recovered

epidemic

model

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.

- http://www.sph.umich.edu/geomed/mods/compart/
- http://www.shodor.org/master/
- http://www.sph.umich.edu/geomed/mods/compart/docjacquez/node1.html