Compartmental Modeling: an influenza epidemic

1. Compartmental Modeling: an influenza epidemic AiS Challenge Summer Teacher Institute 2003 Richard Allen

2. 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.

3. Compartment Modeling • A compartment system is usually open; it exchanges material with its environment I k21 q1 q2 k12 k01 k02

4. 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 Applications

5. 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, … .

6. S I Infecteds Susceptibles SIS Epidemic Model 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

7. SIR Epidemic model  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

8. 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.

9. 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.

10. Susceptibles Infedteds Recovereds 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 I R I R inf*S*I rem*I Infecteds

11. Flu Epidemic (cont.) 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

12. 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.

13. 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