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 k01k02 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, q 2, … 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 SI Infecteds Susceptibles 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 SR Infecteds Susceptible I Recovered U Infected c*S*I ddd 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 = per day, and the removal rate, rec = 0.5 per day - average infectious period of 2 days. S R Infecteds I SusceptiblesRecoveredsInfedteds inf*S*Irem*I S IR

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 = , rec = 0.5 S R Infecteds Susceptible I RecoveredInfected Inf*S*Irem*I epidemic model

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 mpart/ mpart/docjacquez/node1.html