Modern Applications of the SIR Epidemic Model

1. Modern Applications of the SIR Epidemic Model Nik Addleman and Jen Fox

2. Traditional SIR Model • Susceptible, Infected and Recovered S' = - ßSI I' = ßSI - γI R' = γI • Assumptions • S and I contact leads to infection • Infection is a disease, allows for recovery (or death…) • Fixed population

3. Jacobian Analysis S' = - ßSI = 0 I' = ßSI – γI = 0 R' = γI = 0 Equilibrium points: I = S = 0, R = R*

4. Example of SIR Model • Infectious contact rate • β = # daily contacts * transmission probability given a contact • Infectious Period • γ= time until recovered and no longer infectious S t

5. Extensions • Vaccinations • Vaccinated members of susceptible pop. are not as likely to contract disease • Temporary infective/immunity periods

6. Modeling Influenza • Modeling Seasonal Influenza Outbreak in a Closed College Campus. (K. L. Nichol et al.) • Compartmentalized, fixed-population ODE model • Modification of the SIR model • Minimize Total Attack Rate • Experimentally determine parameters

7. Compartments • Students and Faculty • Vaccinated versus Unvaccinated • Symptomatic and Asymptomatic infections • Different β and γ values for various populations • Categories (following slide) • Four susceptible categories • Eight infected • One recovered

9. Constructing Equations • Determining parameters • β varies between students/faculty and symptomatic/asymptomatic • γ has different values for symptomatic/asymptomatic and vaccinated/unvaccinated populations • Vaccine 80% effective • Apply to all compartments

10. Susceptible

11. Infectious … etc

15. Conclusions • Can use SIR model to determine best way to cut down on infections • Stay home when you are sick because you are infectious. Gross. • Get vaccinated! • Even late vaccinations are effective • Vaccine helps you and those around you • 60% vaccination means none of us gets sick