1. SIR Models

2. SIR Models • Model # infected in a population over time • Kermack-McKendrick model (1927) • Assumes (in original form): • Completely homogeneous population(No age, spatial structure) • No latent period • No births/deaths (natural or disease-caused)

3. Kermack-McKendrick Model • Three classes: • S: Susceptible: Naïve • I: Infected • R: Recovered • Infected individuals infect susceptibles • One-way • Similar to metapopulation model

4. Where are the ODEs? • What does the derivative of a function describe? • Rate of change of each class: • dS/dt • dI/dt • dR/dt • Important for equilibria, stability of equilibria, etc. • Parameters: • Beta: Transmission constant • Gamma: Recovery rate • R0: If BS > gamma, disease does not die out • In other words, if dI/dt

5. R example

6. Many ways to modify SIR models: • Vectors (X, Z vector classes) • Vaccination (Susceptibles become immune) • No Immunity (Recovered infecteds are susceptible) • Latent period (Asymptomatic period) • Different classes of infected individuals (Superspreaders) • Recruitment (Birth/death)

7. SIR with Recruitment • Each class has a natural death rate • Keep constant pop: All deaths return to S • Death rate: nu • Example

8. SIR with a Spatial Component • Two patch model • Each patch has S,I,R classes • No movement between patches (for simplicity) • Infection can occur between two groups