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SIR Models

SIR Models. 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). Kermack-McKendrick Model.

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SIR Models

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  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

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