1. General Epidemic SIR Transmission Host Birth, Death Purpose: What fraction of hosts must be vaccinated in order to eradicate disease?

2. Description Susceptible, Infective, Recovered Hosts Background Mortality: All classes Virulence: Extra mortality, infected hosts Reproduction: All classes; Hosts Born S

3. Parameters b Per-capitum Birth  Transmission Rate d Non-disease Mortality D Infective Mortality (D – d): Virulence > 0  Recovery

4. Dynamics

5. Assumptions When Rare, Pathogen Invades Host Pop. R 0 > 1 Endemic Infection Vaccine Available: What level of vaccination (reduction in susceptibility) would prevent disease form advancing when rare?

6. Analysis R 0 : New cases/case; Pathogen rare Infections/time =  S 0 Time: (D +  ) -1 R 0 =  S 0 / (D +  ) > 1 Therefore, S 0 > (D +  )/ 

7. Disease Control via Immunization Reduce S 0  S crit = S 1 S crit : R 0 = 1 Immunize Susceptible Hosts Too Few Hosts for Disease to Advance

8. Disease Control via Immunization More Virulent Disease, Lower R 0

9. Disease Control via Immunization Fraction Still Susceptible Fraction Immunize to Control Disease

10. Disease Control via Immunization Vaccinate p V S 0 Hosts Herd Immunity Vaccine Efficiency: Correction term N-Person Prisoner’s Dilemma ?

11. Vaccination Control (Ricklefs & Miller, 2001)

12. DiseaseLocaleR0R0 pVpV SmallpoxDeveloping World 3 - 50.7 – 0.8 MeaslesEngland130.92 Whooping cough England170.94 RubellaEngland60.83 Chicken PoxUS9 - 100.9 DiphtheriaUS4 - 60.8 Scarlet feverUS5 - 70.8 MumpsUS4 - 70.8 PolioNetherlands60.83 MalariaNigeria800.99 MalariaNigeria160.94 May, 1983; Includes corrections for vaccine efficiencies