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

Compartments & Virulence Susceptible, Infective, Recovered Hosts Background Mortality: All classes Virulence: Extra mortality, Infected hosts Reproduction: All classes; Hosts Born S

Parameters b Per-capitum Birth  Transmission Rate (Mass Action) d Non-Disease Mortality (All) D Infective Mortality (D – d): Virulence > 0  Recovery from Infection

Dynamics of General Epidemic

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

Analysis

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

Disease Control via Immunization More Virulent Disease, Lower R 0

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

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

Vaccination Control (Ricklefs & Miller, 2000)

DiseaseLocaleR0R0 pVpV SmallpoxDeveloping World – 0.8 MeaslesEngland Whooping cough England RubellaEngland60.83 Chicken PoxUS DiphtheriaUS Scarlet feverUS MumpsUS PolioNetherlands60.83 MalariaNigeria MalariaNigeria May, 1983; Includes corrections for vaccine efficiencies