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

dSSIRISbdtdS

IDSIdtdI

dRIdtdR

Assumptions

When Rare, Pathogen Invades Host Pop.

R0 > 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 S0 Scrit = S1

Scrit: Defined by R0 = 1

Immunize Susceptible Hosts

Too Few Hosts for Disease to Advance

Disease Control via Immunization

DSR crit /1 0

More Virulent Disease, Lower R0

/ DScrit

Disease Control via Immunization

0/ SScrit

Fraction Still Susceptible

000

1111 RS

DS

Sp critV

Fraction Immunize to Control Disease

Disease Control via Immunization

Vaccinate pV S0 Hosts

Herd Immunity

Vaccine Efficiency: Correction term

N-Person Prisoner’s Dilemma ?

Vaccination Control (Ricklefs & Miller, 2000)

Disease Locale R0 pV

Smallpox Developing World

3 - 5 0.7 – 0.8

Measles England 13 0.92

Whooping cough

England 17 0.94

Rubella England 6 0.83

Chicken Pox US 9 - 10 0.9

Diphtheria US 4 - 6 0.8

Scarlet fever US 5 - 7 0.8

Mumps US 4 - 7 0.8

Polio Netherlands 6 0.83

Malaria Nigeria 80 0.99

Malaria Nigeria 16 0.94

May, 1983; Includes corrections for vaccine efficiencies